Find the volume of the largest right cylinder that fits in a sphere of radius 4

If the height of a model of the Cape Hatteras Lighthouse given that the scale used is 8 inches to 30 feet. This means that for every 8 inches on the model, the actual lighthouse is 30 feet tall. Therefore, the height of the model is 4.67 feet.

The actual lighthouse is 210 feet tall. To find the height of the model, we need to use a proportion. We can set up the proportion as:

8 inches / 30 feet = x inches / 210 feet

We can cross-multiply to get:

8 inches * 210 feet = 30 feet * x inches

Simplifying, we get:

x inches = (8 inches * 210 feet) / 30 feet = 56 inches

However, the question asks for the height of the model in feet, so we need to convert 56 inches to feet by dividing by 12:

56 inches / 12 inches per foot = 4.67 feet

Therefore, the height of the model is 4.67 feet.

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To find the equation of the plane, we first need to find two vectors that lie on the plane. We can do this by taking the differences between the three given points:

$\vec{v_1} = \begin{pmatrix}6 \\ 0 \\ 3\end{pmatrix} - \begin{pmatrix}0 \\ 0 \\ 0\end{pmatrix} = \begin{pmatrix}6 \\ 0 \\ 3\end{pmatrix}$

$\vec{v_2} = \begin{pmatrix}-3 \\ -1 \\ 5\end{pmatrix} - \begin{pmatrix}0 \\ 0 \\ 0\end{pmatrix} = \begin{pmatrix}-3 \\ -1 \\ 5\end{pmatrix}$

Now we can find the normal vector to the plane by taking the cross product of these two vectors:

$\vec{n} = \vec{v_1} \times \vec{v_2} = \begin{pmatrix}6 \\ 0 \\ 3\end{pmatrix} \times \begin{pmatrix}-3 \\ -1 \\ 5\end{pmatrix} = \begin{pmatrix}3 \\ -27 \\ 6\end{pmatrix}$

Next, we can use the point-normal form of the equation of a plane:

$(\vec{r} - \vec{a}) \cdot \vec{n} = 0$

where $\vec{a}$ is a point on the plane and $\vec{n}$ is the normal vector.

We can choose any of the three given points as $\vec{a}$. Let's use $(0, 0, 0)$:

$(\begin{pmatrix}x \\ y \\ z\end{pmatrix} - \begin{pmatrix}0 \\ 0 \\ 0\end{pmatrix}) \cdot \begin{pmatrix}3 \\ -27 \\ 6\end{pmatrix} = 0$

Simplifying, we get:

$3x - 27y + 6z = 0$

This is the equation of the plane.
To find the equation of the plane passing through points (0, 0, 0), (6, 0, 3), and (-3, -1, 5), first find two vectors in the plane and then compute their cross product to obtain the normal vector of the plane. Finally, use the normal vector and a point on the plane to find the equation of the plane.

Vectors in the plane:
V1 = (6, 0, 3) - (0, 0, 0) = (6, 0, 3)
V2 = (-3, -1, 5) - (0, 0, 0) = (-3, -1, 5)

Cross product (normal vector N):
N = V1 x V2 = (0*(-5) - 3*(-1), 3*(-5) - 6*5, 6*(-1) - 0*(-3))
N = (3, -15, -6)

Equation of the plane using the normal vector and a point on the plane (0, 0, 0):
3x - 15y - 6z = 0

So, the equation of the plane is 3x - 15y - 6z = 0.

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It will take the population 10.5 years to reach 25,000

From the question, we have the following parameters that can be used in our computation:

Inital population, a = 15000

Rate of increase, r = 5%

Using the above as a guide, we have the following:

The function of the situation is

f(x) = a * (1 + r)ˣ

So, we have

f(x) = 15000 * (1 + 5%)ˣ

When it reaches 25000, we have

15000 * (1 + 5%)ˣ = 25000

Divide both sides by 15000

(1 + 5%)ˣ = 1.67

Take the logarithm of both sides

x = log(1.67)/log(1 + 5%)

Evaluate

x = 10.5

Hence, the number of years is 10.5

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Answer:

Step-by-step explanation:

Find area of all shapes and add

A(biggest rectangle) = Lw =  6*3 =18   there are 2 of those A=36

A(long skinny rectangle = Lw = 2*6 =12  there are 2 of those A=24

A(smallest) = Lw = 2*3=6 there are 2 of those A=12

Add it all up

A=36+24+12 = 72

I checked it 5 times.  I keep getting 72

The number of times she should expect to pull a yellow marble is 13

From the question, we have the following parameters that can be used in our computation:

Color of Marble Frequency

Yellow 26

Green 34

Purple 18

Red 22

This means that

Times she should expect to pull a yellow marble is

Yellow = P(Yellow) * 50

So, we have

Yellow = 26/(26 + 34 + 18 + 22) * 50

Evaluate

Yellow = 13

Hence, the number of times she should expect to pull a yellow marble is 13

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If x² + bx - 9 is factorable, then the possible values for b are -6 and 6.

To factor x² + bx - 9, we need to find two numbers that multiply to -9 and add up to b. We can write this as (x + a)(x + b) = x² + bx - 9. Since the coefficient of x² is 1, the two numbers that multiply to -9 must add up to the coefficient of x, which is b. We can find the factor pairs of -9 to see which ones add up to b:

-1 and 9 add up to 8

-3 and 3 add up to 0

-9 and 1 add up to -8

3 and -3 add up to 0

1 and -9 add up to -8

9 and -1 add up to 8

Only -6 and 6 add up to b, so the possible values for b are -6 and 6. Therefore, x² - 6x - 9 and x² + 6x - 9 are both factorable.

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To find the volume and surface area of the composite figure, we need to find the volume and surface area of the right cylinder and the hemisphere separately, and then add them together.

First, let's find the volume of the right cylinder.

The formula for the volume of a right cylinder is V = πr^2h, where r is the radius and h is the height. In this case, the radius is half of the diameter, which is 4/2 = 2 inches. So, the volume of the right cylinder is:

V1 = π(2)^2(12) = 96π cubic inches

Next, let's find the volume of the hemisphere. The formula for the volume of a hemisphere is V = (2/3)πr^3, where r is the radius. In this case, the radius is also 2 inches. So, the volume of the hemisphere is:

V2 = (2/3)π(2)^3 = 16/3π cubic inches

To find the total volume of the composite figure, we just need to add V1 and V2:

Vtotal = V1 + V2 = 96π + 16/3π = 320/3π cubic inches

Now, let's find the surface area of the composite figure.

To do this, we need to find the surface area of the curved surface of the cylinder, the top of the cylinder, and the curved surface of the hemisphere separately, and then add them together.

The formula for the surface area of the curved surface of a cylinder is A = 2πrh, where r is the radius and h is the height. In this case, the radius is still 2 inches, and the height is 12 inches. So, the surface area of the curved surface of the cylinder is:

A1 = 2π(2)(12) = 48π square inches

The formula for the surface area of the top of a cylinder is A = πr^2, where r is the radius. In this case, the radius is still 2 inches. So, the surface area of the top of the cylinder is:

A2 = π(2)^2 = 4π square inches

Finally, the formula for the surface area of the curved surface of a hemisphere is A = 2πr^2, where r is the radius. In this case, the radius is still 2 inches. So, the surface area of the curved surface of the hemisphere is:

A3 = 2π(2)^2 = 8π square inches

To find the total surface area of the composite figure, we just need to add A1, A2, and A3:

Atotal = A1 + A2 + A3 = 48π + 4π + 8π = 60π square inches

So, the volume of the composite figure is 320/3π cubic inches, and the surface area of the composite figure is 60π

square inches.

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The horizontal distance between (16, -24) and (-3, -24) is 19 units.

Explanation:

We can calculate the horizontal distance by finding the difference between the x-coordinates of the two points.

Horizontal distance = difference in x-coordinates = (-3) - 16 = -19

However, the distance is negative, which doesn't make sense as distance is always positive. So we take the absolute value of the difference to get the actual distance.

|Horizontal distance| = |-19| = 19

Therefore, the horizontal distance between the two points is 19 units.

The approximate probability that someone might answer the next call E.J. makes is 0.25 or 25%. The correct answer is C. 0.25.

To find the approximate probability that someone might answer E.J.'s next call, we'll use the information provided and follow these steps:

1. Calculate the total number of calls made: answered calls (40) + unanswered calls (120).
2. Find the proportion of answered calls to the total calls.
3. Express the proportion as a probability (as a percentage or fraction).

Let's do the calculations:

1. Total calls = 40 (answered) + 120 (unanswered) = 160 calls
2. Proportion of answered calls = 40 answered calls / 160 total calls = 1/4
3. Probability = 1/4 = 0.25 (as a decimal) or 25% (as a percentage)

So, the approximate probability that someone might answer the next call E.J. makes is 0.25 or 25%. The correct answer is C. 0.25.

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A population of 600 organisms is expected to increase by 50.

We can use the equation I = k * n to determine whether the anticipated growth (I) of an organism population is directly related to the size of the current population (n).

Where k is a proportionality constant that connects the anticipated growth to the current population.

We can use the details provided in the problem to determine the value of k. We can write: 20 = k * 240 to represent an increase of 20 in a sample of 240 organisms.

We can calculate k as follows: k = 20 / 240 = 1 / 12.

Now that we know the value of k, we can use it to calculate the predicted growth for a population of 600:

I = (1 / 12) * 600 = 50

Therefore, a population of 600 organisms is expected to increase by 50.

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Laptop- 44, Tablet- 94, Desktop- 62, Percentage of tablets- 47%.

To calculate the percentage of devices that are tablets, we need to find the proportion of tablets among all devices. In this case, there are 94 tablets out of a total of 200 devices (44 laptops + 94 tablets + 62 desktops).

To find the proportion, we divide the number of tablets by the total number of devices: 94/200 = 0.47.

To convert this proportion to a percentage, we multiply by 100: 0.47 * 100 = 47%.

47% of the devices owned by the local company are tablets.

This means that nearly half of the devices fall into the tablet category, making it a significant portion of their device inventory.

It's important to note that this calculation assumes the given numbers accurately represent the actual device distribution and that there are no missing or unaccounted devices.

By determining the percentage of tablets, we gain insight into the composition of the device inventory, which can be useful for decision-making and resource allocation within the local company.

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We have evidence to suggest that the true percentage of college degrees in mathematics is different from 1%.

The null hypothesis is a type of hypothesis that explains the population parameter and is used to examine if the provided experimental data are reliable.

To test the claim that the percentage of college degrees in mathematics is not 1%, we can use a hypothesis test. Let's assume the null hypothesis is that the true percentage of college degrees in mathematics is 1%, and the alternative hypothesis is that it is different from 1%.

- Null hypothesis: The percentage of college degrees in mathematics is 1%.

- Alternative hypothesis: The percentage of college degrees in mathematics is different from 1%.

We can use a binomial distribution to model the number of graduates with math degrees in a sample of 12,317. Under the null hypothesis, the expected number of graduates with math degrees is:

Expected value = sample size * probability of math degrees = 12,317 * 0.01 = 123.17

Since we are testing at a 0.10 level of significance, the critical values for a two-tailed test are ±1.645 (using a standard normal distribution table).

The test statistic can be calculated as:

z = (observed value - expected value) / standard deviation

The standard deviation of the binomial distribution can be calculated as:

√(sample size * probability of success * (1 - probability of success))

So,

standard deviation = √(123.17 * 0.01 * 0.99) = 1.109

The observed value is 148.

The test statistic is:

z = (148 - 123.17) / 1.109 = 22.38

Since the absolute value of the test statistic is greater than 1.645, we can reject the null hypothesis at the 0.10 level of significance.

Therefore, we have evidence to suggest that the true percentage of college degrees in mathematics is different from 1%.

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The velocity with which a ball reaches the ground when it is dropped

from a height of 64 m is 35.44m/sec

From the information given, we have that the equation representing the velocity of the ball is expressed as;

V = 4.43√h

Given that the parameters of the formula are;

Since the height of the ball from the ground is 64m, we have to substitute the value, we have;

V = 4.43√64

Find the square root of the value

V= 4.43(8)

Now, multiply both the values to determine the velocity, we get;

V = 35.44m/sec

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The correct answers are:True, True, False.

The first statement is true. The initial investment is represented in the formula as the coefficient of the growth factor[tex](1+0.05)^t[/tex].

The second statement is true. The growth factor in the formula is (1+0.05), which represents a 5% increase or a multiplier of 1.05.

The third statement is false. The annual interest rate is not 50%. The growth factor (1+0.05) represents a 5% increase, but the actual annual interest rate is 5%, not 50%.

Therefore, the correct answers are:

True

True

False

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Answer:

x^9√x

Step-by-step explanation:

Numerator: e^(1 + ln ∞) * (1/∞) = e^(1 + ∞) * 0 = 0, Denominator: 1 + e^∞ = ∞. So, the limit of the expression is: lim (x→∞) [e^(1 + ln x) / ln (1 + e^x)] = 0/∞ = 0

To find the limit of the given function, let's first rewrite the terms using the provided limit notation:

lim (x→∞) [e^(1 + ln x) / ln (1 + e^x)]

To solve this limit, we will apply L'Hôpital's rule, which states that if the limit has the form 0/0 or ∞/∞, we can find the limit by taking the derivative of the numerator and denominator with respect to x:

Numerator: d(e^(1 + ln x))/dx = e^(1 + ln x) * d(1 + ln x)/dx = e^(1 + ln x) * (1/x)
Denominator: d(ln(1 + e^x))/dx = (1/(1 + e^x)) * d(e^x)/dx = (1/(1 + e^x)) * e^x

Now, we will find the limit of the new expression:

lim (x→∞) [(e^(1 + ln x) * (1/x)) / ((1/(1 + e^x)) * e^x)]

Simplify the expression by canceling out the e^x terms:

lim (x→∞) [(e^(1 + ln x) * (1/x)) / (1 + e^x)]

Now, let's substitute x→∞:

Numerator: e^(1 + ln ∞) * (1/∞) = e^(1 + ∞) * 0 = 0
Denominator: 1 + e^∞ = ∞

So, the limit of the expression is:

lim (x→∞) [e^(1 + ln x) / ln (1 + e^x)] = 0/∞ = 0

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According to the given information in the question to find the length of side v in triangle VWX, we can use the Law of Sines. the length of side v is approximately [tex]281.8[/tex] cm.

image for "define centimeters in high-level mathematics." A centimeter is a metric unit used to quantify small distances and the object's length. Cm is used to represent it in writing.

It can also be described as the measure of length in the current metric system, referred to as the International System of Units (SI). It is equal to one-hundredth of a meter.

which states that:

[tex]a/sin(A) = b/sin(B) = c/sin(C)[/tex]

where a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the opposite angles.

In this case, we know the length of side w (600 cm), and the measures of angles V and W.

To find the length of side v, we can use the Law of Sines with sides v and w and angle V:

[tex]v/sin(V) = w/sin(W)[/tex]

[tex]v/sin(26^\circ) = 600/sin(80^\circ)[/tex]

[tex]v = (600 \times sin(26^\circ))/sin(80^\circ)[/tex]

[tex]v \approx 281.8 cm[/tex] (rounded to the nearest 10th of a centimeter)

Therefore, the length of side v is approximately [tex]281.8 cm[/tex].

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If he put 40 red balls, 55 purple balls, 45 yellow balls, and 60 green balls into the ball pit. while his sister is playing, one ball rolls out of the pit. Therefore, the probability that the ball that rolled out of the pit is red is 0.2.

The probability of selecting a red ball from the ball pit can be found by dividing the number of red balls by the total number of balls in the pit.

Total number of balls = 40 + 55 + 45 + 60 = 200

Probability of selecting a red ball = Number of red balls / Total number of balls

Probability of selecting a red ball = 40 / 200

Probability of selecting a red ball = 0.2

Therefore, the probability that the ball that rolled out of the pit is red is 0.2.

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The measure of angle PQR is 90 degrees.

Since PR is the diameter of the circle, it follows that angle POR is a right angle, i.e., it measures 90 degrees.

By the inscribed angle theorem, the measure of angle PQR is half the measure of angle POR. Thus,

angle PQR = 1/2 * angle POR

= 1/2 * 90

= 45 degrees.

However, this is not the final answer since angle PQR is not a stand-alone angle, but rather a part of a right-angled triangle PQR.

Since the three angles in a triangle add up to 180 degrees, and we already know that angle PQR is 45 degrees, it follows that:

angle PRQ + angle PQR + angle QPR = 180 degrees

Since angle PQR = 45 degrees, we have:

angle PRQ + 45 + angle QPR = 180 degrees

Rearranging, we get:

angle PRQ + angle QPR = 135 degrees

Since angles PRQ and QPR are complementary angles (together they form a right angle), their sum is 90 degrees. Therefore,

angle PRQ + angle QPR = 90 degrees

Substituting this into the previous equation, we get:

90 degrees = 135 degrees

This is a contradiction, and hence our assumption that angle PQR measures 45 degrees is false.

Therefore, we conclude that angle PQR must measure 90 degrees, since it is the only angle that can satisfy the given conditions.

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To find the length of border needed to put around a rectangular room, you can follow these steps:

Measure the length and width of the room using a tape measure or ruler. Let's say the length is L and the width is W.

Determine the type and width of the border you want to put around the room. Let's say you want to use a wallpaper border that is 6 inches wide.

Calculate the length of the border needed for the top and bottom of the room.

This can be done by adding the width of the room to twice the width of the border. So the length of the top and bottom borders would be:

Length of top/bottom borders = 2(W + 6 inches)

Calculate the length of the border needed for the left and right sides of the room. This can be done by adding the length of the room to twice the width of the border. So the length of the left and right borders would be:

Length of left/right borders = 2(L + 6 inches)

Add up the length of all four borders to get the total length of border needed to put around the room:

Total length of border = 2(W + 6 inches) + 2(L + 6 inches)

Simplifying the above equation, we get:

Total length of border = 2L + 2W + 24 inches

So the total length of border needed to put around the room is 2L + 2W + 24 inches.

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In the scenario you mentioned, the part of the Federal Reserve's congressional mandate that is triggered is both price stability and maximum sustainable employment.

The Federal Reserve has a dual mandate from Congress which consists of:

1. Price stability: This objective aims to maintain a stable inflation rate, ensuring that the purchasing power of money remains relatively constant over time. Price stability helps households and businesses make informed decisions regarding investments and consumption.

2. Maximum sustainable employment: This objective focuses on maintaining a high level of employment while avoiding unsustainable economic growth that could lead to high inflation. Achieving maximum sustainable employment involves minimizing unemployment while not causing inflation to rise significantly.

In the scenario you mentioned, the Federal Reserve would need to carefully balance its monetary policy to ensure that both objectives, price stability and maximum sustainable employment, are achieved. This might involve adjusting interest rates or using other monetary policy tools to influence economic activity, employment levels, and inflation.

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About 10,676 gallons of paint were used to paint the 157 newly built homes in the subdivision, assuming each house required exactly 68 gallons of paint and 13 paint brushes.

The statement problem states that there are 157 newly built homes in a subdivision, and that 68 gallons of paint and 13 paint brushes were used for each house. This means that each house required 68 gallons of paint and 13 paint brushes.

To find the total amount of paint used for all 157 houses, we need to multiply the amount of paint used per house (68 gallons) by the number of houses (157):

Total amount of paint = 68 gallons/house x 157 houses

Total amount of paint = 10,676 gallons

Therefore, approximately 10,676 gallons of paint were used for the 157 newly built homes in the subdivision.

It's worth noting that this calculation assumes that each house required exactly 68 gallons of paint and 13 paint brushes. In reality, there may be some variation in the amount of paint and brushes used for each house, so the actual total may be slightly different.

However, this calculation provides a reasonable estimate of the total amount of paint used

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Answer: The given polynomial is -2x³.

a) To write the polynomial in standard form, we arrange the terms in descending order of degrees:

-2x³

b) The degree of a polynomial is the highest exponent of the variable in the polynomial. In this case, the degree of the polynomial is 3.

c) The lead coefficient is the coefficient of the term with the highest degree. In this case, the lead coefficient is -2.

Step-by-step explanation:

The analogy that best represents the bold letters for each question is related to relationships and cross-sections.

Some analogies are given below:

Explanation: The relationship between happy and sad is that they are opposite emotions.

Similarly, up and down are opposite directions, so the analogy holds.

Explanation: The relationship between a cat and a kitten is that a kitten is a young cat.

Similarly, a puppy is a young dog, so the analogy holds.

Explanation: The relationship between circle and sphere is that a circle is a two-dimensional shape, while a sphere is a three-dimensional shape that has a circular cross-section.

Similarly, a square is a two-dimensional shape, while a cube is a three-dimensional shape that has a square cross-section, so the analogy holds.

Explanation: The relationship between the oven and baking is that an oven is a tool used to bake food.

Similarly, a grill is a tool used to grill food, so the analogy holds.

Explanation: The relationship between mother and father is that they are the parents of a child.

Similarly, a daughter and a son are the children of their parents, so the analogy holds.

Explanation: The relationship between a book and a chapter is that a book is divided into chapters.

Similarly, a play is divided into acts, so the analogy holds.

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The size  of square tablecloth needed to cover the entire square tabletop is 25 ft^2 (Option B).

To determine the size of the tablecloth needed to cover a square tabletop with a perimeter of 20 feet, we will first find the side length of the square and then calculate the area.

Step 1: Determine the side length of the square.
Since the perimeter of a square is equal to 4 times its side length (P = 4s), we can find the side length by dividing the perimeter by 4.
Side length (s) = Perimeter (P) / 4 = 20 ft / 4 = 5 ft

Step 2: Calculate the area of the square.
The area of a square is equal to the side length squared (A = s^2).
Area (A) = Side length (s) ^ 2 = 5 ft ^ 2 = 25 ft^2

So, the size of the tablecloth needed to cover the entire square tabletop is 25 ft^2 (Option B).

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Answer: The probability of getting heads on a fair coin is 1/2, and the probability of rolling an even number on an eight-sided fair die is 4/8 = 1/2. Therefore, the probability of getting both heads and an even number is:

P(heads and even number) = P(heads) x P(even number) = (1/2) x (1/2) = 1/4

The probability of getting tails on a fair coin is also 1/2, and the prime numbers less than 4 are 2 and 3. The probability of rolling a 2 or a 3 on an eight-sided fair die is 2/8 = 1/4. Therefore, the probability of getting tails and a prime number less than 4 is:

P(tails and prime number less than 4) = P(tails) x P(prime number less than 4) = (1/2) x (1/4) = 1/8

So the probability that Oscar gets heads and an even number is 1/4, and the probability that Oscar gets tails and a prime number less than 4 is 1/8.

Step-by-step explanation: can i get brianliest :D

Answer:

5.6 miles

Step-by-step explanation:

the patteren in 0.3    0.6     0.3       0.6

Let's say we add 'x' red candies and 'y' blue candies to the bag to create the desired ratio of 3 to 2:

Then, the total number of red candies in the bag will be 5 + x, and the total number of blue candies will be 1 + y.

According to the problem, the ratio of red candies to blue candies should be 3 to 2:

(5 + x) / (1 + y) = 3/2

Cross-multiplying this equation, we get:

2(5 + x) = 3(1 + y)

Simplifying this equation, we get:

10 + 2x = 3 + 3y

2x - 3y = -7

We want to find the least number of red and blue candies that can be added to the bag to satisfy this equation.

One way to do this is to try different values of x and y that satisfy the equation until we find the smallest possible values that work.

For example, we can start by setting x = 1 and y = 2:

2(5 + 1) = 3(1 + 2)

12 = 9

This doesn't work, so let's try another set of values, x = 4 and y = 5:

2(5 + 4) = 3(1 + 5)

18 = 18

This set of values works, so we have found the least number of red and blue candies that can be added to the bag to create a ratio of 3 to 2 for the number of red candies to the number of blue candies:

We need to add 4 red candies and 5 blue candies to the bag to create a ratio of 3 to 2 for the number of red candies to the number of blue candies.

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The E is enclosed by the planes z = 0 and z = x + y + 5

Why will E is enclosed by the planes z = 0?

The evaluate the triple integral ∫∫∫_E eˣ dV, where E is enclosed by the planes z = 0 and z = x + y + 5, and by the cylinders x² + y² = 4 and x² + y² = 9,

Follow these steps:

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