Find the volume of the region e that lies between the paraboloid z − 24 2 x 2 2 y 2 and the cone z − 2sx 2 1 y 2

By making the substitution u = a² - x² and using integration by substitution, we can evaluate the integral of (a² - x²[tex])^(^3^/^2^)[/tex] from 0 to a. The resulting value is [tex]-a^5^/^5[/tex].

We can evaluate this integral using the substitution method. Let's substitute u = a² - x². Then du/dx = -2x, which implies dx = -du/(2x). Also, when x = 0, u = a².

Substituting these expressions into the integral, we get:

∫(a² - x²[tex])^(^3^/^2^)[/tex] dx from 0 to a

= ∫(a² - x²[tex])^(^3^/^2^)[/tex] (-du/(2x)) from a² to 0

= (-1/2) ∫[tex]u^(^3^/^2^)[/tex] du from a² to 0

= (-1/2) [2/5 [tex]u^(^5^/^2^)[/tex]] from a² to 0

= (-1/5) [a⁵ - 0⁵]

= [tex]-a^5^/^5[/tex]

Therefore, the value of the integral is [tex]-a^5^/^5[/tex].

Learn more about Substitution method

brainly.com/question/30284922

#SPJ11

Area of the lot = 1.03 acres

The line length of the triangular lot = 700 ft

The height of the triangular lot = 130 ft

Note:

Area of a triangle = 0.5 x base x height

Calculate the base of the triangular lot using the Pythagoras's theorem

[tex]\text{Length}^2=\text{Height}^2+\text{Base}^2[/tex]

     [tex]700^2=130^2+\text{Base}^2[/tex]

     [tex]\text{Base}^2=700^2-130^2[/tex]

   [tex]\text{Base}^2=490000-16900[/tex]

          [tex]\text{Base}^2=473100[/tex]

         [tex]\text{Base}=\sqrt{473100}[/tex]

           [tex]\text{Base}=687.82[/tex]

The base of the triangular lot = 687.82 ft

Area of the triangular lot = 0.5 x 687.82 x 130

Area of the triangular lot = 44708.3 ft²

NB

1 ft² = 2.3 x 10^(-5) Acres

44708.3 ft² = 44708.3 x 2.3 x 10^(-5)

44708.3 ft² = 1.03 acres

Therefore:

Area of the lot = 1.03 acres

Answer:

Area of the lot = 1.03 acres

The line length of the triangular lot = 700 ft

The height of the triangular lot = 130 ft

Note:

Area of a triangle = 0.5 x base x height

Calculate the base of the triangular lot using the Pythagoras's theorem

The base of the triangular lot = 687.82 ft

Area of the triangular lot = 0.5 x 687.82 x 130

Area of the triangular lot = 44708.3 ft²

NB

1 ft² = 2.3 x 10^(-5) Acres

44708.3 ft² = 44708.3 x 2.3 x 10^(-5)

44708.3 ft² = 1.03 acres

Therefore:

Area of the lot = 1.03 acres

Step-by-step explanation:

The given equation is a curve in the Cartesian plane. Therefore, the  exact length of the curve [tex]y^2= 4(x+5)^3 , 0 \leq x \leq 3, y > 0[/tex]  is  [tex]2(3 \sqrt{3} - \sqrt{6} )[/tex] units

To find its length, we can use the formula for the arc length of a curve in terms of its parameterization.

First, we need to rewrite the equation in terms of a parameterization. Let's use x as the parameter, so we have [tex]y = 2\sqrt{(x+5)^3}[/tex]. Then, taking the derivative of y with respect to x, we get:

dy/dx = √(x+5)

Using this, we can calculate the arc length of the curve as:

[tex]L = \int_0^3 \sqrt{(1 + (dy/dx)^2) dx}[/tex]

Substituting dy/dx, we get:

[tex]L = \int_0^3 \sqrt{(1 + x+5) dx}[/tex]

Simplifying the inside of the square root, we get:

[tex]L = \int_0^3 \sqrt{(x+6) dx}[/tex]

Making the substitution u = x+6, we get:

[tex]L = \int_6^9 \sqrt{u \;du}[/tex]

Using the power rule of integration, we get:

[tex]L = (2/3)u^{(3/2)} |_6^9[/tex]

[tex]L = (2/3)(9\sqrt{9} - 6\sqrt{6} )[/tex]

[tex]L = 2(3\sqrt{3} - \sqrt{6})[/tex]

Therefore, the exact length of the curve is [tex]2(3 \sqrt{3} - \sqrt{6} )[/tex] units

To know more about equation refer here:

https://brainly.com/question/31190601#

#SPJ11

The indicated value f(g(4)) has a value of 2 when evaluated

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

f(x)=√x+1, g(x)=2x-5, and h(x) = 3x² - 3.

Calculate g(4)

So, we have

g(4) = 2(4) - 5

Evaluate

g(4) = 3

Next, we have

f(g(4)) = f(3)

Substitute the known values in the above equation, so, we have the following representation

f(g(4))=√3+1

Evaluate

f(g(4)) = 2

Hence, the value is 2

Read more about composite function at

https://brainly.com/question/10687170

#SPJ1

The simplified equivalent of the given expression; (2/3 x15/-16) - (7/12 x -24/35) using PEMDAS guidelines is; -9 / 40.

It follows from the task content that the simplified form of the given expression is to be determined.

Since the given expression is; (2/3 x15/-16) - (7/12 x -24/35); the expression can be simplified by first solving the parentheses so that we have;

( -30 / 48 ) - ( -168 / 420 )

By simplifying the fractions; we have;

(-5 / 8) - ( -2 / 5)

= -5/8 + 2/5

= -9 / 40.

Ultimately, the simplified expression as required is; -9 / 40.

Read more on simplified expression;

https://brainly.com/question/23385237

#SPJ1

Step-by-step explanation:

To answer the question, we can use algebra. Let's assume that the total number of voters is "x". If 40% of the voters chose Shane, then 60% of the voters chose the other candidates. We can set up an equation:

0.6x = 540

Solving for x, we get:

x = 900

Therefore, there were 900 voters in total.

Answer: 900

The distributive law for sets A, B, C can be stated as follows: A union (B intersection C) = (A union B) intersection (A union C).

To prove the distributive law for sets, we need to show that any element x that belongs to A union (B intersection C) also belongs to (A union B) intersection (A union C), and vice versa.

Let x be an arbitrary element in A union (B intersection C). Then, x must belong to either A or (B intersection C) or both.

Case 1: If x belongs to A, then x must belong to A union B and A union C, since A is a subset of both sets. Therefore, x belongs to (A union B) intersection (A union C).

Case 2: If x belongs to B intersection C, then x belongs to both B and C. Therefore, x belongs to A union B and A union C, since A is a subset of both sets. Therefore, x belongs to (A union B) intersection (A union C).

Hence, we have shown that A union (B intersection C) is a subset of (A union B) intersection (A union C), and vice versa. Therefore, the distributive law holds.

To prove the second part, we will use a proof by contradiction.

Assume that A and C are not disjoint, and B and D are not disjoint. Then, there exist elements a and c such that a belongs to both A and C, and there exist elements b and d such that b belongs to both B and D.

Therefore, (a,b) belongs to both A times B and C times D, which contradicts the assumption that A times B and C times D are disjoint.

Hence, either A and C are disjoint, or B and D are disjoint.

For more questions like Distributive law click the link below:

https://brainly.com/question/30339269

#SPJ11

The probability that both students chosen forgot their lunch is 1/45. Therefore, the probability that both students chosen forgot their lunch is 1/45.

To find the probability that both students chosen forgot their lunch, we need to use the formula for calculating probability:

P(A and B) = P(A) x P(B|A)

where P(A) is the probability of event A occurring, and P(B|A) is the probability of event B occurring given that event A has already occurred.

In this case, event A is the first student being chosen as someone who forgot their lunch (which has a probability of 2/10), and event B is the second student also being chosen as someone who forgot their lunch (which has a probability of 1/9, since there is one less student left to choose from).

So, putting it all together:

P(both students forgot their lunch) = P(A and B) = P(A) x P(B|A)
= (2/10) x (1/9)
= 1/45

Therefore, the probability that both students chosen forgot their lunch is 1/45.

In a class of 10 students, there are 2 students who forgot their lunch. If the teacher chooses 2 students, the probability that both of them forgot their lunch is calculated as follows:

First, determine the total number of ways to choose 2 students out of 10. This can be done using combinations:
C(10,2) = 10! / (2! * (10-2)!) = 45 combinations

Now, consider the 2 students who forgot their lunch. There's only 1 way to choose both of these students:
C(2,2) = 2! / (2! * (2-2)!) = 1 combination

The probability that both chosen students forgot their lunch is the ratio of the favorable combinations to the total combinations:
P = 1/45

So, the probability that both students chosen forgot their lunch is 1/45.

Visit here to learn more about probability  : https://brainly.com/question/30034780

#SPJ11

The area of the largest circle is 153.86 yard square.

The measurement that is closest to the area of the largest circle can be calculated as follows:

Therefore,

area of the largest circle = πr²

where

Therefore,

radius = 10 + 4 ÷ 2

radius = 14 / 2

radius = 7 yards

Hence,

area of the largest circle = 3.14 × 7²

area of the largest circle = 3.14 ×49

area of the largest circle = 153.86 yard²

learn more on area here: https://brainly.com/question/15751058

#SPJ1

Part A: The factored expression is 8x(5x - 2)

Part B: The expression is 4x - 16

Note that algebraic expressions are described as expressions that consists of coefficients, factors, constants, terms and variables.

They are also made up of arithmetic operations such as addition, subtraction, bracket, parentheses, multiplication and division

From the information given, we have that;

40x² and 16x

40x² - 16x

factorize the values

8x(5x - 2)

To add the expressions;

(10x+8) - 3(2x + 8)

expand the bracket, we have;

10x + 8 - 6x - 24

collect the like terms

4x - 16

Learn about algebraic expressions at: https://brainly.com/question/4344214

#SPJ1

A 95% confidence interval estimate of , the mean procrastination scale for first-year students at this college is between 34.881 and 38.919.

We know that:

Sample size (n) = 65

Sample mean (x) = 36.9

Sample standard deviation (s) = 6.41

Confidence level = 95%

Degrees of freedom = n - 1 = 64

To calculate the confidence interval, we use the formula:

CI = x ± tα/2 * (s/√n)

where tα/2 is the t-score with (n-1) degrees of freedom and α/2 = (1 - confidence level)/2.

Using a t-table or a calculator, we find that tα/2 for a 95% confidence level and 64 degrees of freedom is 1.997.

Plugging in the values, we get:

CI = 36.9 ± 1.997 * (6.41/√65)

Simplifying the expression, we get:

CI = (34.881, 38.919)

Therefore, we can be 95% confident that the true mean procrastination scale for first-year students at this college falls between 34.881 and 38.919.

Learn more about confidence interval at brainly.com/question/15738214

#SPJ11

Triple integration is a powerful tool for computing volumes of complex regions in three-dimensional space and is widely used in mathematical modeling, physics, and engineering.

For the first problem, the volume of the region U can be computed using triple integration in cylindrical coordinates

The bounds of integration for r, θ and z must be determined based on the shape of the region.

For the second problem, the volume of the region U can also be computed using triple integration in cylindrical coordinates, but with different bounds of integration due to the different shape of the region.

In both cases, cylindrical coordinates are used because the regions have cylindrical symmetry, making it easier to integrate over the region.

Triple integration is a powerful tool for computing volumes of complex regions in three-dimensional space and is widely used in mathematical modeling, physics, and engineering.

For more similar questions on topic Coordinate systems.

brainly.com/question/10930749

#SPJ1

For a loan amount taken by Timmy from the bank on simple interest, the interest rate advertised by the bank is equals to the 2.7% per year.

Simple interest defines to the interest calculated only based on the principal. With simple interest method, a borrower only pays interest on the principal. It is calculated by the principal amount multiplied by the interest rate, multiplied by the number of periods and then resultant is divided by 100. Formula is written as [tex]Simple \: interest = \frac{P \times r \times t}{100}[/tex]

Where, P--> principal amount

t --> time period

r -> simple interest rate

We have Timmy takes out a loan on simple interest. The amount of loan that is principal = $750

Time periods = 15 months

The received amount by him = $725

So, simple interest = 750 - 725 = $25

We have to determine the simple interest rate advertised by the bank. Using the above formula, substitute all known values in formula, 25 = [tex] \frac{ 750 × 15 × r}{12×100}[/tex]

[tex]r = \frac{ 1200× 25}{750× 15}[/tex]

= 2.66% per year

Hence, required interest rate is 2.7 % per year.

For more information about simple interest, visit :

https://brainly.com/question/25845758

#SPJ4

Shaunice's average time per lap based on the 6 laps she recorded is approximately 66.5 seconds.

Let's start with Shaunice's data. From the table provided, we can see that Shaunice ran 6 laps and recorded the time it took her to complete each lap. To find Shaunice's average time per lap, we need to add up the times for all 6 laps and then divide by 6. This is the formula for finding the average:

average = sum of all values / number of values

Using this formula, we can calculate Shaunice's average time per lap:

average = (68 + 65 + 65 + 64 + 67 + 70) / 6

average = 399 / 6

average ≈ 66.5 seconds per lap

To know more about average here

https://brainly.com/question/16956746

#SPJ1

Complete Question:

Shaunice, Joshua, and Juan ran several laps around the track. They recorded some data based on 6 of the laps that they ran. The table shows the amount of time that it took Shaunice to complete 6 of the laps that she ran.

Shaunice

Lap   Time (seconds)

1               68

2              65

3             65

4             64

5              67

6               70

Joshua determined his average pace to be 63 seconds per lap.

Answer:

66.5 seconds I believe

Step-by-step explanation:

The answer choice which represents a function with the ordered pair (4, 8) as a solution is; Choice C; 2x = y.

It follows from the task content that the function which has the given ordered pair; (4, 8) as a solution is to be determined.

On this note, by observation; the answer choice C represents an equation whose solution includes (4, 8).

By checking; we have; 2x = y;

2 (4) = 8; 8 = 8 which holds true.

Consequently, answer choice C is correct.

Read more on function solutions;

https://brainly.com/question/8668041

#SPJ1

The slope of the tangent to the curve x² + y³ = 12 at the point when x = 2 is

To find the slope of the tangent to the curve x² + y³ = 12 at the point when x = 2, we need to find the derivative of y with respect to x using implicit differentiation.

Taking the derivative of both sides with respect to x, we get: 2x + 3y²(dy/dx) = 0

We want to find the slope when x = 2, so we substitute x = 2 into the equation above: 2(2) + 3y²(dy/dx) = 0 4 + 3y²(dy/dx) = 0 3y²(dy/dx) = -4 dy/dx = -4/(3y²)

Now, we need to find the value of y when x = 2. Substituting x = 2 into the original equation, we get: 2² + y³ = 12 y³ = 8 y = 2 So, when x = 2, y = 2. Substituting this into the equation for dy/dx, we get: dy/dx = -4/(3(2²)) = -4/12 = -1/3

Visit here to learn more about Slope:

brainly.com/question/3493733

#SPJ11

The letters of the word "are" can be arranged in 6 different ways. These arrangements are: are, aer, rae, rea, ear, era. To calculate the number of arrangements, we use the formula for permutations of n objects, which is n!. In this case, n = 3, so there are 3! = 6 ways to arrange the letters.

If the letters of "are" are arranged in a random order, the probability that the result will be "era" is 1/6. This is because there is only one way to get "era" out of the 6 possible arrangements, and each arrangement is equally likely to occur.

In other words, the probability of an event happening is equal to the number of ways that event can occur, divided by the total number of possible outcomes. In this case, the event is getting the word "era" and the total number of outcomes is 6.

I hope this helps answer your question. Let me know if you have any more questions!
Hello! It seems that you've missed providing the specific letters and the result you're looking for in your question. However, I can explain the process using a general example.

Let's say you have the letters A, B, and C. To determine the number of different arrangements, you can use the formula for permutations, which is n! (n-factorial), where n represents the number of unique items.

For this example:

n! = 3!
  = 3 × 2 × 1
  = 6

So, there are 6 different ways to arrange the letters A, B, and C.

Now, if you're looking for the probability of getting a specific arrangement (for example, "ABC"), you can calculate it by dividing the number of desired outcomes by the total number of possible outcomes. Since there's only 1 way to get the "ABC" arrangement and there are 6 possible arrangements in total:

Probability = (Desired outcomes) / (Total outcomes)
           = 1 / 6
           ≈ 0.1667

This means there's approximately a 16.67% chance of getting the "ABC" arrangement when arranging these letters randomly.

Please provide the specific letters and the result you want to calculate the probability for, and I'd be happy to help you with your question.

For more about permutations

https://brainly.com/question/30498967

#SPJ11

Answer:

4

Step-by-step explanation:

The question is asking us to find what number multiplied by x, the input, will give us y, the output.

We can see that the first input is 5, and the output is 20, so we can set up an equation:

20=_5

_=4

So, the equation would represent:

y=4x

We can check our work with the second set of inputs and outputs:

60=(4)15, which is true, so 4 is the right number.

Hope this helps!

The odds ratio of earning an A for seniors vs. sophomores is 1.36. To calculate the odds ratio, we first need to find the odds of earning an A for each group.

For seniors: - The proportion of seniors earning an A is 21% or 0.21. - The odds of earning an A for seniors is 0.21 / (1 - 0.21) = 0.266
For sophomores:
- The proportion of sophomores earning an A is 16% or 0.16.
- The odds of earning an A for sophomores is 0.16 / (1 - 0.16) = 0.190
Next, we calculate the odds ratio:
- Odds ratio = (odds of seniors earning an A) / (odds of sophomores earning an A)
- Odds ratio = 0.266 / 0.190 = 1.400
Rounding to two decimal places, the odds ratio is 1.36.
To calculate the odds ratio of earning an A for seniors vs. sophomores in a regression course, follow these steps:
Step 1: Find the odds of earning an A for each group.
- Seniors: Historically, 21% earn an A, so the odds for seniors is 0.21 / (1 - 0.21) = 0.21 / 0.79 ≈ 0.266
- Sophomores: Historically, 16% earn an A, so the odds for sophomores is 0.16 / (1 - 0.16) = 0.16 / 0.84 ≈ 0.190
Step 2: Calculate the odds ratio by dividing the odds for seniors by the odds for sophomores.
Odds ratio = Odds for seniors / Odds for sophomores ≈ 0.266 / 0.190 ≈ 1.40
Therefore, the odds ratio of earning an A for seniors vs. sophomores is approximately 1.40 when rounded to 0.01.

learn more about sophomores here: brainly.com/question/26221849

#SPJ11

The predicted population of the dying town in 2023 is approximately 8,200

To predict the population in 2023 using the exponential law, p(t) = P0e^(kt) or p(t) = P0b^t, we first need to find the constants P0 and k (or b). We know the population was 10,000 in 2016 and 9,500 in 2018.

Step 1: Set up the equations using the given information.
For the year 2016 (t=0), p(0) = P0e^(k*0) = 10,000
For the year 2018 (t=2), p(2) = P0e^(k*2) = 9,500

Step 2: Solve for P0 and k.
From the first equation, P0 = 10,000.
Substitute P0 in the second equation: 9,500 = 10,000e^(2k)

Step 3: Solve for k.
Divide both sides by 10,000: 0.95 = e^(2k)
Take the natural logarithm of both sides: ln(0.95) = 2k
Divide by 2: k = ln(0.95) / 2 ≈ -0.0253

Step 4: Predict the population in 2023 (t=7).
p(7) = P0e^(kt) = 10,000e^(-0.0253*7) ≈ 8,200

So, the predicted population of the dying town in 2023 is approximately 8,200, rounded to the nearest whole number.

Learn more about Exponents: https://brainly.com/question/3009444

#SPJ11

Answer:

Step-by-step explanation:

4x2=8

8x8=64 cm cube

Appeals that focus on projecting the appealing traits of the writer are known as ethos-based appeals.

Ethos is one of the three modes of persuasion identified by Aristotle and refers to the credibility and trustworthiness of the speaker or writer. Ethos-based appeals attempt to establish the author's character and authority on a subject, using various strategies such as emphasizing their expertise, experience, or moral character.

By projecting appealing traits of the writer, such as honesty, intelligence, or likability, ethos-based appeals aim to win the audience's confidence and persuade them to accept the argument.

Ethos is particularly important in situations where the audience may be skeptical or distrustful of the writer, such as in political or advertising contexts. Overall, ethos-based appeals are a powerful tool for writers looking to persuade their audience by establishing their credibility and building trust.

To know more about ethos, refer here:

https://brainly.com/question/28850394#

#SPJ11

It will take 3.6 hours or 3 hours and 36 minutes for Ana and her apprentice to build a wall together.

If constructing a brick wall is one unit of work, Ana may complete one sixth of it in an hour, while her apprentice can complete one ninth. They can do 1/6 + 1/9 of the task in an hour while working jointly. By determining the common denominator of 6 and 9, which is 18, we can determine how much work they can complete in an hour.

1/6 + 1/9

= 3/18 + 2/18

= 5/18

They can complete 5/18 of the task in an hour, according to this. We can build up a percentage to determine how long it would take them to do the assignment collectively.

5/18 = 1/x, the time it takes for them to complete the work together is x. Solving for x, we get,

x = 18/5

x = 3.6 hours

Therefore, it would take Ana and her apprentice 3.6 hours, or 3 hours and 36 minutes, to build the wall together.

To know more about concept of work, visit,

https://brainly.com/question/25514572

#SPJ4

The true statement about stepwise regression is that it is a step-by-step method that adds independent variables one by one in order to build a more efficient regression equation.

Regression is a statistical method used to analyze the relationship between one or more independent variables (also known as predictor variables) and a dependent variable (also known as the response variable). The goal of regression analysis is to estimate the strength and direction of the relationship between the independent and dependent variables.

Regression analysis is often used in forecasting, where the independent variables are used to predict future values of the dependent variable. There are many different types of regression analysis, including linear regression, logistic regression, polynomial regression, and multiple regression.

Linear regression is a common type of regression analysis that assumes a linear relationship between the independent and dependent variables. In this type of regression, a straight line is fitted to the data in order to estimate the relationship between the variables. Logistic regression, on the other hand, is used when the dependent variable is binary (i.e., it can only take on two values, such as yes or no), and is used to predict the probability of the dependent variable taking on one of these values based on the independent variables.

Visit here to learn more about regression brainly.com/question/28178214

#SPJ11

In summary, the equation for the tangent plane can be written as z - z0 = ∂f/∂x(x0, y0)(x - x0) + ∂f/∂y(x0, y0)(y - y0), where (x0, y0, z0) represents the coordinates of the given point. This equation represents a linear approximation of the surface near the point of tangency.

To understand the equation for the tangent plane, we start by considering the first-order partial derivatives of the function f(x, y) with respect to x and y. These partial derivatives, denoted as ∂f/∂x and ∂f/∂y, represent the rates of change of the surface with respect to x and y, respectively. At the point (x0, y0), the tangent plane approximates the behavior of the surface near that point.

The equation for the tangent plane is derived by using the point-slope form of a linear equation, where the slope of the plane in the x-direction is given by ∂f/∂x(x0, y0) and in the y-direction by ∂f/∂y(x0, y0). The equation is then written as z - z0 = ∂f/∂x(x0, y0)(x - x0) + ∂f/∂y(x0, y0)(y - y0), which relates the changes in x and y coordinates to the change in z-coordinate on the tangent plane.

Learn more about Equations:

brainly.com/question/29657983

#SPJ11

If a = c is a critical value for f(), where c is a real number and f"(c) = 0, this means that the Second Derivative Test fails. We cannot determine whether f(c) has a local maximum or local minimum at x = c using the Second Derivative Test.

It is possible that f(x) may have a local minimum at x = c, but we cannot confirm this using the Second Derivative Test. However, we do know that f(x) is concave down at x = c since f"(c) = 0 and a critical point with f"(x) < 0 corresponds to a local maximum, Based on the given information, if a = c is a critical value for f(x), where c is a real number and f''(c) = 0.

This means that the Second Derivative Test fails. The reason is that the Second Derivative Test relies on the sign of f''(c) to determine the concavity of the function at the critical point c. If f''(c) > 0, the function is concave up and has a local minimum at x = c. If f''(c) < 0, the function is concave down and has a local maximum at x = c. However, since f''(c) = 0, we cannot determine the concavity or whether the function has a local minimum or maximum at x = c using the Second Derivative Test.

To know more about Second Derivative Test:- https://brainly.com/question/29753185

#SPJ11

Let r be the common ratio. The correct answer is Rounding to the nearest dollar, the value of Madame Dumas's art collection 12 years after she started tracking its worth is $9,498,559.

A) Let the value of the collection three years ago be the first term, a = $450,000.

Then we can write:

Second term: [tex]ar = $810,000[/tex]

Third term: [tex]ar^2 = $1,458,000[/tex]

To find the common ratio r, we can divide the second term by the first term and the third term by the second term:

[tex]ar/a[/tex]= [tex]\frac{810,000}{450,000}[/tex]

[tex]r = 1.8[/tex]

[tex]ar^2/ar[/tex] = [tex]\frac{450000}{810000}[/tex]

[tex]r = 1.8[/tex]

So the explicit rule for the value of Madame Dumas's art collection n years after she started tracking its worth is:[tex]a_n = ar^(n-3)[/tex]

B) To find the value of her collection 12 years after she started tracking its worth, we can use the explicit rule:

[tex]a_12 = ar^(12-3) = ar^9[/tex]

We already know that [tex]a = $450,000[/tex] and [tex]r = 1.8[/tex], so we can substitute those values:[tex]a_12 = $450,000(1.8)^9 = $9,498,558.57[/tex]

To learn more about common ratio, visit here

https://brainly.com/question/31291016

#SPJ4

The slope (m) of the line y = x + 3 is 1, while the y-intercept of the line is 3.

If an equation of a line is expressed in slope-intercept form as y = mx + b, we can easily determine its slope and the y-intercept which are:

m is the slope

b is the y-intercept.

Given the equation y = x + 3, therefore:

the coefficient of x is the slope (m), which is 1.

the y-intercept (b) of the line is 3.

Learn more about the equation of a line on:

https://brainly.com/question/24907633

#SPJ1

The latest that activity B can start is at the end of the 35th day.To determine the latest that activity B can start, we must first understand the sequence and dependencies of the activities. Since the durations of activities A, B, C, and D are given as 35, 5, 6, and 7 days respectively.

let's assume that activity B must follow activity A and activity C and D follow activity B.

In this scenario, activity B can start once activity A is completed, which is after 35 days. Following activity B, which takes 5 days, activity C will take 6 days and activity D will take 7 days. Thus, the total duration of all activities is 35 + 5 + 6 + 7 = 53 days.

To find the latest possible start time for activity B, we need to consider the total time available and the durations of the subsequent activities. Since activity B takes 5 days and the following activities C and D together take 13 days (6 + 7), we can subtract their combined durations from the total time to find the latest possible start time for activity B.

The calculation would be: 53 (total time) - 5 (activity B) - 13 (activity C and D) = 35 days.

Therefore, the latest that activity b can start if a lasts 35 days, b lasts 5, days c lasts 6 days, and d lasts 7 days,  the latest that activity B can start is at the end of the 35th day.

learn more about sequence here: brainly.com/question/12246947

#SPJ11

If two poles are connected by a wire that is also connected to the ground then The wire should be anchored to the ground at a distance of 20 feet from the first pole A to minimize the amount of wire needed.

To minimize the length of the wire, we need to find the point P that minimizes the length of the wire APB.

Let's assume that the wire is perfectly straight, which means that the line segment connecting A and P and the line segment connecting B and P are both perpendicular to the ground.

Let's also call the distance from point P to the first pole A as x. Then the distance from point P to the second pole B is 96 - x.

Using the Pythagorean theorem, we can express the length of the wire AB as: AB^2 = (20 - x)^2 + 12^2

Simplifying this expression, we get: AB^2 = x^2 - 40x + 784

To minimize AB, we need to find the value of x that minimizes AB^2. To do that, we take the derivative of AB^2 with respect to x and set it equal to 0: d/dx (AB^2) = 2x - 40 = 0

Solving for x, we get: x = 20

Therefore, the wire should be anchored to the ground at a distance of 20 feet from the first pole A to minimize the amount of wire needed.

To learn more about “distance” refer to the https://brainly.com/question/26550516

#SPJ11