A tank containing 6000 L of water drains out in 30 min. The volume V of water in the tank after t min of draining is V = 6000(1 – t/30)?. Find the instantaneous time rate of change of V after 15 min of draining. (Book: Technical Mathematics by Allyn J. Washington (2014)) dV = dt -210 L min dV dt = -100 L min O None dV = -200 L dt min dV dt = =-400 L min

The value of the integral is π/2 - 2ln|1+sin(x)/cos(x)| + ln|cos(x)| + C.

To evaluate the integral, we can use the beta function and make a substitution. Let's start by writing the integral in terms of sine and cosine:

∫sins^5(x)/(tan(x)+tan^3(x)) dx = ∫sin^4(x)cos(x)/(sin(x)/cos(x)+sin^3(x)/cos^3(x)) dx

Now, let's make the substitution u = sin(x)/cos(x), which gives us:

∫sin^4(x)cos(x)/(sin(x)/cos(x)+sin^3(x)/cos^3(x)) dx = ∫u^4/(u+u^3) du

Next, we can use the beta function to write this integral in terms of gamma functions. Recall that the beta function is defined as:

B(x, y) = ∫t^(x-1)(1-t)^(y-1) dt from 0 to 1

Using this definition, we can write:

∫u^4/(u+u^3) du = ∫u^2/(1+u)^2 * u^2/(1-u+u^2)^2 du

Now, we can use the substitution v = 1/(1+u) to get:

∫u^2/(1+u)^2 * u^2/(1-u+u^2)^2 du = ∫v^2(1-v)/(1-v^2)^2 dv

Using partial fractions, we can write:

v^2(1-v)/(1-v^2)^2 = 1/(1-v^2) - 1/(1-v)^2

Substituting this back into the integral, we get:

∫v^2(1-v)/(1-v^2)^2 dv = ∫(1/(1-v^2) - 1/(1-v)^2) dv

Using the beta function, we can write:

∫1/(1-v^2) dv = B(1/2, 1/2) * tan^(-1)(v) = π/2

And:

∫1/(1-v)^2 dv = B(1, 1/2) * (1-v)^(-1) = 2/(1-v)

Substituting these back into the integral and simplifying, we get:

∫sins^5(x)/(tan(x)+tan^3(x)) dx = π/2 - 2ln|1+sin(x)/cos(x)| + ln|cos(x)| + C

Therefore, the value of the integral is π/2 - 2ln|1+sin(x)/cos(x)| + ln|cos(x)| + C.

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Answer:

dbd

Step-by-step explanation:

Mia is fostering 10 kittens. She weighed each kitten to the nearest 14 of a pound. The results are recorded in this frequency table.Create a line plot to display the data.To create a line plot, hover over each number on the number line. Then click and drag up to plot the data.

Answer: Im pretty sure that its d

Step-by-step explanation:

The volume of the piñata is

The volume is solved using the formula

= area x thickness

The shape is a composite one and the area is solved by

= area of rectangle + area of triangle

= 12 x 12 + 1/2 x 8 x 12

= 144 + 48

= 192 square in

The volume

= 192 x 6

= 1152 cubic in

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Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

from the angles we understand that they are similar, therefore in proportion, we solve, in fact, with a proportion between the corresponding sides

24 : x = 32 : 38

x = 24 x 38 : 32

x = 912 : 32

x = 28.5 units

-------------------------

check

24 : 28.5 = 32 : 38

0.84 = 0.84

The answer is good

The radius of the cylinder is 2 feet.

The volume of this cylinder is 37. 68 cubic feet. Therefore, the radius of the cylinder can be found as follows:

Therefore,

volume of a cylinder = πr²h

where

Therefore,

volume of a cylinder = 3.14 × r² × 3

37.68 = 9.42r²

divide both sides by 9.42

r² = 37.68 / 9.42

r² = 4

square root both sides of the equation

r = √4

radius = 2 feet

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Answer:

The answer to your problem is ↓

Part A: 110.8 feet

Part B: $77

Step-by-step explanation:

Calculation: Part A.

6.2 x 4 = 24.8 

24.8 x 2 = 49.6

4 x 3 = 12

12 x 2 = 24

6.2 x 3 = 18.6

18.6 x 2 = 37.2

49.6 + 24 + 37.2 = 110.8

Calculation: Part B.

Same as the beginning of Part A:

6.2 x 4 = 24.8 

24.8 x 2 = 49.6

4 x 3 = 12

12 x 2 = 24

6.2 x 3 = 18.6

49.6 + 24 + 18.6 = 92.2

92.2 x .83 = 76.526

We then need to round ‘ 76.526 ‘

Rounded = 77

Thus the answer to your problem is ↓

Part A: 110.8 feet

Part B: $77

Answer:

$76.526

Step-by-step explanation:

Front and back: length = 6.2 feet, width = 4 feet

Left and right: length = 3 feet, width = 4 feet

Top and bottom: length = 6.2 feet, width = 3 feet

The area of each face is:

Front and back: A = lw = (6.2)(4) = 24.8 square feet

Left and right: A = lw = (3)(4) = 12 square feet

Top and bottom: A = lw = (6.2)(3) = 18.6 square feet

The total surface area is the sum of the areas of all six faces:

SA = 2(24.8) + 2(12) + 2(18.6) SA = 49.6 + 24 + 37.2 SA = 110.8 square feet

Part B: To find the cost of covering the bench in ceramic tiles, we need to find the surface area of the bench excluding the bottom face that is on the ground. This is the same as the total surface area minus the area of the bottom face:

SA’ = SA - A(bottom) SA’ = 110.8 - 18.6 SA’ = 92.2 square feet

The cost of covering one square foot of the bench is $0.83, so the total cost is:

C = SA’ x $0.83 C = 92.2 x $0.83 C = $76.526

Rounding to the nearest cent, the cost is $76.53.

Received message. Part A: To find the total surface area of the bench, we need to find the area of each face of the rectangular prism and add them up. The formula for the area of a rectangle is A = lw, where l is the length and w is the width. The dimensions of the bench are 6.2 feet by 3 feet by 4 feet, so we can label the faces as follows: - Front and back: length = 6.2 feet, width = 4 feet - Left and right: length = 3 feet, width = 4 feet - Top and bottom: length = 6.2 feet, width = 3 feet The area of each face is: - Front and back: A = lw = (6.2)(4) = 24.8 square feet - Left and right: A = lw = (3)(4) = 12 square feet - Top and bottom: A = lw = (6.2)(3) = 18.6 square feet The total surface area is the sum of the areas of all six faces: SA = 2(24.8) + 2(12) + 2(18.6) SA = 49.6 + 24 + 37.2 SA = 110.8 square feet Part B: To find the cost of covering the bench in ceramic tiles, we need to find the surface area of the bench excluding the bottom face that is on the ground. This is the same as the total surface area minus the area of the bottom face: SA' = SA - A(bottom) SA' = 110.8 - 18.6 SA' = 92.2 square feet The cost of covering one square foot of the bench is $0.83, so the total cost is: C = SA' x $0.83 C = 92.2 x $0.83 C = $76.526 Rounding to the nearest cent, the cost is $76.53.

Michael should expect that the quantity of visitors that will buy more than one costume is  23.

In order to calculate the quantity of expected visitors who will buy more than one costume amongst a projected 1800 attendees next week, we can utilize the proportion between the individuals who purchased multiple costumes and the overall number of people who bought at least a single costume.

3 / 230 = 0.013

We can determine the potential number of multiple costume buyers among the anticipated 1800 visitors by utilizing a straightforward calculation: multiplying the quantity of one-costume purchasers by the ratio of those who obtained more than one costume.

0.013 x 1800 = 23.4

= 23 visitors

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The point where the line through (5, 2) with slope 4 crosses the vertical axis is (0, -18).

To do this, we can use the point-slope form of a line equation:

y - y1 = m(x - x1)

Here, (x1, y1) is the given point (5, 2) and m is the slope, which is 4. Let's plug in these values:

y - 2 = 4(x - 5)

Now, we need to find the point where the line crosses the vertical axis (y-axis). When a point is on the y-axis, its x-coordinate is 0. So, we will substitute 0 for x and solve for y:

y - 2 = 4(0 - 5)
y - 2 = -20
y = -20 + 2
y = -18

Therefore, the point where the line through (5, 2) with slope 4 crosses the vertical axis is (0, -18).

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The sampling distribution of sample means, especially for samples with n>30, refers to the distribution of means obtained from repeated random sampling from the same population.

According to the Central Limit Theorem, this distribution will have the same mean as the population from which the samples are drawn, and it will be normally distributed regardless of the population's distribution shape. The statement is true. The sampling distribution of sample means is a distribution of the means of all possible samples of a certain size that can be drawn from a population. When the sample size is greater than 30, the Central Limit Theorem states that the sampling distribution will be approximately normal, regardless of the underlying population distribution.

Additionally, the mean of the sampling distribution of sample means will be equal to the population mean, assuming that the samples are drawn randomly and independently from the population. This makes it a useful tool for making inferences about the population mean based on a sample mean.

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The probability that the item will last longer than 4 years is approximately 0.8236 or 82.36%.

To find the probability that the item will last longer than 4 years, we'll use the z-score formula and then look up the corresponding probability in a standard normal distribution table (also known as a z-table).

1. Calculate the z-score: z = (X - μ) / σ where X is the value of interest (4 years), μ is the mean (5.4 years), and σ is the standard deviation (1.5 years). z = (4 - 5.4) / 1.5 z = -1.4 / 1.5 z ≈ -0.93

2. Look up the probability in a z-table: A z-table gives the probability that a value from a standard normal distribution is less than the z-score.

Since we want to find the probability that the item lasts longer than 4 years (greater than the z-score), we need to find the complement of the probability from the z-table. P(Z < -0.93) ≈ 0.1764

3. Calculate the complement: P(Z > -0.93) = 1 - P(Z < -0.93) P(Z > -0.93) = 1 - 0.1764 P(Z > -0.93) ≈ 0.8236

Your answer: The probability that the item will last longer than 4 years is approximately 0.8236 or 82.36%.

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a) The quadratic regression function that models this data is [tex]y = -20x^2 + 920x - 5200[/tex].

b) To find the ticket price that would maximize revenue, we need to find the x-value of the vertex of the quadratic function. The x-value of the vertex is given by [tex]\frac{-b}{2a}[/tex], where a = -20 and b = 920. So, the ticket price that would maximize revenue is [tex]x=\frac{-b}{2a} = \frac{-920}{2(-20)} = $23[/tex]

The vertex of the quadratic function is (23, 11,630), which means that if the ticket price is set at $23, the revenue will be maximized.

c) The maximum revenue is given by the y-value of the vertex of the quadratic function, which is 11,630 dollars. This means that if the ticket price is set at $23, the maximum revenue that can be generated is $11,630.

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Answer:

[tex]m = \frac{2 - ( - 4)}{1 - ( - 1)} = \frac{6}{2} = \frac{3}{1} = 3[/tex]

Answer: your mum has all the answers just ask her kidding if i am correct its 29

(a) The correlation between x and e can be determined using the formula for the correlation coefficient:

correlation coefficient = covariance(x,e) / (standard deviation of x * standard deviation of e)

Since the covariance between x and e is given as 0.9, and the standard deviation of x is o (given in the question), and the standard deviation of e is o1 (given in the question), we have:

correlation coefficient = 0.9 / (o * o1)

(b) Given y = 3 + xx + e and B1 = 3 and B2 = 1, we can solve for e as:

e = y - B1 - B2x

Substituting the values, we get:

e = y - 3 - x

Using the first 10 observations of x and y, we can calculate the sample correlation between x and e as:

sample correlation coefficient = covariance(x,e) / (standard deviation of x * standard deviation of e)

Using the formula, we can calculate the sample covariance as:

covariance(x,e) = SUM[(xi - x_bar)*(ei - e_bar)] / (n - 1)

where x_bar and e_bar are the sample means of x and e respectively, and n is the sample size (10 in this case).

Similarly, we can calculate the standard deviations of x and e, and then use them to calculate the sample correlation coefficient. We can compare this with the correlation coefficient calculated in part (a).

(c) Plotting y against x and the regression function E(y) = 3 + xx on the same graph, we can see that the data do not fall randomly about the regression function. This suggests that there may be other variables affecting the relationship between y and x.

(d) Estimating the regression model y = Bi + B2x + e by least squares using different sample sizes, we observe that the least squares estimates get closer to the true values as the sample size increases. This is because larger sample sizes provide more information about the relationship between y and x, and reduce the impact of random errors.

(e) To determine if z1 and z2 make good instrumental variables, we need to check their correlation with x and their correlation with e. Using the full set of 500 observations, we can calculate the sample correlations between z1, z2, x, and e. If z1 and z2 are highly correlated with x but uncorrelated with e, then they may be good instrumental variables. Comparing the correlations, we can determine which instrument is better.

(f) Estimating the model y = Bi + B2x + e by instrumental variables using z1 and different sample sizes, we observe that the IV estimates are getting closer to the true values as the sample size increases. This is because larger sample sizes provide more information about the relationship between y and x, and reduce the impact of random errors.

(g) Estimating the model y = Bi + B2x + e by instrumental variables using z2 and different sample sizes, we observe that the IV estimates are getting closer to the true values as the sample size increases. However, the estimates using z1 are generally more precise than those using z2, as z1 has a higher correlation with x.

(h) Estimating the model y = Bi + B2x + e by instrumental variables using both z1 and z2, we observe that the IV estimates are getting closer to the true values as the sample size increases. Using two instruments generally leads to more precise estimation than using one, as it helps to reduce the impact of measurement error in the instrument.

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The answers are explained in the solution.

Part A:

To calculate how much fabric is needed to make 3 aprons, we need to multiply the amount of fabric needed for one apron by 3.1 apron requires 1 1/2 yards of fabric for the front and 1/8 yards of fabric for the tie.1 1/2 yards + 1/8 yards = 15/8 yards.

Now we can multiply the total fabric needed for one apron by 3 to get the fabric needed for 3 aprons:

3 x 15/8 yards = 45/8 yards

So, the total fabric needed to make 3 aprons = 45/8 yards.

Part B:

If Susan originally has 7 yards of fabric and she uses 45/8 yards to make 3 aprons, we can subtract the amount used from the original amount to find out how much fabric is left over.

7 yards - 45/8 yards = 56/8 yards - 45/8 yards

= 11/8 yards

So, after making the aprons, Susan will have 11/8 yards of fabric left over.

Part C:

To determine if Susan has enough fabric left to make another apron, we need to compare the amount of fabric left (11/8 yards) with the amount of fabric needed for one apron (1 1/2 yards + 1/8 yards = 15/8 yards).

Since 15/8 yards is greater than 11/8 yards, Susan does not have enough fabric left to make another apron.

She is short by 4/8 yards (or 1/2 yard) of fabric.

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The given figure is a right triangular prism, with 2 parallel and congruent triangular faces and 3 rectangular faces.

The triangular faces have sides 13 ft, 13ft and 24 ft and the height of 5 ft.

Two of the rectangular faces are 13 ft x 30 ft and the remaining face is 24 ft x 30 ft.

Surface area is the sum of areas of all 5 faces.

Area formula for triangle is A = bh/2 and for rectangle is A = ab.

Let's verify the steps of calculation.

Step 1

This is right

Step 2

This is wrong

Step 3

This is right

Step 4

Correction in step 2, it should be 720 but not 1440.

Correction in last step, the sum:

To show that an element e of a matroid M is a coloop of M if and only if e is in every basis of M, we need to prove both directions of the statement.

First, let's assume that e is a coloop of M. By definition, a coloop is an element that is not in any basis of M, but adding it to any circuit of M creates a new basis. Since e is not in any basis, it must be in every circuit of M. Now, suppose that e is not in some basis B of M. Then we can remove an element f from B and add e to obtain a new basis B', which contradicts the definition of a coloop. Therefore, e must be in every basis of M.

Conversely, let's assume that e is in every basis of M. We want to show that e is a coloop of M, i.e., that adding e to any circuit of M creates a new basis. Let C be any circuit of M, and suppose that adding e to C does not create a new basis. Then there must exist some element f in C such that removing f and adding e still gives a basis. But this means that e is not necessary for the independence of C, contradicting the assumption that e is in every basis of M. Therefore, e must be a coloop of M.

As for Exercise 6 of Section 1.4, it provides alternative characterizations of coloops in a matroid M, including:

- An element e is a coloop of M if and only if it is the unique maximal element of M that is not in any basis.
- An element e is a coloop of M if and only if there exists a basis B of M such that B\{e} is not a basis.
- An element e is a coloop of M if and only if M\e has a unique basis.
- An element e is a coloop of M if and only if for any basis B of M, there exists an element f in B such that B\{f} U {e} is also a basis.

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For a cost function, C(x) = 36100 + 800x + x²

a) The cost at the production level 1250 is equal to 2,598,600.

b) The average cost at the production level 1250 is equal to 2,078.88.

c) The marginal cost at the production level 1250 is equal to 3300 $/unit.

d) The production level, x = 60 that will minimize the average cost.

e) The minimal average cost is equals the 1,461.67.

Let consider C(x) be a total cost function where x is quantity of the product, then,

We have a cost function is written as C(x) = 36100 + 800x + x²

a) The cost at production level 1250, that is x = 1250 is equals to

=> C( 1250) = 36100 + 800× 1250 + 1250²

= 2,598,600

b) The average cost at the production level 1250, that is AC(x) [tex]= \frac{36100 + 800x + x²}{x}[/tex]

[tex]= \frac{36100}{x} + 800 + x[/tex]

Plug the value x = 1250

[tex]= \frac{36100}{1250} + 800 + 1250[/tex]

= 2,078.88

c) The marginal cost at the production level 1250 is equal to the derivative of

[tex]\frac{dC(x)}{dx }[/tex], evaluated for x = 1250,

[tex]\frac{dC(x)}{dx }[/tex] = C'(x)

= 800 + 2x

C'(1250) = 800 + 2× 1250 = 3300$/unit

d) As we know the average cost of the total cost function is,

[tex] A C(x) = \frac{36100}{x} + 800 + x[/tex]

Compute the critical point for minimizing the average cost, differentating the above equation, [tex]AC′(x)= \frac{ d(\frac{36100}{x} + 800 + x)}{dx}[/tex]

[tex]= \frac{- 36100}{x²} + 1[/tex]

For critical value plug AC'(x) = 0

[tex]\frac{- 36100}{x²} + 1 = 0[/tex]

=> x² - 3600 = 0

=> x = ± 60

As the quantity must be positive so x = 60.

e) Now we will compute the minimum average value at x = 60,

[tex] A C(60) = \frac{36100}{60} + 800 + 60[/tex]

= 1,461.67

Hence, required value is 1,461.67.

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The net force on a vehicle is directly proportional to its acceleration and mass, according to Newton's Second Law of Motion. Therefore, we can use the equation F = ma, where F is the net force, m is the mass of the vehicle, and a is the acceleration.

We know that the net force on the vehicle is 1800 and its acceleration is 1.5. Substituting these values into the equation, we get:
1800 = m × 1.5

To solve for m, we need to isolate it on one side of the equation. Dividing both sides by 1.5, we get:

m = 1800 ÷ 1.5

m = 1200

Therefore, the mass of the vehicle is 1200 kilograms to the nearest kilogram

Net force = mass × acceleration

In this case, the net force on the vehicle is 1800 N (Newtons), and it is accelerating at a rate of 1.5 m/s² (meters per second squared). We can rearrange the formula to solve for mass:

Mass = net force ÷ acceleration

Now, plug in the given values:

Mass = 1800 N ÷ 1.5 m/s²

Mass ≈ 1200 kg

To the nearest kilogram, the mass of the vehicle is approximately 1200 kg.

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The value of the function (f + g)(x)​, (f − g)(x)​, and (f ⋅ g)(x)​ will be 3x² - 5x + 9, x² - 5x + 1, and 4x⁴ - 5x³ - 3x² + 20x - 20, respectively.

Given that:

Function, f(x) = 2x² - 5x + 5 and g(x) = x² + 4

The function (f + g)(x) is calculated as,

(f + g)(x) = f(x) + g(x)

(f + g)(x) = 2x² - 5x + 5 + x² + 4

(f + g)(x) = 3x² - 5x + 9

The function (f − g)(x) is calculated as,

(f − g)(x) = f(x) - g(x)

(f − g)(x) = 2x² - 5x + 5 - x² - 4

(f − g)(x) = x² - 5x + 1

The function (f ⋅ g)(x)​ is calculated as,

(f ⋅ g)(x) = f(x) ⋅ g(x)

(f ⋅ g)(x) = (2x² - 5x + 5) ⋅ (x² - 4)

(f ⋅ g)(x) = 4x⁴ - 5x³ + 5x² - 8x² + 20x - 20

(f ⋅ g)(x) = 4x⁴ - 5x³ - 3x² + 20x - 20

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The complete question is given below.

The functions are f(x) = 2x² - 5x + 5 and g(x) = x² + 4. Find:

(f+g)(x)​, (f−g)(x)​, and (f⋅g)(x)​

Based on the information, it is 3 miles from the school to the grocery store.

Looking at the map, we can see that the school is located at (-4, 5) and the grocery store is located at (-5, 4). The horizontal distance between them is 1 unit, and the vertical distance is also 1 unit.

Therefore, the total distance between the school and the grocery store is:

Distance = (horizontal distance) x (distance per unit) + (vertical distance) x (distance per unit)

Distance = 1 x 1.5 miles + 1 x 1.5 miles

Distance = 3 miles

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So, there are 20 possible outfits.

An outfit like t-shirts is a group of garments that have been specifically chosen or created to be worn together. A firm, organisation, or group that collaborates closely is referred to as an outfit. It may be used as a verb to signify to supply with the right tools.

The term outfit can be used to refer to coordinated clothing, such as a shirt and trousers that you usually wear to job interviews. From out- + fit (v.), "act of fitting out (a ship, etc.) for an expedition," 1769. The broader sense of "articles and equipment required for an expedition" is documented in American English from 1787.

To calculate the number of outfits possible, we need to multiply the number of options for each item.

Number of options for jeans = 2 pairs = 2

Number of options for t-shirts = 5

Number of options for shoes = 2 pairs = 2

Therefore, the total number of possible outfits is:

2 x 5 x 2 = 20

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Petra and Jonah are traveling separately and not meeting up in Poole. In this case, the 47 minutes that Jonah mentions could refer to the total travel time from his home to his destination (which might not be Poole).

It's possible that Petra and Jonah are not starting their journey from the same location, or that they are using different modes of transportation to get to the train station. Here are a few possible scenarios that could explain how Petra and Jonah could get from home to Poole in less than 60 minutes:

Petra lives closer to the train station than Jonah, so she only needs to travel a short distance to get there. Jonah, on the other hand, lives farther away and needs to take a bus or drive to the train station. Petra could arrive at the train station in a few minutes, take the 12-minute train ride to Poole, and get there in under 30 minutes total. Jonah, who has a longer journey to the train station, might take 40-50 minutes to get there, but could still arrive in Poole in less than 60 minutes if he catches a train shortly after arriving at the station.

Petra and Jonah live in the same area, but Petra prefers to walk or bike to the train station while Jonah takes a bus or drives. If Petra's home is closer to the train station than Jonah's, she could arrive in 10-15 minutes and take the 12-minute train ride to Poole, arriving in under 30 minutes total. Jonah might take longer to get to the station, but could still arrive in Poole in less than 60 minutes if he catches a train shortly after arriving.

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Petra and Jonah has this information home to the train station. 12 minutes train to Poole 47 minutes jonah says it will take less that 60 minutes in total to go from home to Poole. ​

How does this occur?

The best prediction for the possible number of times both coins will land on tails would be = 1/2

To calculate the possible outcome of the event the formula that should be used is given as follows:

probability = possible outcome/sample space.

sample space for a coin tossed 44 times = 44×2 = 88.

for two coins = 88×2 = 176

Possible sample space = 176/2 = 88

probability = 88/176

= 1/2

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The ending view of the story is that the human made a vow to return to the land that God had created and to preserve its beauty and bounty for generations to come.

Once upon a time, we have beautiful country with green forests, crystal clear rivers and snow-capped mountains. It was a paradise with fresh air and abundant wildlife because he created the land with His own hands and it was a sight to behold.

But, as time passed, people began to leave the countryside and move to the cities in search of work and prosperity. They built towering skyscrapers and sprawling suburbs which leaves the countryside behind. The towns grew and prospered but had problems of pollution, traffic, and crime and people longed for the peace and simplicity of the countryside. They had forgotten that God had made a country, and man had made the town.

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The radius and height of a real soda can with a double thickness in the top and bottom surfaces that minimize its surface area are approximately 2.89 cm and 13.15 cm, respectively. These dimensions are closer to the dimensions of a real soda can compared to the dimensions obtained in part (a).

a. To minimize the surface area of a cylindrical soda can, we need to find the values of radius and height that minimize the surface area equation.

Let's denote the radius of the can as r and the height as h. The volume of the can is given as 354 cm^3, so we have:

πr^2h = 354

Solving for h, we get:

h = 354 / (π[tex]r^2[/tex])

The surface area of the can can be calculated as follows:

A = 2πr^2 + 2πrh

Substituting the expression for h in terms of r, we get:

A = 2πr^2 + 2πr(354 / πr^2)

Simplifying:

A = 2πr^2 + 708 / r

To minimize the surface area, we need to find the value of r that makes the derivative of A with respect to r equal to zero:

dA/dr = 4πr - 708 / r^2

Setting dA/dr = 0, we get:

4πr = 708 / r^2

Multiplying both sides by r^2, we get:

4πr^3 = 708

Solving for r, we get:

r = (708 / 4π)^(1/3) ≈ 3.64 cm

Substituting this value of r back into the expression for h, we get:

h = 354 / (π(3.64)^2) ≈ 9.29 cm

Therefore, the radius and height of the cylindrical soda can with minimum surface area and volume of 354 cm^3 are approximately 3.64 cm and 9.29 cm, respectively.

b. Real soda cans do not seem to have an optimal design because their dimensions are not the same as the ones obtained in part (a). The radius of a real soda can is 3.1 cm and the height is 12.0 cm. However, real soda cans have a double thickness in their top and bottom surfaces, which means that their dimensions are not directly comparable to the dimensions of the cylindrical can we calculated in part (a).

To find the dimensions of a real soda can with a double thickness in the top and bottom surfaces that minimize its surface area, we can use the same approach as in part (a), but with the appropriate modification to the surface area equation:

A = 4πr^2 + 708 / r

Setting dA/dr = 0, we get:

8πr^3 = 708

Solving for r, we get:

r = (708 / 8π)^(1/3) ≈ 2.89 cm

Substituting this value of r back into the expression for h, we get:

h = 354 / (π(2.89)^2) ≈ 13.15 cm

Therefore, the radius and height of a real soda can with a double thickness in the top and bottom surfaces that minimize its surface area are approximately 2.89 cm and 13.15 cm, respectively. These dimensions are closer to the dimensions of a real soda can compared to the dimensions obtained in part (a).

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Solving an exponential equation we can see that it will take 23.86 years.

We know that the population is modeled by the exponential equation:

P= 100,000•10^(0.02t)

The initial population is 100,000, so it will triple when P = 300,000

Then the equation we need to solve is:

300,000 = 100,000•10^(0.02t)

Now we can solve this for t.

300,000/100,000 = 10^(0.02t)

3 = 10^(0.02t)

Apply the natural logarithm in both sides:

ln(3) = 0.02*t*ln(10)

t = ln(3)/(0.02*ln(10)) = 23.86

It will take 23.86 years.

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Complete question.

"The population of a town can be modeled using the formula P=20,000e^0.02t , where t is the number of years after 2012 and P is the town's population. Which of the following equations can be used to find the number of years after 2012 that the population will triple to  300,000?"