Nina purchased apples and strawberries. She purchased a total of 9 pounds of fruit and spent a total of $16.35. Strawberries cost $1.60 per pound and apples cost $ 1.99 per pound. How many pounds of each type of fruit did she buy?

Answer:

  see below

Step-by-step explanation:

The formula is the same whether the rate of change is positive or negative. Here, it is negative.

  A = Pe^(rt)

for P = 4000, r = -0.10. Then you have ...

  A = 4000e^(-0.10t)

Answer:

Step-by-step explanation:

Let x represent the distance that the truck was driven.

Buck rented a truck for $39.95 plus $0.32 per mile. This means that the total cost of driving x miles with the truck would be

39.95 + 0.32x

Before returning the truck, he Filled the tank with gasoline, which cost $9.85. This means that the total amount that he spent is

39.95 + 0.32x + 9.85

If the total cost was $70.23, it means that

39.95 + 0.32x + 9.85 = 70.23

49.8 + 0.32x = 70.23

0.32x = 70.23 - 49.8 = 20.43

x = 20.43/0.32 = 63.8

Approximately 64 miles

Answer:

S = 128πx² + 64πx + 8π

Step-by-step explanation:

Suraface area of a cylinder is given by:

S = 2πrh + 2πr²

We know that the height is 3 times as big as the radius, hence:

h = 3r

so we can plug in the new h value and rewrite the S equation as:

S = 2πrh + 2πr²

S = 2πr(3r) + 2πr²

S = 6πr² + 2πr²

S = 8πr²

We're given in the question that the radius is ​(4x + 1​) inches, so plug that into r.

Given: r = 4x + 1​

Therefore,

S = 8πr²

S = 8π(4x + 1​)²

S = 8π(16x²+8x+1)

S = 128πx² + 64πx + 8π

Answer:

The answer to your question is 128πx² + 64πx + 8π

Step-by-step explanation:

Data

radius = (4x + 1)

height = 3(4x + 1)

Formula

Area = 2πrh + 2πr²

Substitution

Area = 2π(4x + 1)(3)(4x + 1) + 2π(4x + 1)

Simplification

Area = 6π(4x + 1)² + 2π(4x + 1)²

Area = 6π(16x² + 8x + 1) + 2π(16x² + 8x + 1)²

Area = 96πx² + 48πx + 6π + 32πx² + 16πx + 2π

Area = 128πx² + 64πx + 8π

Answer:

321.24 ft²

Step-by-step explanation:

If 3in = 5ft

   1in = x

5/3 = 1.6

So 1in = 1.67 ft

Actual dimensions will be;

L = 19.2*1.67 = 32.06ft

W = 6*1.67 = 10.02ft

Area = 32.06 * 10.02 = 321.24 ft²

To find the length and width of the actual studio, set up a proportion to convert the given dimensions from the scale to the actual dimensions. Then, find the area of the actual studio by multiplying the length and width.

To find the length and width of the actual studio, we need to convert the given dimensions from the scale to the actual dimensions. Since the scale is 3 in.: 5 ft, we can set up the proportion:

3 in / 5 ft = 19.2 in / x ft

Cross multiplying and solving for x, we get:

x = 32 ft

The length of the actual studio is 32 ft. Similarly, we can find the width:

3 in / 5 ft = 6 in / y ft

Solving for y, we get:

y = 10 ft

The width of the actual studio is 10 ft.

To find the area of the actual studio, we multiply the length and width:

Area = length x width = 32 ft x 10 ft = 320 ft²

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Answer: 343 in³

Explanation: In this problem, we're asked to find the volume of a cube.

It's important to understand that a cube is a type of rectangular prism and the formula for the volume of a rectangular prism is shown below.

Volume = length × width × height

In a cube however, the length, width, and height are all the same. So we can use the formula side × side × side instead.

So the formula for the volume of a cube is side × side × side or s³.

So to find the volume of the given cube, since each side has a length of 7 inches, we can plug this information into the formula to get (7 in.)³ or (7 in.)(7 in.)(7 in).

7 x 7 is 49 and 49 x 7 is 343.

So we have 343 in³.

So the volume of the given cube is 343 in³.

Answer:

343 in³

Step-by-step explanation:

Answer:

Step-by-step explanation:

The diagonals of the parallelogram are A(-5, -1), C(-1, 5) and B(-9, 6), D(3, -2).

Slope of diagonal AC = (5 - (-1)) / (-1 - (-5)) = (5 + 1) / (-1 + 5) = 6 / 4 = 3/2

Slope of diagonal BD = (-2 - 6) / (3 - (-9)) = -8 / (3 + 9) = -8 / 12 = -2/3

For the parallelogram to be a rhombus, the intersection of the diagonals are perpendicular.

i.e. the product of the two slopes equals to -1.

Slope AC x slope BD = 3/2 x -2/3 = -1.

Therefore, the parallelogram is a rhombus.

Answer: 7

Step-by-step explanation:

Given : The list shows the number of viewers of an online music video each day for 5 consecutive days.

5;  35;  245;  1,715;  12,005;

In exponential growth equation : [tex]y=Ab^x[/tex] , A = initial value , b= growth factor and x = time period.

In the given table A = 5  , at x = 0

Growth factor is the common ratio between the consecutive terms.

So [tex]b=\dfrac{35}{5}=7[/tex]

and we can check that it is common between each consecutive terms is 7.

Hence, the factor by which the number of viewers change each day to the first day to the fifth day = 7

The number of viewers changed by a factor of 2401 from the first day to the fifth day.

The number of viewers of an online music video changed each day by multiplying the previous day's viewers by a constant factor. To find this factor, we divide the viewers on the fifth day by the viewers on the first day. By doing this, we get a factor of 12005/5 = 2401.

So, the number of viewers changed by a factor of 2401 from the first day to the fifth day.

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The measure of the first, second and third angles are 20°, 27° and 133°.

An exterior angle of a triangle is the sum of two opposite interior angles.

According to given question

The residents of a downtown neighbourhood designed a​ triangular-shaped park as part of a city beautification program.

Let us assume the first angle to be x°.

Therefore from the given data second angle is (x + 7)° and the third angle is (7x - 7)°.

We know that the sum of all the interior angles of a given triangle is 180°.

∴ x + (x + 7) + (7x - 7) = 180°

  9x = 180°

    x = 180°/9

    x = 20°.

So, The first angle of the triangular park is 20°, The second angle is 27° and the third angle is 133°.

Learn more about triangles here :

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t= 2.25 secs

Step-by-step explanation:

Step 1 :

Equation for motion with uniform acceleration is v = u+ at

where v is the final velocity

          u is the initial velocity

          a is the acceleration due to gravity

          and t is the time

Step 2 :

Here , v = 0 because at the highest point final velocity is 0.

u = 72 feet/sec

a = -32 ft/sec^2

We need to find the time t.

Substituting in the equation we have,

0 = 72 -32 * t

=> 32 t = 72

=> t = 72/32 = 2.25 secs

Answer:

Superintendent of the region will select approximately 4 Science teachers in the board.

Step-by-step explanation:

Given:

Number of math teachers = 40

Number of English teachers = 45

Number of Science teachers = 27

Number Social Studies teachers = 32

Number of teachers to be selected in the board =20

We need to find the number Science teachers should be selected in the board.

Solution:

First we ill find the Total number of teachers.

Now we can say that;

Total number of teachers is equal to sum of Number of math teachers, Number of English teachers,Number of Science teachers and Number Social Studies teachers

framing in equation form we get;

Total number of teachers = [tex]40+45+27+32=144[/tex]

Probability of Science teacher can be calculated by Dividing Number of Science teacher by Total number of teachers.

[tex]P(S)=\frac{27}{144}=0.1875[/tex]

Now  the number of Science teachers should be selected in the board we need to multiply P(S) with Number of teachers to be selected in the board.

framing in equation form we get;

number of Science teachers should be selected in the board = [tex]0.1875\times 20=3.75\approx4[/tex]

Hence We can say that;

Superintendent of the region will select approximately 4 Science teachers in the board.

Calculate the proportion of science teachers out of the total and apply it to the board size to determine the number of science teachers to select.

To determine how many science teachers the superintendent should select for a board of 20 teachers:

Option D: [tex]20 \div 6-4[/tex] is the correct solution to the equation [tex]6(x+4)=20[/tex]

Option E: [tex]\frac{1}{6} (20-24)[/tex] is the correct solution to the equation [tex]6(x+4)=20[/tex]

Option F: [tex](20-24) \div 6[/tex] is the correct solution to the equation [tex]6(x+4)=20[/tex]

Explanation:

The expression is [tex]6(x+4)=20[/tex]

Let us find the value of x.

[tex]\begin{aligned}6(x+4) &=20 \\6 x+24 &=20 \\6 x &=-4 \\x &=-\frac{2}{3}\end{aligned}[/tex]

Now, we shall find the expression that is equivalent to the value [tex]x=-\frac{2}{3}[/tex]

Option A: [tex](20-4) \div 6[/tex]

Simplifying the expression, we have,

[tex]\frac{16}{6}=\frac{8}{3}[/tex]

Since, [tex]\frac{8}{3}[/tex] is not equivalent to [tex]x=-\frac{2}{3}[/tex], the expression [tex](20-4) \div 6[/tex] is not equivalent to the equation [tex]6(x+4)=20[/tex]

Hence, Option A is not the correct answer.

Option B: [tex]16(20-4)[/tex]

Simplifying the expression, we have,

[tex]16(16)=256[/tex]

Since, 256 is not equivalent to [tex]x=-\frac{2}{3}[/tex], the expression [tex]16(20-4)[/tex] is not equivalent to the equation [tex]6(x+4)=20[/tex]

Hence, Option B is not the correct answer.

Option C: [tex]20-6-4[/tex]

Simplifying the expression, we have,

[tex]20-10=10[/tex]

Since, 10 is not equivalent to [tex]x=-\frac{2}{3}[/tex], the expression [tex]20-6-4[/tex] is not equivalent to the equation [tex]6(x+4)=20[/tex]

Hence, Option C is not the correct answer.

Option D: [tex]20 \div 6-4[/tex]

Using PEMDAS and simplifying the expression, we have,

[tex]$\begin{aligned}(20 \div 6)-4 &=\frac{10}{3}-4 \\ &=\frac{10-12}{3} \\ &=-\frac{2}{3} \end{aligned}$[/tex]

Thus, [tex]-\frac{2}{3}[/tex] is equivalent to [tex]x=-\frac{2}{3}[/tex], the expression [tex]20 \div 6-4[/tex] is equivalent to the equation [tex]6(x+4)=20[/tex]

Hence, Option D is the correct answer.

Option E: [tex]\frac{1}{6} (20-24)[/tex]

Simplifying the expression, we have,

[tex]$\begin{aligned} \frac{1}{6}(20-24) &=\frac{1}{6}(-4) \\ &=-\frac{4}{6} \\ &=-\frac{2}{3} \end{aligned}$[/tex]

Thus, [tex]-\frac{2}{3}[/tex] is equivalent to [tex]x=-\frac{2}{3}[/tex], the expression [tex]\frac{1}{6} (20-24)[/tex] is equivalent to the equation [tex]6(x+4)=20[/tex]

Hence, Option E is the correct answer.

Option F: [tex](20-24) \div 6[/tex]

Simplifying the expression, we have,

[tex]$\begin{aligned}(20-24) \div 6 &=-\frac{4}{6} \\ &=-\frac{2}{3} \end{aligned}$[/tex]

Thus, [tex]-\frac{2}{3}[/tex] is equivalent to [tex]x=-\frac{2}{3}[/tex], the expression [tex](20-24) \div 6[/tex] is equivalent to the equation [tex]6(x+4)=20[/tex]

Hence, Option F is the correct answer.

To solve 6(x + 4) = 20, we isolate x to get x = -2/3. The correct expressions representing the solution are E. 1/6(20 - 24) and F. (20 - 24) / 6.

To solve the equation 6(x + 4) = 20, we need to isolate x.

First, distribute the 6 on the left-hand side:

6x + 24 = 20

Next, subtract 24 from both sides:

6x = -4

Finally, divide by 6:

x = -4/6 = -2/3

Now, let's analyze the given choices:

Therefore, the correct choices are E and F.

Answer:

Part A) see the explanation

Part B) see the explanation

Step-by-step explanation:

Let

x ----> the distance from one side of the tunnel in feet:

f(x) ---> the height of a tunnel in feet

Part A: What do the x-intercepts and maximum value of the graph represent? What are the intervals where the function is increasing and decreasing, and what do they represent about the distance and height?

1)

we know that

The x-intercepts are the values of x when the value of the function is equal to zero

In the context of the problem, the x-intercepts are the distances from one side of the tunnel when the height of the tunnel is equal to zero

2)

The maximum value of the graph is the vertex

The x-coordinate of the vertex represent the distance from one side of the tunnel when the height is a maximum value

The y-coordinate of the vertex represent the maximum height of the tunnel

In this problem

the vertex is (18,32)

That means

The maximum height of the tunnel is 32 feet and occurs when the distance from one side of the tunnel is 18 feet

3)

we know that

The function is increasing at the interval [0,18)

That means

As the distance from one side of tunnel increases the height of the tunnel increases too

The function is decreasing at the interval (18,36]

That means

As the distance from one side of tunnel increases the height of the tunnel decreases

Part B: What is an approximate average rate of change of the graph from x = 5 to x = 15, and what does this rate represent?

we know that

To find the average rate of change, we divide the change in the output value by the change in the input value

the average rate of change using the graph is equal to

[tex]\frac{f(b)-f(a)}{b-a}[/tex]

In this problem we have

[tex]a=5[/tex]

[tex]b=15[/tex]

[tex]f(a)=f(5)=15[/tex]  ----> see the attached figure

[tex]f(b)=f(15)=31[/tex]  ----> see the attached figure

Substitute

[tex]\frac{31-15}{15-5}=1.6[/tex]

That means

The height of the tunnel increases 1.6 feet as the distance from one side of tunnel increases 1 foot in the interval from x=5 to x=15

Answer:

The amount of acid in third container is =42%

Step-by-step explanation:

Given , one container is filled with a mixture that is 30% acid a second container filled with a mixture that is 50% acid and the second container 50% larger than the first .

Let, the volume of first container is = x

Then , the volume of second container = (x+ x of 50%)

                                                                 = x + 0.5 x

                                                                 = 1.5 x

Therefore the amount of acid in first container [tex]=x \times \frac{30}{100}[/tex] = 0.3 x

The amount of acid in second container [tex]=1.5x \times \frac{50}{100}[/tex]  = 0.75x

Total amount of acid= 0.3x + 0.75x = 1.05 x

Total amount  of solution = x+1.5x = 2.5x

The amount of acid in third container is = [tex]\frac{1.05x}{2.5x} \times 100%[/tex] %

                                                                     = 42%

Final answer:

By calculating the total acid content and total volume when two containers with different acid concentrations are mixed, we find that the third container has an acid concentration of 42%.

Explanation:

The question involves mixing two solutions of different acid concentrations and calculating the final concentration of acid. Let's assume the first container has 1 unit of volume. Since the second container is 50% larger, it has 1.5 units of volume. The first container has 30% acid, so it contains 0.3 units of acid. The second container has 50% acid, resulting in 0.75 units of acid. Together, the total volume of the solution is 2.5 units, and the total amount of acid is 1.05 units.

Therefore, the percentage of acid in the third container is calculated by dividing the total acid by the total volume: (1.05 units of acid / 2.5 units of volume) x 100% = 42%.

Answer:

564,480

Step-by-step explanation:

The appliocation of factorial n! = n(n-1)(n-2)(n-3).....

Arrangement when french and English delegates are to seat together ;

since we have 10 delegates, Freench and english will be treated as 1 delegates as such we have 9groups. The number of ways of arranging 9groups = 9! and if the french and English are to be treated as a group = 2!

Hence, number of ways of arrangement = 9! x 2! = 725,760

In this case, English and french are to be seated next to each other while russia and US delegates are not to seat nect to each other ; both groups will be treated as 2 as such we have 8groups, the number of ways of arranging 8groups = 8!

french and english together = 2!

Russia and US together = 2!

The number of ways of arrangement = 8! x 2! x 2! = 161,280

hence the number of ways of arranging 10 delegates if french and english are to be seated next to each other and Russia and US are not to be seated next to each other;

N = 725,760 - 161,280 = 564,480

There are 282,240 different seating arrangements possible if the French and English delegates are to be seated next to each other and the Russian and U.S. delegates are not to be next to each other.

To determine the number of different seating arrangements, we can treat the French and English delegates as a single entity. Therefore, we have 9 entities overall (Russia, (France and England), United States, and the 6 remaining countries).

The number of arrangements of these 9 entities is 9! (9 factorial) which is equal to 362,880. However, since the Russian and U.S. delegates cannot be seated next to each other, we need to subtract the number of arrangements where they are together.

Let's consider the Russian and U.S. delegates as a single entity as well. Now we have 8 entities to arrange (Russian and U.S., (France and England), and the 6 remaining countries). The number of arrangements of these 8 entities is 8! which is equal to 40,320.

Finally, we need to consider the arrangements of the Russian and U.S. delegates within their entity. The Russian and U.S. delegates can be arranged in 2! (2 factorial) ways, which is equal to 2.

Therefore, the total number of different seating arrangements is 362,880 - (40,320 * 2) = 362,880 - 80,640 = 282,240.

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Answer:

  85 m

Step-by-step explanation:

The diagonal of a square is √2 times the length of the side. The park will have a side length of 120/√2 m ≈ 84.85 m, about 85 meters.

_____

The relations are ...

  diagonal = (√2)×(side length)

  side length = diagonal/√2 . . . . . . . . . divide the above equation by √2

Answer:

Step-by-step explanation:

Triangle ABC is a right angle triangle.

From the given right angle triangle

BC represents the hypotenuse of the right angle triangle.

With m∠B as the reference angle,

AB represents the adjacent side of the right angle triangle.

AC represents the opposite side of the right angle triangle.

To determine BC , we would apply the Sine trigonometric ratio which is expressed as

Tan θ = opposite side/hypotenuse. Therefore,

Sin 44 = 10/BC

0.695 = 10/BC

BC = 10/0.695

BC = 14.4 to 1 decimal place.

The total number of people surveyed is 134. The number of people who listened to either rap or alternative rock is 76. The number of people who listened to heavy metal only is 1.

To solve this problem, we can use a Venn diagram to organize the given information. Let's start by filling in the total number of people surveyed, which is the sum of those who listened to rap, heavy metal, alternative rock, and none of the three channels. From the given information, we have:

To find the total number of people surveyed, we add these numbers:

37 + 30 + 49 + 18 = 134.

Therefore, 134 people were surveyed.

b. To find the number of people who listened to either rap or alternative rock, we need to add the number of people who listened to rap and the number of people who listened to alternative rock, while making sure we don't double count those who listened to both:

37 + 49 - 10 = 76.

Therefore, 76 people listened to either rap or alternative rock.

c. To find the number of people who listened to heavy metal only, we need to subtract those who listened to heavy metal and either rap or alternative rock from the total number of people who listened to heavy metal:

30 - 13 - 16 = 1.

Therefore, 1 person listened to heavy metal only.

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Answer:

10 hrs

Step-by-step explanation:

sally earns $5/hr while Adam earns $4/hr

let the number of Hour sally works be 'x'

From question Adam work 2 hrs more than sally,

therefore Adam works (x + 2)hrs also the both earn same amount per week

therefore;

4 x [tex](x + 2)[/tex] = 5 x [tex]x[/tex]

4x + 8 = 5x

5x - 4x = 8

x = 8hrs

Adam works for (8+2)hrs = 10hrs

Answer:

55% and 24%

Step-by-step explanation:

Firstly, we need an arbitrary number to serve as the number of students.

Let the total number of students be 120.

Firstly we need college-educated percentage.

Men are two-thirds, women are one-third

Men = 2/3 * 120 = 80

Women = 1/3 * 120 = 40

5/8 of men are college educated = 5/8 * 80 = 50

2/5 of women are college educated = 2/5 * 40 = 16

Total college educated = 50+ 16 = 66

% college educated = 66/120 * 100 = 55%

Percentage of college educated that are women.

Total college educated = 66

The % is 16/66 * 100 = 24%

Answer:

Step-by-step explanation:

Let n represent the number of nickels in the jar.

The jar contains only​ pennies, nickels,​ dimes, and quarters. There are 14 ​pennies, 28 ​dimes, and 16 quarters. This means that the total number if pennies, dimes and quarters would be

14 + 28 + 16 = 58

This means that the number of coins that are not nickel is 58

The total number of coins in the jar is 70. Therefore, the equation that can be used to find n is

n + 58 = 70

n = 70 - 58

n = 12 nickels

Answer:

Insufficient information

Step-by-step explanation:

The information provided us isn't sufficient to determine the range for the measurements in list X. Because we need to be given the range for the measurements in Y at least. We also need the number of measurements in both X and Y.

The answer is 15,

because F(X)'s minimum value is (-3, -3)

and G(X)'s maximum value is (12, 3)

So when you subtract -3 from 12, you get a number that is greater then both. (because of the negative 3)

12 - (-3) = 15

Answer:

A.  15

Step-by-step explanation:

I got it rigth on the test :)

Have a great day fam!

The value of given expression is 16

Solution:

Given expression is:

[tex]\sqrt{x^2} + \sqrt{x} \times \sqrt{x} + 6[/tex]

We have to find the value of expression when x = 5

Substitute x = 5

[tex]\sqrt{x^2} + \sqrt{x} \times \sqrt{x} + 6\\\\\sqrt{5^2} + (\sqrt{5} \times \sqrt{5}) + 6[/tex]

We know that,

[tex]\sqrt{a} \times \sqrt{a} = a[/tex]

Therefore, the above equation becomes,

[tex]\sqrt{5^2} + 5 + 6\\\\\sqrt{25} + 5 + 6\\\\5 + 5 + 6 = 10 + 6 = 16[/tex]

Thus the value of given expression is 16

Answer:

Option 5) 23.5% is a parameter and 28% is a statistic

Step-by-step explanation:

We are given the following in the question:

"According to the U.S. Census bureau, 23.5% of people in the United States are under the age of 18. In a random sample of 250 residents of a small town in Ohio, 28% of the sample was under 18."

Population of interest:

People in the United States

Parameter:

23.5% of people in the United States are under the age of 18

Sample:

Random sample of 250 residents of a small town in Ohio

Statistic:

28% of the sample was under 18

Thus, the correct answer is

Option 5) 23.5% is a parameter and 28% is a statistic

Answer: The total amount that he paid for the suit is $170

Step-by-step explanation:

The original price of the suit was $260.

Mr.Nguyen bought a suit that was on sale for 40% off the original price. This means that the amount of discount in the suit is

40/100 × 260 = 0.4 × 260 = $104

The sale price would be

260 - 104 = $156

He paid 9% sales tax on the sale price. This means that the amount of sales tax that he paid is

9/100 × 156 = 0.09 × 156 = $14.04

The total amount that he paid for the suit, including tax is

156 + 14.04 = $170 to the nearest dollar

The probability of A = 1/4 = 0.25

The probability of B = 4/5 = 0.80

P(A ∩ B) = P(A) * P (B) = (0.25) * (0.80) = 0.20

P(A ∩ B') = P(A) * P (B') = (0.25) * (1-0.80) = 0.05

P(A' ∩ B') = P(A') * P (B') = (1-0.25) * (1-0.80) = 0.15

Answer:

A. P(A U B) = P(A) + P(B) - P(A ∩ B) = 0.25 + 0.80 - 0.20 = 0.85

B. P(A U B') = P(A) + (1 - P(B)) - P(A ∩ B') = 0.25 + (1-0.80) - 0.05 = 0.40

C. P(A' U B') = (1 - P(A)) + (1 - P(B)) - P(A' ∩ B') = (1-0.25) + (1-0.80) - 0.15 = 0.80

The probabilities using the addition rule of probabilities are P(A ∪ B) = 17/20, P(A ∪ B') = 2/5, and P(A' ∪ B') = 4/5.

In solving this problem, we will use the addition rule for probabilities to find the likelihood of different combinations of randomly generated web ad designs based on given design elements.

A. P(A ∪ B)

We'll start by finding the probability of event A (the design color is red) and B (the font size is not the smallest one) occurring independently. Since there are four different colors, the probability of A is 1/4. When it comes to event B, since there are five different font sizes and the event is that the font size is not the smallest one, there are four font sizes that meet the criteria.

Therefore, the probability of B is 4/5. We can now calculate P(A ∪ B) using the formula P(A) + P(B) - P(A AND B). Assuming that the color choice and font size are independent of each other (the choice of one does not affect the choice of the other), the probability of both A and B occurring (A AND B) is simply the product of their separate probabilities: P(A) x P(B) = (1/4) x (4/5) = 1/5.

So, P(A ∪ B) = P(A) + P(B) - P(A AND B) = (1/4) + (4/5) - (1/5) = 1/4 + 16/20 - 4/20 = 1/4 + 12/20 = 5/20 + 12/20 = 17/20.

B. P(A ∪ B')

To find P(A ∪ B'), we first need to calculate P(B'). Since the probability of B is 4/5, the probability of B' (the font size is the smallest one) is the remaining fraction of 1, which is 1/5. We then apply the addition rule, which becomes P(A \∪\ B') = P(A) + P(B') - P(A AND B'), with P(A AND B') being the probability of choosing the red color and the smallest font size. Because A and B' are independent, P(A AND B') = P(A) x P(B') = (1/4) x (1/5) = 1/20.

Therefore, P(A ∪ B') = P(A) + P(B') - P(A AND B') = 1/4 + 1/5 - 1/20 = 5/20 + 4/20 - 1/20 = 8/20 or 2/5.

C. P(A' ∪ B')

To find P(A' ∪ B'), we need to determine the probabilities of A' and B' occurring separately. P(A') is the probability of not choosing the red color, which is the complement of P(A), meaning P(A') = 1 - P(A) = 3/4. P(B') as we calculated before, is 1/5. As these events are independent, P(A' AND B') = P(A') x P(B') = (3/4) x (1/5) = 3/20.

Thus, P(A' ∪ B') = P(A') + P(B') - P(A' AND B') = 3/4 + 1/5 - 3/20 = 15/20 + 4/20 - 3/20 = 16/20 or 4/5.

Final answer:

By defining equations based on the relationships between Jen's, Carrie's, and Fran's numbers and solving them, we find that Carrie's number is 64, Jen's number is 73, and Fran's number is 70.

Explanation:

To solve for Fran's number, we need to use the information given: Jen's number is 9 more than Carrie's, and Fran's number is 3 less than Jen's number. Their total sum is 207. We can set up equations to solve this. Let Carrie's number be c, Jen's number be c + 9 (since it is 9 more than Carrie's), and Fran's number be c + 9 - 3 (since it is 3 less than Jen's).

The equation to express the sum of their numbers will be: c + (c + 9) + (c + 9 - 3) = 207.

Now let's solve the equation:

Now that we have Carrie's number, we can find Fran's number:

Therefore, Fran's number is 70.

Answer:

The volume of cube is increasing at a rate 7.5 cubic meter per hour.  

Step-by-step explanation:

We are given the following in the question:

Surface area of cube = 24 square meters.

Let l be the edge of cube.

Surface area of cube =

[tex]6l^2 = 24\\l^2 = 4\\l = 2[/tex]

Thus, at that instant the edge of cube is 2 meters.

[tex]\dfrac{dS}{dt} = 15\text{ square meters per hour}\\\\\dfrac{d(6l^2)}{dt} = 15\\\\12l\dfrac{dl}{dt} = 15\\\\\dfrac{dl}{dt} = \dfrac{15}{12\times 2} = \dfrac{15}{24}[/tex]

We have to find the rate of change in volume.

Volume of cube =

[tex]l^3[/tex]

Rate of change of volume =

[tex]\dfrac{dV}{dt} = \dfrac{d(l^3)}{dt} = 3l^2\dfrac{dl}{dt}\\\\\dfrac{dV}{dt} = 3(2)^2\times \dfrac{15}{24} \\\\\dfrac{dV}{dt} = 7.5\text{ cubic meter per hour}[/tex]

Thus, the volume of cube is increasing at a rate 7.5 cubic meter per hour.

The rate of change of the volume of the cube at the given instant is; dV/dt = 7.5 m³/h

Surface area is increasing at a rate of 15 m²/h

Thus, dS/dt = 15 m²/h

Now, at a certain instant the surface area of the cube is 24 m².

Formula for cube surface area is; S = 6l²

Thus;

6l² = 24

l² = 24/6

l² = 4

l = √4

l = 2 m

From S = 6l², differentiating both sides with respect to t gives;

dS/dt = 12l(dl/dt)

We know that dS/dt = 15

Thus;

12l(dl/dt) = 15

dl/dt = 15/(12l)

putting 2 for l gives;

dl/dt = 15/(12 * 2)

dl/dt = 0.625 m/h

Now formula for volume of a cube is; V = l³

Differentiating both sides with respect to t gives;

dV/dt = 3l²(dl/dt)

plugging in then relevant values gives;

dV/dt = 3 × 2² × (0.625)

dV/dt = 7.5 m³/h

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Answer:

The equation of the normal line to the curve y = x² - 3x at the point (7, 28) is 11y + x - 315 = 0

Step-by-step explanation:

First, we need to find the slope of the line tangent to the curve at the point (7, 28).

To do this, we need to differentiate y with respect to x and evaluate at the point x0 = 7.

Given y = x² - 3x

dy/dx = 2x - 3

So, the slope, m of the tangent line is dy/dx at x = 7:

m = 2(7) - 3 = 11

Slope, m = 11

The slope, m2 of the line perpendicular to the tangent is given as m2 = -1/m

m2 = −1/11

Finally, given the slope m of a line, and a point (x0, y0) on the line, we can use the point-slope form of the equation of a line:

y - y0=m(x - x0)

The perpendicular line has slope m2 = -1/11, and (7, 28) is a point on that line, the desired equation is:

y - 28 = (-1/11)(x - 7)

Or multiplying by 11, we have

11y - 308 = -x + 7

11y + x - 315 = 0

The slope of the curve at point (7,28) is 11, found by differentiating the given function. The slope of the normal line is the negative reciprocal of that, -1/11. Inserting these into the line equation, we get y - 28 = -1/11 * (x - 7).

To find the equation of the line perpendicular to the tangent line, we first find the slope of the tangent line. The derivative of the given function y = x^2 - 3x tells us the slope of the curve at any point, so let's find the derivative:

y' = 2x - 3

Now, let's find the slope of the tangent line at the point (7,28) by plugging 7 into the derivative:

y'(7) = 2*7 - 3 = 11

The slope of the line perpendicular to this (the normal line) is the negative reciprocal of the tangent line's slope, so it's -1/11.  

The equation of the line with slope m that goes through the point (x1, y1) is:

y - y1 = m*(x - x1)

Substitute the slope of -1/11 and the point (7,28) into this equation to get the equation of the normal line:

y - 28 = -1/11 * (x - 7)

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Answer:

E) 13

Step-by-step explanation:

∫₀⁴ f'(t) dt = f(4) − f(0)

8 = f(4) − 5

f(4) = 13