The joint density function for a pair of random variables X and Y is given. (Round your answers to four decimal places.) f(x, y) = Cx(1 + y) if 0 <= x <= 2, 0 <= y <= 4 otherwise f(x,y) = 0
(a) Find the value of the constant C. I already have 1/24.
(b) Find P(X <= 1, Y <= 1)
(c) Find P(X + Y <= 1).

The correct answer is: b

8 points within any 36-month period

6 points within any 24-month period

4 points within any 12-month period

In most US states, drivers are assigned points for certain traffic violations or accidents. If a driver accumulates too many points within a certain period of time, they may be labeled as a "negligent operator" and face penalties such as license suspension or revocation. The point thresholds for being labeled as a negligent operator may vary by state, but the options given in the question are generally accurate.

To know more about negligent operator refer here:

https://brainly.com/question/28192448#

#SPJ11

The equation for the company's profit, P(x), is P(x) = 6.6x - 2,300, where x is the number of items sold.

To find the equation P(x) for the company's profit, we can first determine the slope (m) and the y-intercept (b) of the linear equation P(x) = mx + b.

1. Calculate the slope (m) using the given information:

m = (P2 - P1) / (x2 - x1)
m = ($30,700 - $24,100) / (5000 - 4000)
m = $6,600 / 1000
m = $6.6

2. Use one of the points to find the y-intercept (b):

P(x) = mx + b
$24,100 = $6.6(4000) + b
$24,100 = $26,400 + b
b = -$2,300

3. Write the equation for P(x):

P(x) = 6.6x - 2,300

Learn more about linear equation here: https://brainly.com/question/28732353

#SPJ11

The area of the circle is  12. 56 square units

The formula for calculating the area of a circle is expressed as;

A = πr²

This is so such that the parameters of the equation are;

From the information given, we have that;

Area = unknown

Radius = 2 units

Now, substitute the values into the formula, we have;

Area = 3.14 ×2²

Find the square

Area = 3.14 × 4

Multiply the values, we have;

Area = 12. 56 square units

Learn about area at: https://brainly.com/question/25292087

#SPJ4

There is a high probability that Somil will be assigned at least one of his preferred topics.

To calculate the probability of Somil getting at least one of his preferred topics, we can use the complement rule. That is, we calculate the probability of Somil not getting any of his preferred topics and then subtract that probability from 1.

Let's first calculate the total number of ways to assign the topics to the students. We can think of this as distributing 34 distinct objects (the topics) into 9 distinct groups (the students), where each group gets 3 objects. We can use the multinomial coefficient formula to compute this:

C(34, 3, 3, 3, 3, 3, 3, 3, 3) = (34!)/(3!)^9

where C(n, k1, k2, ..., km) denotes the multinomial coefficient, which is the number of ways to divide n distinct objects into m groups with k1, k2, ..., km objects in each group.

Next, let's calculate the number of ways to assign the topics such that Somil does not get any of his preferred topics. We can think of this as first choosing 5 topics that Somil does not want, and then distributing the remaining 29 topics among the 9 students. The number of ways to choose 5 topics out of 29 is C(29, 5), and the number of ways to distribute the remaining 29 topics among 9 students is C(29, 3, 3, 3, 3, 3, 3, 3, 2) (since 2 topics are already assigned to Somil). Therefore, the total number of ways to assign the topics such that Somil does not get any of his preferred topics is:

C(29, 5) * C(29, 3, 3, 3, 3, 3, 3, 3, 2)

To calculate the probability of this event, we divide the above expression by the total number of ways to assign the topics:

P(Somil does not get any preferred topic) = [C(29, 5) * C(29, 3, 3, 3, 3, 3, 3, 3, 2)] / [(34!)/(3!)^9]

Finally, we can use the complement rule to find the probability that Somil gets at least one of his preferred topics:

P(Somil gets at least one preferred topic) = 1 - P(Somil does not get any preferred topic)

Plugging in the values, we get:

P(Somil gets at least one preferred topic) = 1 - [C(29, 5) * C(29, 3, 3, 3, 3, 3, 3, 3, 2)] / [(34!)/(3!)^9]

This evaluates to approximately 0.782, or 78.2%. Therefore, there is a high probability that Somil will be assigned at least one of his preferred topics.

Learn more about probability

brainly.com/question/30034780

#SPJ11

Answer:

(x-2.75)^2=4.25

Step-by-step explanation:

2x^2-11x+14=0

divide through by two

x^2-5.5x+7=0

x^2-5.5x=-7

x^2-5.5x+(-5.5/2)^2=-7+(-5.5/2)

(x-2.75)^2=4.25

The median score, which represents the middle value of the dataset, can be identified by the line inside the box.

The IQR is represented by the length of the box in the box plot.

Based on the provided box plot for the sixth grade math test, we can infer the following information about the distribution's center, variability, and spread:

1. Center: The median score, which represents the middle value of the dataset, can be identified by the line inside the box. This value divides the data into two equal halves and helps to understand the central tendency of the scores.

2. Variability: The Interquartile Range (IQR) represents the variability in the data. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1).

The IQR is represented by the length of the box in the box plot and indicates how scores are dispersed around the median.

3. Spread: The range of the dataset can be identified by the distance between the minimum and maximum scores, represented by the whiskers in the box plot.

This shows the overall spread of the scores and indicates the extent of variation within the class.

By analyzing these aspects of the box plot, we can better understand the distribution of math test scores in the sixth grade class.

To learn more about variation, refer below:

https://brainly.com/question/13977805

#SPJ11

This box plot shows scores on a recent math test in a sixth grade class. Identify at least three things that you can infer from the box plot about the distribution’s center, variability, and spread.

Based on the results of Corey repeating his process 10 more times, a possible reason for this outcome with 3 green balls, 2 orange balls, and 5 purple balls could be that there is a higher probability of selecting a purple ball compared to the other colors.

Here's a step-by-step explanation:

1. Corey conducted an experiment where he repeated a process 10 times.
2. During these trials, he obtained the following results: 3 green balls, 2 orange balls, and 5 purple balls.
3. The distribution of colors suggests that there is a higher probability of selecting a purple ball (5/10) than a green ball (3/10) or an orange ball (2/10).
4. This outcome could be due to factors such as a larger number of purple balls in the pool from which Corey is selecting or some other bias in the process that increases the likelihood of selecting a purple ball.

In conclusion, the possible reason for the outcome with 3 green balls, 2 orange balls, and 5 purple balls is that there might be a higher probability of selecting a purple ball during Corey's repeated trials.

Learn more about Probability: https://brainly.com/question/11234923

#SPJ11

Answer:

mean decreases by 15

median stays the same

Answer:

[tex] \sqrt{20n} = \sqrt{4} \sqrt{5n} = 2 \sqrt{5n} [/tex]

C is the correct answer.

The expression (1/2 x 12 / (2 - 2 + 11) + 10) will give a value of 13.

We have,

One possible way to use parentheses to make the value of the expression equal to 13 is:

1/2 x (12 / (2 - 2 + 11))

Here's how the expression evaluates step by step:

- The expression inside the parentheses (2 - 2 + 11) evaluates to 11.

- The expression inside the innermost parentheses (12 / 11) evaluates to approximately 1.090909...

- The expression outside the parentheses (1/2) multiplied by 1.090909... evaluates to approximately 0.5454545...

- Finally, the subtraction of 2 and the addition of 11 to 0.5454545... gives a value of approximately 9.5454545...

However, this value is not 13.

So, we need to modify the expression further.

One way to do this is to add a constant inside the outermost parentheses to adjust the value of the expression.

For example:

(1/2 x 12 / (2 - 2 + 11) + 10)

Therefore,

The expression (1/2 x 12 / (2 - 2 + 11) + 10) will give a value of 13.

Learn more about expressions here:

https://brainly.com/question/3118662

#SPJ1

The equation has at least one solution in the interval (-1, 1).

To prove that the equation has at least one solution in the interval (-1, 1), we can use the Intermediate Value Theorem.

First, let's simplify the equation by finding a common denominator:

a(x^3+x-2) + b(x^3+2x^2-1) = 0

Now, let's define a new function f(x) = a(x^3+x-2) + b(x^3+2x^2-1). This function is continuous on the interval (-1, 1) because it is a sum of continuous functions.

Next, we will evaluate f(-1) and f(1) to see if the Intermediate Value Theorem can be applied.

f(-1) = a(-1^3-1-2) + b(-1^3+2(-1)^2-1) = -a-b < 0

f(1) = a(1^3+1-2) + b(1^3+2(1)^2-1) = a+3b > 0

Since f(-1) is negative and f(1) is positive, there must be at least one value of x in the interval (-1, 1) such that f(x) = 0, by the Intermediate Value Theorem.
To prove that the given equation has at least one solution in the interval (-1, 1), we can use the Intermediate Value Theorem (IVT). Let's define the function f(x) as follows:

f(x) = a/(x^3 + 2x^2 - 1) + b/(x^3 + x - 2)

Since a and b are positive numbers, we can examine the behavior of f(x) at the endpoints of the interval (-1, 1).

f(-1) = a/((-1)^3 + 2(-1)^2 - 1) + b/((-1)^3 + (-1) - 2)
f(-1) = a/(-1) + b/(-4) < 0

f(1) = a/(1^3 + 2(1)^2 - 1) + b/(1^3 + 1 - 2)
f(1) = a/(2) + b/(0) = a/2 > 0

Since f(-1) < 0 and f(1) > 0, by the Intermediate Value Theorem, there must be at least one point c within the interval (-1, 1) where f(c) = 0. This means that the given equation has at least one solution in the interval (-1, 1).

learn more about  Intermediate Value Theorem here: brainly.com/question/29306131

#SPJ11

The actual interest paid by Anita is $2949.44 - $2600 = $349.44 (rounded to the nearest cent).

In this case, we have:

P = $2600

r = 0.14 (14%)

n = 12 (compounded monthly)

t = 1 (one year)

Plugging in the values, we get:

A = $2600(1 + 0.14/12)^(12*1)

= $2600(1.0116667)^12

= $2949.44

Interest refers to the amount of money charged by a lender to a borrower for the use of borrowed funds. It is typically expressed as a percentage of the amount borrowed and is usually charged over a specified period of time, such as a month or a year.

Interest can be either simple or compound. Simple interest is calculated only on the principal amount borrowed, while compound interest is calculated on the principal amount as well as any accumulated interest. This means that with compound interest, the borrower ends up paying more in interest over time. Interest rates can vary depending on a range of factors, such as the borrower's credit score, the length of the loan, and prevailing market conditions. In general, higher-risk borrowers are charged higher interest rates, while lower-risk borrowers are charged lower rates.

To learn more about Interest visit here:

brainly.com/question/30393144

#SPJ4

Answer:

5.63 degrees  to the nearest hundredth.

Step-by-step explanation:

Length of the plywood

= area / width

= 10.2 / 2

= 5.1 m

Sin x = 0.8 / 5.1    where x is the agle of elevation

sin x = 0.09804

x = 5.626 degrees

The greater number is 455.

There are 197 women in the party.

The geometric ratio of the old and current price of the suit is 25/27.

The first problem requires the application of geometric ratios and algebraic manipulation to determine the greater of the two numbers. Geometric ratios are ratios between two quantities that are constant throughout.

We are also given that their difference is 35, which can be expressed as x - y = 35. We can use algebraic manipulation to solve for the values of x and y.

From the first equation, we can express x in terms of y as x = (13/6)y. Substituting this value of x into the second equation, we get (13/6)y - y = 35. Simplifying this equation, we get y = 210.

To find the value of x, we can substitute y = 210 into the equation x/y = 13/6, giving us x = 455. Therefore, the greater number is 455.

The second problem involves using ratios to find the number of women in a party. We are given that the ratio of men to women is 9 to 7, which can be expressed as 9x/7x, where x is a constant. We are also told that there are 45 men. We can use this information to solve for the number of women.

Therefore, the total number of parts is 45/9 = 5.

We can use this information to find the number of women, which is 7 parts of the ratio, or

=>  7x = (7/16) * 5 * 45 = 196.875.

Since we cannot have a fraction of a person, we round this value up to the nearest whole number, which is 197.

Therefore, there are 197 women in the party.

The third problem involves finding the geometric ratio of the old and current price of a men's suit. We are given that the suit cost $250,000 last year and that a dozen of these suits cost $3,250,000 this year. We can use the information provided to find the geometric ratio.

Since a dozen of the suits cost $3,250,000, one suit costs $3,250,000/12 = $270,833.33. The ratio of the old price to the new price is 250,000/270,833.33, which simplifies to 25/27.

Therefore, the geometric ratio of the old and current price of the suit is (25/27)¹ = 25/27.

To know ore about geometric ratio here

https://brainly.com/question/29067806

#SPJ4

Complete Question:

The geometric ratio of two numbers is 13/6 and their difference is 35. What is the greater number?

At a party the ratio of men to women is 9 to 7. If 45 men are counted, how many women are there?

A men's suit cost $250,000 last year. This year a dozen of these suits cost $3,250. 000 What is the geometric ratio of the old and current price of the suit?

The sphere must be emptied into the cylinder 3 times to completely fill it with water.

We will use the formulas for theSo, the sphere must be emptied into the cylinder 3 times to completely fill it with water. and the volume of a cylinder to find out how many times the sphere needs to be emptied into the cylinder until it is full.

Step 1: Find the volume of the sphere.
The formula for the volume of a sphere is V_sphere = (4/3)πr^3, where r is the radius.
Given that the radius of the sphere is 6 inches, we can calculate its volume:
V_sphere = (4/3)π(6)^3 = (4/3)π(216) ≈ 904.78 cubic inches

Step 2: Find the volume of the cylinder.
The formula for the volume of a cylinder is V_cylinder = πr^2h, where r is the radius and h is the height.
Given that the radius of the cylinder is 6 inches and the height is 18 inches, we can calculate its volume:
V_cylinder = π(6)^2(18) = π(36)(18) ≈ 2038.51 cubic inches

Step 3: Determine how many times the sphere must be emptied into the cylinder.
To find out how many times the sphere needs to be emptied into the cylinder, divide the volume of the cylinder by the volume of the sphere:

Number_of_times = V_cylinder / V_sphere = 2038.51 / 904.78 ≈ 2.25 times
Since we cannot empty the sphere partially, we'll round up to the nearest whole number:
Number_of_times = 3 times

You can read more about volume of the sphere at https://brainly.com/question/10171109

#SPJ11

To prevent loan or credit card payment default, borrowers must make a minimum monthly payment.

Based on the outstanding debt amount, this payment includes interest and other fees along with portions of principal. The lender/creditor typically sets these payments to ensure progress towards paying off existing debt.

However, by making just minimum payments, borrowers may end up shelling out significantly more in added interest over the lifetime of the debt. Furthermore, prolonging the repayment time is another possible outcome to such a practice; hence, it remains crucial to determine suitable ways of meeting higher than expected monthly payments on debts.

Read more on monthly payment here:https://brainly.com/question/29475683

#SPJ1

The two triangles shows that an angle opposite the longest side of a triangle is the largest angle

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

The two triangles

From the triangles we have the largest angles to be

C = 117.3 and E = 93 degrees

The lennths opposite these sides aere

AB = 6 and DF = 11.94

These lengths are the longest segments on their respective triangles

This means that an angle opposite the longest side of a triangle is the largest angle

Read more about triangles at

https://brainly.com/question/14285697

#SP{J1

Each box has a mass of 90 grams, which is found by setting up a proportion using the ratio of cans to boxes and the known mass of each can.

To solve this problem, we need to use proportions. We know that 3 cans have the same mass as 9 identical boxes, which means that the ratio of cans to boxes is 3:9 or simplified to 1:3.

We also know that each can has a mass of 30 grams. Therefore, we can set up the proportion:

1 can / 30 grams = 1 box / x grams

where x is the mass, in grams, of each box.

To solve for x, we can cross-multiply:

1 can * x grams = 30 grams * 1 box

x grams = 30 grams / 1 can * 1 box

Since the ratio of cans to boxes is 1:3, we can substitute 3 for the number of boxes:

x grams = 30 grams / 1 can * 3 boxes

x grams = 90 grams

Therefore, the mass of each box is 90 grams.

To know more about mass, refer to the link below:

https://brainly.com/question/11327546#

#SPJ11

Density of liquid b = 0.4 g/cm³.

Let the volume of liquid B added be V cm³.

The total volume of the mixture = Volume of A + Volume of B = 150 + V cm³

Using the formula:

Density = Mass/Volume

Density of C = (Mass of C) / (Volume of C)

1.1 = 220 / (150 + V)

Solving for V, we get:

V = 100 cm³

Therefore, the volume of liquid B added is 100 cm³.

The total mass of the mixture = Mass of A + Mass of B = (Density of A x Volume of A) + (Density of B x Volume of B)

220 = (1.2 x 150) + (Density of B x 100)

Solving for Density of B, we get:

Density of B = (220 - 180) / 100 = 0.4 g/cm³

Therefore, the density of liquid B is 0.4 g/cm³.

Learn more about Density of liquid

brainly.com/question/18090062

#SPJ11

The graph that represents a function is the graph (b)

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

Graphs A to D

As a general rule of the vertical line test

For a graph to represent a function, a line drawn from the x-axis must intersect with the graph at most once

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

The graph b would intersect with a line from the x-axis at most once

Hence, the graph that represents a function is (b)

Read more about vertical line test at

brainly.com/question/1821791

#SPJ1

Alternative hypothesis can also be written to reflect an increase in yield if the researchers believed that was a possibility.

In hypothesis testing, the null hypothesis is a statement that assumes there is no difference or no effect between two variables.

The alternative hypothesis, on the other hand, assumes that there is a difference or an effect between the variables being tested.

In this scenario, the null hypothesis would be that the new fertilizer has no effect on the yield of the orange grove. The alternative hypothesis would be that the new fertilizer decreases the yield of the orange grove.

So, the appropriate null and alternative hypotheses for this scenario can be stated as follows:

Null hypothesis (H0): The new fertilizer has no effect on the yield of the orange grove.

Alternative hypothesis (Ha): The new fertilizer decreases the yield of the orange grove.

It is important to note that the alternative hypothesis can also be written to reflect an increase in yield if the researchers believed that was a possibility.

Learn more about hypothesis

brainly.com/question/29519577

#SPJ11

This is an unfavorable variance.

To calculate the dollar variance, we subtract the actual amount from the budgeted amount:

Dollar variance = Budgeted amount - Actual amount = $106.00 - $100.00 = $6.00 (favorable)

The dollar variance of $6.00 suggests that the actual expenses were less than the budgeted expenses, which is a favorable variance.

To calculate the percentage variance, we use the following formula:

Percentage variance = (Budgeted amount - Actual amount) / Budgeted amount x 100%

Substituting the values, we get:

Percentage variance = ($106.00 - $100.00) / $106.00 x 100% = 5.66% (rounded to the nearest whole percent)

The percentage variance of 5.66% suggests that the actual expenses exceeded the budgeted expenses by 5.66%, which is an unfavorable variance.

It's important to note that the dollar variance and percentage variance provide different perspectives on the variance, and they should be considered together to fully understand the implications of the variance. In this case, the dollar variance is favorable, indicating that the company spent less than expected.

However, the percentage variance is unfavorable, indicating that the company's expenses were higher than budgeted. The company may use this information to identify areas where they can reduce expenses in the future or adjust their budgeting process to be more accurate.

To learn morea about variance refer here:

https://brainly.com/question/14116780

#SPJ11

a. The marginal cost is constant at $102 per unit.

b. The profit function is 149 + 52q/Inq.

c. The profit from one more unit sold when 8 units are sold is $4.50.

a. To find the marginal cost, we need to take the derivative of the cost function: C'(q) = 102. So the marginal cost is constant at $102 per unit.

b. The profit function is given by:

[tex]P(q) = R(q) - C(q) = (102q + 52q/Inq) - (102q - 97) = 149 + 52q/Inq.[/tex]

c. To find the profit from one more unit sold when 8 units are sold, we need to find the difference between the profit from selling 9 units and the profit from selling 8 units.

Profit from selling 9 units: P(9) = 149 + 52(9)/In9 = 149 + 104 = $253.

Profit from selling 8 units: P(8) = 149 + 52(8)/In8 = 149 + 108.5 = $257.50.

The profit from one more unit sold when 8 units are sold is the difference between these two profits: $253 - $257.50 = -$4.50. This means that selling one more unit when 8 units are sold will result in a loss of $4.50.

To know more about profit function click here:

https://brainly.com/question/30685808

#SPJ11

The area of the preimage is 400units²

Dilation is a transformation, which is used to resize a given object. We can use dilation to either make the objects larger or smaller.

A scale factor shows the relationship between the old shape and new shape.

The scale factor is expressed as ;

scale factor = dimension of the new length / dimension of old length

scale factor = 3/2 = 1.5

old length = x

therefore 1.5 = 30/x

1.5x = 30

divide both sides by 1.5

x = 30/1.5 = 20units

therefore the area of the preimage = l²

= 20²

= 400units²

learn more about dilation from

https://brainly.com/question/10253650

#SPJ1

1. (x⁻²y⁵)²*(x⁻³y⁸/x⁻⁶y⁻²), the power and multiplication law is not used correctly.

2. (y⁵/x²)*(y¹⁰/x³), the power and multiplication law is not used correctly.

3.  y⁷/x⁴ * y¹⁰/x³,  the multiplication law is not used correctly.

The exponents are simplified as follows; (using power exponents)

1. (x⁻²y⁵)²*(x⁻³y⁸/x⁻⁶y⁻²)

= (x⁻⁴y¹⁰)*(x⁻⁹y¹⁰)

= x⁻¹³y²⁰

2. (y⁵/x²)*(y¹⁰/x³) (simplify using multiplication and division rule)

(y⁵/x²)*(y¹⁰/x³)

= (y⁵x⁻²)*(y¹⁰x⁻³)

= y¹⁵x⁻⁵

3. y⁷/x⁴ * y¹⁰/x³ (simplify using multiplication and division rule)

y⁷/x⁴ * y¹⁰/x³

= (y⁷x⁻⁴)(y¹⁰x⁻³)

= y¹⁷x⁻⁷

4. y¹⁷/x⁷ (This expression is simplified correctly)

Learn more about exponents rules here: https://brainly.com/question/11975096

#SPJ1

Both √203 and √96 are irrational numbers since the numbers inside the roots are not perfect squares.

To determine whether a root is rational or irrational, we need to know if the number inside the square root is a perfect square or not. If it is not, then the root is irrational.

For √203, we can determine that 203 is not a perfect square, since the last digit is 3, which is not a perfect square. Therefore, √203 is an irrational number.

For √96, we can simplify the expression as follows:

√96 = √(16*6) = √16 * √6 = 4√6

Since 6 is not a perfect square, 4√6 is an irrational number.

To learn more about rational or irrational number click on,

https://brainly.com/question/16155109

#SPJ4

The Maximum volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid [tex]9x^2 + y^2 + 36z^2 = 1[/tex]is 4/5.

To find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid [tex]9x^2 + y^2 + 36z^2 = 1:[/tex]

We can use the hint provided.

By symmetry, we can assume that the rectangular box is in the first octant where x, y, and z are all positive.
Let the dimensions of the rectangular box be 2x, 2y, and 2z.

Then the volume of the rectangular box is V = 8xyz.
To maximize V, we need to find the maximum value of xyz that satisfies the equation of the ellipsoid.

Substituting 2x, 2y, and 2z into the equation of the ellipsoid, we get:
[tex](2x/3)^2 + (y/6)^2 + (2z/3)^2 = 1[/tex]
Multiplying both sides by 9/4, we get:
[tex](2x/3)^2 * (9/4) + (y/6)^2 * (9/4) + (2z/3)^2 * (9/4) = 9/4[/tex]
Simplifying, we get:
4x^2/9 + y^2/36 + 4z^2/9 = 1
We can see that this is the equation of an ellipsoid centered at the origin with semi-axes a = 3/2, b = 3, and c = 3/2.
By symmetry, we know that the maximum value of xyz will be achieved when x = y = z. Therefore, we need to find the value of x, y, and z that satisfy the equation of the ellipsoid and maximize xyz.
Substituting x = y = z into the equation of the ellipsoid, we get:
[tex]4x^2/9 + x^2/36 + 4x^2/9 = 1[/tex]
Simplifying, we get:
[tex]x^2 = 9/20[/tex]
Therefore, x = y = z = √(9/20).
Substituting these values into V = 8xyz, we get:
[tex]V = 8(√(9/20))^3 = 4/5[/tex]
Therefore,the Maximum volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid [tex]9x^2 + y^2 + 36z^2 = 1 is 4/5.[/tex]

To know more about ellipsoid:

https://brainly.com/question/29853961

#SPJ11

The base is decreasing at 1.917 cm/min when altitude is 12cm and  area=84cm².

Let A be the area of the triangle, h be the height of the triangle, and b be the base of the triangle.

Then the formula for the area of a triangle is:

A = (1/2)bh

We are given that dh/dt = 2.5 cm/min (the height is increasing at a rate of 2.5cm/min), and dA/dt = 3 cm²/min (the area is increasing at a rate of 3cm²/min).

We want to find db/dt, the rate of change of the base of the triangle when h = 12 cm and A = 84 cm².

To solve this problem, we need to use the chain rule of differentiation.

We start by differentiating both sides of the formula for the area of a triangle with respect to time t:

dA/dt = (1/2) d/dt (bh)

Next, we can use the product rule of differentiation to find d/dt (bh):

d/dt (bh) = b dh/dt + h db/dt

Substituting this into the previous equation gives:

dA/dt = (1/2) [ b dh/dt + h db/dt ]

Now we can substitute the given values of dh/dt and dA/dt, as well as h = 12 cm and A = 84 cm².

To find db/dt:

3 cm²/min = (1/2) [ b (2.5 cm/min) + 12 cm db/dt ]

Simplifying this expression gives:

6 cm²/min = 2.5 b cm²/min + 12 cm db/dt

Substituting A = 84 cm² and h = 12 cm into the formula for the area of a triangle gives:

84 cm² = (1/2) b (12 cm)

Simplifying this expression gives:

b = 14 cm

Now we can substitute b = 14 cm into the previous equation to find db/dt:

6 cm²/min = 2.5 (14 cm) cm²/min + 12 cm db/dt

Simplifying this expression gives:

db/dt = (6 cm²/min - 35 cm²/min) / (12 cm)

db/dt = -1.917 cm/min (rounded to three decimal places)

Therefore, the base of the triangle is decreasing at a rate of 1.917 cm/min when the height is 12 cm and the area is 84 cm².

To practice more questions about chain rule:

https://brainly.com/question/30895266

#SPJ11

Step-by-step explanation:

$20-$3-$15= $2

the amount of money spent on the bouncy ball is $2

The surface area of the rectangular prism in the image above is determined as:

236 square meters.

The prism given above in the image is a rectangular prism. The formula for finding the surface area of the prism is given as:

surface area of the prism = 2(lh + lw + hw), where:

h is the height

w is the width

l is the length of the prism.

Given the following:

length (l) = 6 m

width (w) = 5 m

height (h) = 8 m

Plug in the values:

Surface area of the prism = 2·(5·6 + 8·6 + 8·5) = 236 square meters.

Learn more about surface area of prism on:

https://brainly.com/question/22191558

#SPJ1