Mathematics High School

## Answers

**Answer 1**

To multiply (2.36 x 10^8) by (108.4 x 10^5), we can multiply the numerical parts and add the exponents of 10:

(2.36 x 10^8) * (108.4 x 10^5) = (2.36 * 108.4) x (10^8 * 10^5) = 255.664 x 10^(8+5) = 255.664 x 10^13

The final answer, written in scientific notation, is 2.55664 x 10^14.

## Related Questions

the cost, in dollars, of producing x units of a certain item is given by c(x)=5x−8x−2−−−−√. find the production level that minimizes the average cost per unit.

### Answers

The production level that minimizes the **average cost** per unit is 0.64 units.

To find the **production level **that minimizes the average cost per unit, we need to first find the average cost function.

The average cost function is given by:

AC(x) = c(x)/x

Substituting c(x) = 5x - 8√x - 2, we get:

AC(x) = (5x - 8√x - 2)/x

To minimize the average cost per unit, we need to find the value of x that minimizes the average cost function.

To do this, we need to take the derivative of the **average **cost function with respect to x and set it equal to 0:

d/dx AC(x) = (5 - 4/√x)/x^2 = 0

Solving for x, we get:

5 = 4/√x

√x = 4/5

x = (4/5)^2

x = 0.64

Therefore, the production level that minimizes the average cost per unit is 0.64 units.

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The radius of a cylindrical construction pipe is 2.5 ft . If the pipe is 28 ft long, what is its volume?

Use the value 3.14 for pi , and round your answer to the nearest whole number.

Be sure to include the correct unit in your answer.

### Answers

The **volume** of the cylindrical construction pipe given the height and **radius** to the nearest whole number is 550 cubic feet.

what is the volume of the cylindrical construction pipe?

Volume of the **cylindrical** construction pipe = πr²h

Where,

π = 3.14

r = radius = 2.5 ft

h = **height** = 28 ft

Volume of the cylindrical construction pipe = πr²h

= 3.14 × 2.5² × 28

= 3.14 × 6.25 × 28

= 549.50 cubic ft

Approximately to the nearest whole number,

= 550 cubic ft

Hence, the cylindrical **construction pipe** has a volume of 550 cubic ft

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A company, that has a store within a major shopping center, wants to conduct

a survey of a population. Because the population is large the company selects

a sample asking customers who walk in the store if they would be willing to

take part in a survey, What type error has the company made in selecting the

sample?

Convenience sampling

O Sample size error

Random errors

O None of the above

Submit Answer

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MO

### Answers

**Answer:**

Many errors were shown in the text

**Step-by-step explanation:**

How to explain the word problem It should be noted that to determine if Jenna's score of 80 on the retake is an improvement, we need to compare it to the average improvement of the class. From the information given, we know that the class average improved by 10 points, from 50 to 60. Jenna's original score was 65, which was 15 points above the original class average of 50. If Jenna's score had improved by the same amount as the class average, her retake score would be 75 (65 + 10). However, Jenna's actual retake score was 80, which is 5 points higher than what she would have scored if she had improved by the same amount as the rest of the class. Therefore, even though Jenna's score increased from 65 to 80, it is not as much of an improvement as the average improvement of the class. To show the same improvement as her classmates, Jenna would need to score 75 on the retake. Learn more about word problem on; brainly.com/question/21405634 #SPJ1 A class average increased by 10 points. If Jenna scored a 65 on the original test and 80 on the retake, would you consider this an improvement when looking at the class data? If not, what score would she need to show the same improvement as her classmates? Explain.

Determine the independent and dependent variable from the following situation. Quincy was given 3 video games for his new game system. Every month he saves enough to get 2 more video games.

independent variable is?

dependent variable is?

### Answers

In the given situation:

The independent variable is: Time or months. Quincy's saving and acquisition of additional video games depend on the passage of time.

The dependent variable is: Number of video games. The number of video games Quincy has is dependent on the amount of time that has passed and his ability to save money.[tex][/tex]

Question 2(Multiple Choice Worth 2 points)

(Creating Graphical Representations LC)

A teacher was interested in the cafeteria food that students preferred in a particular school. She gathered data from a random sample of 200 students in the school and wanted to create an appropriate graphical representation for the data.

Which graphical representation would be best for her data?

Stem-and-leaf plot

Line plot

Histogram

Box plot

### Answers

Answer:

a histogram

Step-by-step explanation:

This way of classifying data I a good method as it helps identify the pattern of data.

consider the function f(x)=5x4−5x3−2x2−5x 8. using descartes' rule of signs, what is the maximum possible number of positive roots?

### Answers

According to **Descartes' **rule of signs, the maximum possible number of positive roots of a polynomial is equal to the number of sign changes in the coefficients of its terms, or less than that by an even number.

In the given **polynomial function** f(x) = 5x^4 - 5x^3 - 2x^2 - 5x + 8, there are two sign changes in the coefficients, from positive to negative after the second term and from negative to positive after the third term.

Therefore, the maximum possible number of positive roots of this polynomial is either 2 or 0 (less than 2 by an even number).

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Please help please please please please

### Answers

**Answer:36**

**Step-by-step explanation:**

im done typing the explanations lol

there are good pythagorean theorem calculators, just search for them

For this question, please leave your answer in "choose" notation - please do not write any factorials or simplify in any way. The pet store has 6 puppies, 9 kittens, 4 lizards, and 5 snakes. c. If you select five pets from the store randomly, what is the probability that at least one of the pets is a puppy?

### Answers

The **probability** equation will be : (at least one puppy) = 1 - P(no puppies selected)

To find the probability that at least one of the pets selected is a **puppy**, we can subtract the probability of selecting no puppies from 1.

The total number of pets in the store is 6 + 9 + 4 + 5 = 24. The number of ways to select 5 pets out of 24 is C(24, 5).

The number of ways to select no puppies is C(18, 5) because we need to choose all 5 pets from the remaining 18 non-puppy pets.

Therefore, P(no puppies **selected**) = C(18, 5) / C(24, 5).

Finally, we can calculate P(at least one puppy) = 1 - P(no puppies selected).

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find the area of the surface obtained by rotating the curve y=x36 12x,12≤x≤1,y=x36 12x,12≤x≤1, about the xx-axis

### Answers

The area of the surface obtained by rotating the curve y = x^3 - 6x, 1 ≤ x ≤ 2, about the x-axis is π units **squared**.

What is the area of the surface formed by rotating the curve y = x^3 - 6x, 1 ≤ x ≤ 2, about the x-axis?

To find the **area **of the surface obtained by rotating the curve y = x^3 - 6x, 1 ≤ x ≤ 2, about the x-axis, we can use the method of cylindrical shells. This involves dividing the curve into infinitely thin strips, each of which acts as a cylindrical shell when rotated around the x-axis. The height of each shell is given by the function y = x^3 - 6x, and the circumference of each shell is determined by the interval of x-values.

Using the formula for the surface area of a **cylindrical **shell, which is given by 2πrh, where r represents the distance from the axis of rotation (in this case, the x-axis) and h represents the height of the shell, we integrate this expression over the given interval. In this case, the interval is from x = 1 to x = 2.

By evaluating the **integral **and simplifying, we obtain the area of the surface as π units squared.

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sophie needs 420 g of flour to bake a cake. her scales only weigh in ounces. how many ounces of flour does she need? 1 ounce

### Answers

Sophie needs approximately 14.82 **ounces** of flour to bake her cake .

To convert **grams** to ounces, we can use the conversion factor that 1 ounce is approximately equal to 28.35 grams . The mass m in grams (g) is equal to the mass m in ounces (oz) times 28.34952

1 ounces = 28.35 gram

So, to find the number of ounces of flour Sophie needs, we can **divide** the weight in grams by the **conversion** factor .

420 g × 1 ounces / 28.35 g

420 g / 28.35 g = 14.82 ounces

Therefore, Sophie needs approximately 14.82 ounces of flour to bake her cake .

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Pease help with this question

### Answers

The **weight** of liquid in the **hemisphere** is 129408.2 pounds.

How to find the total weight of liquid in the hemisphere?

The tank is in the shape of an** hemisphere** and has a diameter of 18 feet. If the liquid fills the tank, it has a density of 84.8 pounds per cubic feet.

Therefore, total weight of the liquid can be found as follows:

density = mass / volume

Therefore,

volume of the liquid in the hemisphere tank = 2 / 3 πr³

Therefore,

r = 18 / 2 = 9 ft

volume of the liquid in the hemisphere tank = 2 / 3 × 3.14 × 9³

volume of the liquid in the hemisphere tank = 4578.12 / 3

**volume** of the liquid in the hemisphere tank = 1526.04 ft³

Hence,

weighty of the liquid in the tank = 526.04 × 84.8 = 129408.192

weighty of the liquid in the tank = 129408.2 pounds

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Investigate each pattern below a) 2.4.6.8...... 1. Investigate how the pattern progresses to the next term(s) (1) 2. Continue the pattern with the next three terms 3. Describe the rule used to generate the pattern. 4. Use the rule to find term 50. (2) (2)

### Answers

The 50th term in the pattern is 100, obtained by applying the rule of adding 2 to the previous term for each **Subsequent term**.

1. Investigation of the Pattern **Progression**:

In the given pattern, the sequence starts with the number 2 and then increments by 2 for each subsequent term. So, the first term is 2, the second term is 4, the third term is 6, and so on. The pattern progresses by adding 2 to the previous term to obtain the next term.

2. Continuing the **Pattern**:

To continue the pattern with the next three terms, we need to apply the rule mentioned above. Starting from the last term given in the pattern, which is 8, we add 2 to it successively to find the next three terms. Following this rule, the next term is 10, then 12, and finally 14. Therefore, the next three terms in the pattern are 10, 12, and 14.

3. Rule to Generate the Pattern:

The rule used to generate the pattern is to add 2 to the previous term to obtain the next term. In **mathematical notation**, it can be represented as: Tn = Tn-1 + 2, where Tn represents the nth term in the sequence.

4. Finding Term 50:

Using the rule mentioned above, we can find the 50th term in the pattern. We know that the first term is 2, and for each subsequent term, we add 2. Therefore, to find the 50th term, we can use the formula: T50 = T1 + (50 - 1) * 2.

Substituting the values, we have: T50 = 2 + (50 - 1) * 2 = 2 + 49 * 2 = 2 + 98 = 100.

Hence, the 50th term in the pattern is 100, obtained by applying the rule of adding 2 to the previous term for each subsequent term.

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The estimated regression equation for these data is Y=7.6+.9x . Compute SSE, SST, and SSR (to 1 decimal).

xi 2 6 9 13 20

yi 7 18 9 26 23

SSE =

SST =

SSR = What percentage of the total sum of squares can be accounted for by the estimated regression equation (to 1 decimal)? What is the value of the sample correlation coefficient (to 3 decimals)?

### Answers

The value of SSE = 97.9, SST = 380, SSR = 282.1, the percentage of the total sum of squares accounted for by the estimated regression equation is approximately 74.24%, and the sample **correlation coefficient** is approximately 0.872.

To solve this problem, we first need to find the **predicted values** of y using the given regression equation

yi-hat = 7.6 + 0.9xi

Using the given values of xi, we get:

yi-hat = 7.6 + 0.9(2) = 9.4

yi-hat = 7.6 + 0.9(6) = 12.4

yi-hat = 7.6 + 0.9(9) = 16.3

yi-hat = 7.6 + 0.9(13) = 20.5

yi-hat = 7.6 + 0.9(20) = 24.4

Now we can calculate SSE, SST, and SSR

SSE = Σ(yi - yi-hat)² = (7-9.4)² + (18-12.4)² + (9-16.3)² + (26-20.5)² + (23-24.4)² = 97.9

SST = Σ(yi - ȳ)² = (7-16)² + (18-16)² + (9-16)² + (26-16)² + (23-16)² = 380

SSR = SST - SSE = 380 - 97.9 = 282.1

The percentage of the total sum of squares that can be accounted for by the estimated **regression equation** is

R² = SSR/SST x 100% = 282.1/380 x 100% ≈ 74.24%

To find the sample correlation coefficient (r), we need to first calculate the sample covariance (sxy) and the sample standard deviations (sx and sy)

sxy = Σ(xi - x)(yi - y)/n = [(2-10)(7-16) + (6-10)(18-16) + (9-10)(9-16) + (13-10)(26-16) + (20-10)(23-16)]/5 = 82

sx = √[Σ(xi - x)²/n] = √[((2-10)² + (6-10)² + (9-10)² + (13-10)² + (20-10)²)/5] ≈ 6.66

sy = √[Σ(yi - y)²/n] = √[((7-16)² + (18-16)² + (9-16)² + (26-16)² + (23-16)²)/5] ≈ 7.78

Now we can calculate r is

r = sxy/(sx sy) = 82/(6.66 x 7.78) ≈ 0.872

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This scatter plot shows the relationship between the average study time and the quiz grade. The line of

best fit is shown on the graph.

Need Help ASAP!

Explain how you got it please

### Answers

The approximate **value** of b is 40.

The **slope** of the **line** **of** **best** **fit** is 4/3.

We have,

From the **scatter** **plot**,

The y-intercept is (0, b).

This means,

The y-values when x = 0.

We can see that,

y = 40 when x = 0.

Now,

There are two points on the **scatter** **plot**.

B = (20, 70) and C = (35, 90)

So,

The slope.

= (90 - 70) / (35 - 20)

= 20/15

= 4/3

Thus,

The approximate **value** of b is 40.

The **slope** of the **line** **of** **best** **fit** is 4/3.

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fill in the blank. two samples are ________________ if the sample values are paired. question content area bottom part 1 two samples are ▼ if the sample values are paired.

### Answers

your answer is independent

Two samples are **paired **if the sample values are paired.

**Paired samples** are a type of dependent samples where each observation in one sample is uniquely paired or matched with an observation in the other sample. The pairing is usually based on a natural association, such as measuring the same variable on the same** subject **before and after a treatment, or measuring two variables on the same subject at the same time. Paired samples are often analyzed using methods such as paired t-test or Wilcoxon signed-rank test, which take into account the dependency between the samples. Pairing can also help to reduce variability and increase statistical power in the analysis.

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The relative density of steel is 7.8. Find: 1. the mass of a solid steel cube of side 10cm.

2. the volume of the steel that has a mass of 8kg

### Answers

a) The **mass **of the solid steel is M = 7800 kg

b) The volume of the **steel **is V = 1.0256 cm³

Given data ,

The relative **density **of steel is 7.8

Now , Mass = Density x Volume

a)

The side length of the solid **steel **is s = 10 cm

So , the volume of solid is V = 10³ = 1000 cm³

The mass of the solid is M = 7.8 x 1000

M = 7800 kg

b)

The mass of steel is M = 8 kg

So, the volume of **steel **is V = M / D

V = 8 / 7.8

V = 1.0256 cm³

Hence , the **density**,mass and volume of the **steel **is solved.

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A is an n x n matrix. Mark each statement True or False. Justify each answer.

i. If A

x

=

λ

x

for some vectors, then λ

is an eigenvalue of A.

ii. A matrix A is not invertible if and only if 0 is an eigenvalue of A.

iii. A number c is an eigenvalue of A if and only if the equation (A - cI)x = 0 has a nontrivial solution.

iv. Finding an eigenvector of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy.

v. To find the eigenvalues of A, reduce A to echelon form.

### Answers

i. True. If A⋅x = λ⋅x for some non-zero vector x, then λ is an eigenvalue of A. This follows from the definition of **eigenvalues **and **eigenvectors**.

ii. True. A matrix A is not invertible if and only if its determinant is zero. The determinant of A being zero is equivalent to having at least one eigenvalue equal to zero. Therefore, 0 being an eigenvalue implies that A is not invertible, and vice versa.

iii. True. A number c is an eigenvalue of A if and only if the equation (A - cI)⋅x = 0 has a nontrivial solution, where I is the identity matrix. This equation represents the condition for the existence of non-zero solutions for the hom*ogeneous system of equations. Therefore, c being an eigenvalue implies the existence of nontrivial solutions to the equation.

iv. False. Finding an eigenvector of A can be difficult, especially for larger matrices or when the eigenvalues are complex. The process usually involves solving a system of linear **equations **or using other **numerical **methods. Checking whether a given vector is an eigenvector is straightforward by verifying if it satisfies the definition of eigenvectors.

v. False. Reducing A to echelon form does not directly provide the eigenvalues of A. The **echelon **form of a matrix is used to determine other properties, such as rank or invertibility, but it does not directly reveal the eigenvalues. To find the eigenvalues of A, one typically needs to compute the characteristic polynomial and solve for its roots.

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3. What percentage of the shirt cost is the discount?

shirt cost

discount

### Answers

The shirt cost is 100% and the discount is 33.33%

find the value of X what is the value of X?

### Answers

[tex] \sqrt{36 - 25} = \sqrt{11} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ [/tex]

use the laplace transform to solve the given initial-value problem. y'' − 17y' 72y = scripted capital u(t − 1), y(0) = 0, y'(0) = 1 y(t) = scripted capital u t −

### Answers

The solution to the given** initial value** problem is y(t) = -e^(8t) + e^(9t)u(t-1).

To solve the given initial value problem using the Laplace transform, we first take the **Laplace transform **of both sides of the differential equation:

L[y''(t)] - 17L[y'(t)] + 72L[y(t)] = L[scripted capital u(t-1)]

Using the property L[**derivatives **of y(t)] = sY(s) - y(0) - y'(0)s and L[scripted capital u(t-a)] = e^(-as)/s, we get:

s^2 Y(s) - sy(0) - y'(0) - 17sY(s) + 17y(0) + 72Y(s) = e^(-s)/s

Substituting y(0) = 0 and y'(0) = 1, we simplify and solve for Y(s):

Y(s) = 1/(s-9)(s-8)

Using partial fraction decomposition, we can write Y(s) as:

Y(s) = -1/(s-8) + 1/(s-9)

Taking the inverse Laplace transform of Y(s), we get:

y(t) = -e^(8t) + e^(9t)u(t-1)

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This exercise explores the effect of linear transformations f :R² + R2. (a) For points v, w E R², let l be the line segment joining them (i.e., I consists of the convex linear combinations tv + (1 – t)w with 0 0, so the third is at (a,b) with b + 0 (as the third vertex cannot be on the line through the other two vertices); any triangle can be arranged to be such a T by sliding and rotating it in RP

### Answers

This exercise explores the effect of linear transformations on points in R² to R². It considers the** line segment** between two points and the concept of a "**triangle inequality**" for any three points on a plane.

The exercise focuses on the effect of** linear transformations **on points in R² (a 2-dimensional space) to R². It starts by considering two points, v and w, in R² and defines the line segment l that joins them. This line segment is characterized by the **convex** linear combinations of v and w, where t ranges from 0 to 1. These combinations represent the points along the line segment.

The exercise then introduces the concept of a "**triangle inequality**" for any three points on a plane. It states that for any three points, v, w, and u, on a plane, the distance between v and u is less than or equal to the sum of the distances between v and w, and between w and u. This inequality helps establish the relationship between the points in the triangle formed by v, w, and u.

To further explore this concept, the exercise introduces a triangle T with vertices v, w, and u. It states that the first two vertices, v and w, are at (0,0) and (1,0) respectively. The third vertex, u, is at (a,b) with b > 0. This condition ensures that the third vertex cannot lie on the line passing through the other two** vertices**. The exercise suggests that any triangle can be transformed to such a T by sliding and rotating it in RP, the real projective plane.

Overall, the exercise delves into the impact of linear transformations on points in R² and emphasizes the triangle inequality as a fundamental concept for analyzing the relationships between points on a plane.

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A biologist created the following graph to show the relationship between the temperature of water (x), in degrees Celsius, and the number of insect larvae (y) in the water:

graph of y equals minus 2 times the square of x plus 20 times x plus 400

What does the peak of the graph represent?

The number of larvae in the water is greatest at 450 degrees Celsius.

The number of larvae in the water is greatest at 5 degrees Celsius.

The least number of larvae in the water is 450.

The least number of larvae in the water is 5.

### Answers

The **peak** of the **graph** represents the least number of larvae in the water is at 5 degrees Celsius."

What does the peak of the graph represent?

The given **quadratic** **equation** is y = -2x² + 20x + 400.

The coefficient of the x² term is negative (-2) meaning that the **graph** opens downwards.

This indicates that the peak will occur at the vertex of the parabola.

The x-coordinate of the vertex can be found using the formula: x = -b / (2a),

where;

a and b are the coefficients of the x² and x terms, respectively.

In this case, a = -2 and b = 20, so the x-coordinate of the vertex is:

x = -20 / (2 * -2)

x = -20 / -4

x = 5

Therefore, the **peak** of the graph, where the number of larvae is greatest, occurs at 5 degrees Celsius.

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suppose that you are dealt 5 cards from a well shuffled deck of cards. what is the probability that you receive a hand with exactly three suits

### Answers

**Probability **of receiving a hand with exactly three suits [tex]= (4 * (13^3)) / 2,598,960[/tex]

**What is Combinatorics?**

**Combinatorics **is a branch of **mathematics **that deals with counting, arranging, and organizing objects or elements. It involves the study of combinations, permutations, and other related concepts. Combinatorics is used to solve problems related to **counting **the number of possible outcomes or **arrangements **in various scenarios, such as selecting items from a set, arranging objects in a specific order, or forming groups with specific properties. It has applications in various fields, including probability, **statistics**, computer science, and optimization.

To calculate the probability of receiving a hand with exactly three suits when dealt 5 cards from a well-shuffled deck of cards, we can use combinatorial principles.

There are a total of 4 suits in a standard deck of cards: hearts, diamonds, clubs, and spades. We need to calculate the probability of having exactly three of these suits in a 5-card hand.

First, let's calculate the number of **favorable **outcomes, which is the number of ways to choose 3 out of 4 suits and then select one card from each of these suits.

Number of ways to choose 3 suits out of 4: C(4, 3) = 4

Number of ways to choose 1 card from each of the 3 suits[tex]: C(13, 1) * C(13, 1) * C(13, 1) = 13^3[/tex]

Therefore, the number of favorable outcomes is [tex]4 * (13^3).[/tex]

Next, let's calculate the number of **possible outcomes**, which is the total number of 5-card hands that can be dealt from the deck of 52 cards:

Number of possible outcomes: C(52, 5) = 52! / (5! * (52-5)!) = 2,598,960

Finally, we can calculate the probability by dividing the number of favorable outcomes by the number of possible outcomes:

Probability of **receiving **a hand with exactly **three** suits =[tex](4 * (13^3)) / 2,598,960[/tex]

This value can be simplified and expressed as a decimal or a percentage depending on the desired format.

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a basket of fruits contains 5 apples and 3 pears. sharon took two fruits at random. what is the probability that both fruits are apples? write your answer in the simplest form of fraction

### Answers

The **probability** that Sharon **randomly** selects two apples from the basket of fruits, given that there are 5 apples and 3 pears, can be expressed as a fraction.

To find the probability, we need to consider the total **number** of possible outcomes and the number of favorable **outcomes**.

The total number of **possible outcomes** is the number of ways Sharon can select any two fruits from the basket, which can be calculated using combinations. In this case, there are 8 fruits in total, so the total number of possible outcomes is C(8, 2) = 28.

The number of **favorable outcomes** is the number of ways Sharon can select two apples from the five available in the basket, which is C(5, 2) = 10.

Therefore, the probability that both fruits Sharon selects are apples is 10/28, which can be simplified to 5/14.

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If A is a 4x6 matrix, what is the largest possible value for the rank of A?

a.4 b.6 c.2 d.3

### Answers

A** 4x6 matrix** is a rectangular array of numbers with** 4 rows and 6 columns**. The elements of the matrix are typically denoted by a letter with subscripts indicating the row and column.

The rank of a matrix is the **dimension** of the vector space spanned by its columns or rows. It is also equal to the number of linearly independent columns or rows of the matrix.

Since A is a 4x6 matrix, the largest possible value for the rank of A is min(4, 6), which is 4x4 identity matrix or 4 if there are 4 linearly **independent** rows or columns in A.

To find the rank of A, we can perform row **operations** on A to reduce it to row echelon form or reduced row echelon form. Row operations include adding a multiple of one row to another row, multiplying a row by a **non-zero scalar**, and swapping two rows.

After performing the row operations, the number of non-zero rows in the resulting matrix is the rank of A. Since the rank of a matrix is equal to the rank of its transpose, we can also perform column operations to find the rank of A.

Therefore, the answer is (a) 4, as it is the largest possible value for the rank of a 4x6 matrix.

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33. SAT test scores are normally distributed with a mean of 500 and standard deviation of 100. Find the probability that a randomly chosen test-taker will score below 450. (Round your answer to four decimal place). 35. Using the information in question 33, what is the probability that a random chosen test- taker will score above 600? (Round your answer to four decimal place). For questions 33-35, first find the corresponding z-values by hand, then you may use your calculator or a z-table to find your results. Clearly state the method you used and how you calculated your results if you used a calculator.

### Answers

The **probability **that a randomly chosen test-taker will **score **below 450 on the SAT is approximately 0.1587, and the probability of scoring above 600 is approximately 0.0228.

To find the **probability **that a randomly chosen test-taker will score below 450 on the SAT, we need to calculate the corresponding z-value and use a z-table or calculator to find the probability.

Step 1: Calculate the z-value using the formula z = (x - μ) / σ, where x is the score, μ is the mean, and σ is the** standard deviation.** In this case, x = 450, μ = 500, and σ = 100.

z = (450 - 500) / 100

z = -0.5

Step 2: Use a z-table or calculator to find the cumulative probability associated with the z-value. The cumulative probability represents the area under the standard normal distribution curve up to the given z-value. In this case, we want the **area **to the left of z = -0.5.

Using a z-table or calculator, the cumulative probability for z = -0.5 is approximately 0.3085.

Step 3: Subtract the cumulative probability from 0.5 to find the probability below 450. Since the standard normal distribution is symmetric, the probability below the z-value is equal to 0.5 minus the cumulative probability.

Probability below 450 = 0.5 - 0.3085

Probability below 450 ≈ 0.1915

Therefore, the probability that a randomly chosen test-taker will score below 450 on the SAT is approximately 0.1915, rounded to four decimal places.

For the second question, we need to find the probability that a randomly chosen test-taker will score above 600 on the SAT.

Step 1: Calculate the z-value using the formula z = (x - μ) / σ. In this case, x = 600, μ = 500, and σ = 100.

z = (600 - 500) / 100

z = 1

Step 2: Use a z-table or calculator to find the cumulative probability associated with the z-value. We want the area to the left of z = 1.

Using a z-table or calculator, the cumulative probability for z = 1 is approximately 0.8413.

Step 3: Subtract the cumulative probability from 1 to find the probability above 600. Since the standard normal distribution is **symmetric**, the probability above the z-value is equal to 1 minus the cumulative probability.

Probability above 600 = 1 - 0.8413

Probability above 600 ≈ 0.1587

Therefore, the probability that a randomly chosen test-taker will score above 600 on the SAT is approximately 0.1587, rounded to four decimal places.

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Name a time where the two “hands” of an analog clock would form a right angle. (BONUS: How many times does a right angle form on the clock face each day?)

### Answers

There are a total of 2 x 2 = 4 instances where the two "hands" of an analog clock form a** Right angle**.

The two "hands" of an analog clock form a right angle at two specific times during a 12-hour period. The first occurrence is at 3:15, where the minute hand **points **to the 3 and the hour hand points to the 9, forming a right angle. The second occurrence is at 9:45, where the minute hand points to the 9 and the hour hand points to the 3, forming another right angle.

To determine how many times a right angle forms on the clock face each day, we need to consider both the AM and PM periods. In a 24-hour day, there are 12 hours in the AM (from 12:00 AM to 11:59 AM) and 12 **hours **in the PM (from 12:00 PM to 11:59 PM).

For each 12-hour **period**, there are two instances where the hands form a right angle. Therefore, in a full day, there are a total of 2 x 2 = 4 instances where the two "hands" of an analog clock form a right angle.

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(1 point) convert the following rectangular coordinates into polar coordinates. always choose 0≤θ<2π. (a) (0,5)

### Answers

The **polar coordinates **for the rectangular coordinates (0, 5) are (r, θ) = (5, π/2).

To convert **rectangular coordinates** (x, y) to polar coordinates (r, θ), we use the formulas r = √(x² + y²) and θ = arctan(y/x). In this case, x = 0 and y = 5. We can apply the formulas as follows:

STEP 1. Calculate r: r = √(0² + 5²) = √(25) = 5

STEP 2. Calculate θ: Since x = 0, we cannot use the arctan(y/x) formula directly. Instead, we determine the angle based on the quadrant in which the point lies. The point (0, 5) lies on the positive y-axis, which corresponds to an angle of π/2 radians.

So, the polar coordinates for the rectangular coordinates (0, 5) are (r, θ) = (5, π/2).

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ACGF is a parallelogram.

image

If ∠CAG has a measure of (a+20)° , and ∠ACF has a measure of (2a+10)° find the measure of ∠ACF.

### Answers

The **measure **of ∠ACF is 110° for the given **parallelogram**.

Given that ∠CAG has a **measure **of (a+20)°, and ∠ACF has a measure of (2a+10)°.

As we know that the sum of the interior angle is always 360 degrees in** a quadrilateral.**

So, 2(a+20)° + 2(2a+10)° = 360

2a + 40 + 4a + 20 = 360

6a = 360 - 60

6a = 300

a = 50

Therefore, the value of a is 50.

To find the measure of ∠ACF, we substitute the value of a back into the **equation**:

∠ACF = 2a + 10

∠ACF = 2(50) + 10

∠ACF = 100 + 10

∠ACF = 110°

So, the **measure **of ∠ACF is 110°.

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a manufacturer of cell phones would like to estimate how much longer the battery lasts in their model 10 phone than in their model 9 phone. to estimate this difference, they randomly select 40 cell phones of each model from the production line. they subject each phone to a standard battery life test. the 40 model 10 phones have a mean battery life of 14.4 hours with a standard deviation of 2.1 hours. the 40 model 9 phones have a mean battery life of 12.8 hours with a standard deviation of 2.3 hours. what is the appropriate inference procedure to be used to estimate how much longer the battery lasts in their model 10 phone than in their model 9 phone? t confidence interval for a mean z confidence interval for a proportion t confidence interval for a difference in means z confidence interval for a difference in proportions

### Answers

The required, we can be 95% confident that the true difference in battery life between the model 10 and model 9 phones is between 0.25 and 2.95 hours longer for model 10 phones.

The appropriate inference procedure to be used to estimate how much longer the battery lasts in their model 10 phone than in their model 9 phone is a t-confidence interval for a difference in means.

The reason we use a t-test is that we are dealing with small sample sizes (n₁ = n₂ = 40) and do not know the population** standard** **deviations**.

We use a **confidence interval **instead of a hypothesis test because the question is asking for an estimate of the difference in battery life, rather than testing a specific hypothesis.

We can use the following formula to calculate the **confidence interval**:

( X₁ - X₂ ) ± t* ( Sqrt( s₁²/n₁ + s₂²/n₂ ) )

where:

X₁ and X₂ are the sample means of the battery life for model 10 and model 9, respectively

s₁ and s2 are the sample standard deviations of the battery life for model 10 and model 9, respectively

n₁ and n₂ are the sample sizes for model 10 and model 9, respectively

t is the critical t-value for the desired confidence level (degrees of freedom = n₁ + n₂ - 2)

Plugging in the given values, we get:

( 14.4 - 12.8 ) ± t* ( √( 2.1²/40 + 2.3²/40 ) )

= 1.6 ± t* 0.573

To find the critical t-value, we need to determine the degrees of freedom:

df = n₁ + n₂ - 2 = 78

Using a t-table or a calculator, for a 95% confidence level with 78 degrees of freedom, the critical t-value is approximately 1.99.

Plugging this into the formula above, we get:

1.6 ± 1.99 * 0.573

= ( 0.25, 2.95 )

Therefore, we can be 95% **confident **that the true difference in battery life between the model 10 and model 9 phones is between 0.25 and 2.95 hours longer for model 10 phones.

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