Evaluate the integrals in Exercises 31–56. Some integrals do notrequire integration by parts. ∫(1+2x^2)e^x^2 dx

Answer:

the correct is D the graph is a decreasing function

dA/dt = πa(db/dt) + πb(da/dt)

b is not changing at this time.

(a) To find dA/dt, we can use the product rule of differentiation:

A = πab
dA/dt = π(db/dt)a + πb(da/dt)
dA/dt = πa(db/dt) + πb(da/dt)  (since a and b can be interchanged)


(b) When a = 880 inches, da/dt = -2 inches/min (since a is decreasing by 2 inches per minute) and A is constant. We can use the formula for A and plug in the given values:

A = πab
π(880)(175) = constant
b = constant/(πa)
db/dt = (-πa constant')/(πa^2)  (using the quotient rule of differentiation)

Substituting the given values, we get:

db/dt = (-π(880)(175)(0))/(π(880)^2)
db/dt = 0 inches/min

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The events A and B are not independent. This can be answered by the concept of Probability.

Two events A and B are considered independent if the occurrence of one event does not affect the probability of the other event occurring. In this case, we are given that the probability of event A occurring is 0.2, the probability of event B occurring is 0.3, and the probability of both events occurring is 0.05.

To determine if the events A and B are independent, we can check if the probability of event A occurring is the same whether or not event B has occurred. Similarly, we can check if the probability of event B occurring is the same whether or not event A has occurred.

Let's start with the probability of event A occurring given that event B has occurred. We can use conditional probability to calculate this:

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

Substituting the given probabilities, we get:

P(A|B) = 0.05 / 0.3 = 1/6

Now, let's compare this to the probability of event A occurring without any knowledge of event B, which is simply given as P(A) = 0.2. Since P(A|B) is not equal to P(A), we can conclude that event A is dependent on event B.

Similarly, we can check if event B is dependent on event A:

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

Substituting the given probabilities, we get:

P(B|A) = 0.05 / 0.2 = 1/4

Again, since P(B|A) is not equal to P(B) = 0.3, we can conclude that event B is also dependent on event A.

Therefore, we can conclude that events A and B are not independent.

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We can say with 99% confidence that the proportion of all students at the college who will vote for Jennifer is between 0.729 and 0.797.

To construct a confidence interval for the proportion of all students at the college who will vote for Jennifer, we can use the following formula:

[tex]CI = p + z\times \sqrt{(p\times(1-p)/n)}[/tex]

where p is the sample proportion, z is the z-score for the desired confidence level, and n is the sample size.

First, we need to calculate the sample proportion:

p = 599/785 = 0.763

Next, we need to find the z-score for a 99% confidence level. From the standard normal distribution table, the z-score for a 99% confidence level is 2.576.

Now we can plug in the values and calculate the confidence interval:

[tex]CI = 0.763 + 2.576\times \sqrt{ (0.763\times (1-0.763)/785)}[/tex]

  = 0.763 ± 0.034

  = (0.729, 0.797)

Therefore, we can say with 99% confidence that the proportion of all students at the college who will vote for Jennifer is between 0.729 and 0.797.

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The amplitude of vibrations after a very long time is 0.

a) The equation of motion of a mass-spring system is given by

m x'' + kx = f(t)

where m is the mass, k is the spring constant and f(t) is the external force. Substituting the given values, we get

6x'' + 54x = 30e−4t cos 5t

The solution of this equation is given by

x(t) = A cos (ωt + θ)

where A is the amplitude, ω is the angular frequency and θ is the phase angle.

Substituting the given values, we get

x(t) = A cos (5t + θ)

At t = 0, x(0) = A cos θ

At t = π, x(π) = A cos (5π + θ)

Therefore, the position of the mass when t = π is given by

x(π) = A cos (5π + θ)

b) The amplitude of vibrations after a very long time is given by

A = x(0) = A cos θ

Therefore, the amplitude of vibrations after a very long time is 0.

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Answer: 180

Step-by-step explanation: 1440 divided by 8 is 180.

To determine if a sequence converges or diverges, we need to find its general term and analyze its behavior as n approaches infinity. The given sequence has the general term:
a(n) = (n + p)(n + 2p)(n^2 + p)


ii. To find the first five terms of the sequence, we will plug in n = 1, 2, 3, 4, and 5:

a(1) = (1 + p)(1 + 2p)(1 + p^2)
a(2) = (2 + p)(2 + 2p)(4 + p^2)
a(3) = (3 + p)(3 + 2p)(9 + p^2)
a(4) = (4 + p)(4 + 2p)(16 + p^2)
a(5) = (5 + p)(5 + 2p)(25 + p^2)

These are the first five terms of the sequence, but their exact values will depend on the value of p.

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The volume of the solid of revolution is approximately 23.5619 cubic units.

To find the volume of the solid of revolution obtained by rotating the region bounded by y = r² and y = 1 about the horizontal line y = 1, you can use the disk method. Here's the step-by-step explanation:

1. First, determine the limits of integration. Since y = r², r = sqrt(y). The curve intersects y = 1 when r² = 1, so r = 1. The limits of integration are from 0 to 1.

2. Next, find the radius of each disk, which is the distance from the curve y = r² to the horizontal line y = 1. The radius is (1 - r²).

3. Now, find the area of each disk. The area is given by A(r) = π(radius)² = π(1 - r²)².

4. Finally, integrate the area function from 0 to 1 to find the volume of the solid of revolution: V = ∫[0,1] π(1 - r²)² dr.

Evaluating the integral, you get V ≈ 23.5619.

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Answer:

Step-by-step explanation:

easy its a

Household earning $55000 or more will act as larger potential customer base .

Given,

Earnings of household.

Here,

Earning are categorised on two incomes.

1st : Household earning $55000 or more will likely go to new shoe store .

2nd : Household earning $25000 or more will likely go to coffee shop .

Thus the household that earns more money will become potential customers for more number of things rather than household earnings less amount .

So, The households having more income will become larger potential customer .

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The indefinite integral of ∫(¹¹√x + ¹²√x)dx is (2/3)[tex]x^{\frac{3}{2}[/tex] + C₁ + (2/3)[tex]x^{\frac{5}{2}[/tex] + C₂ where C₁ and C₂ are constants of integration.

To find the indefinite integral of ∫(¹¹√x + ¹²√x)dx, we can use the linearity property of integration which states that the integral of a sum of functions is equal to the sum of their integrals.

Using this property, we can break down the given expression into two separate integrals:

∫(¹¹√x)dx + ∫(¹²√x)dx

To evaluate these integrals, we can use the power rule of integration, which states that the integral of xⁿ is equal to (1/(n+1))x^⁽ⁿ⁺¹⁾ + C, where C is the constant of integration.

Using this rule, we get:

∫(¹¹√x)dx = (2/3)[tex]x^{\frac{3}{2}[/tex] + C₁

∫(¹²√x)dx = (2/3)[tex]x^{\frac{5}{2}[/tex] + C₂

Therefore, the indefinite integral of ∫(¹¹√x + ¹²√x)dx is:

(2/3)[tex]x^{\frac{3}{2}[/tex] + C₁ + (2/3)[tex]x^{\frac{5}{2}[/tex] + C₂

where C₁ and C₂ are constants of integration.

In summary, to find the indefinite integral of a sum of functions, we can break it down into separate integrals and use the power rule of integration to evaluate each integral.

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Answer:

S = 2π(7^2) + π(7)(15) = 637.42 square cm.

D is correct.

The exact surface area is 203π square cm., or about 637.74 square cm.

A = 1/2 ∫[(3sinθ)² - (1 + sinθ)²] dθ from θ = π/6 to θ = 5π/6

To find the area of the region that lies inside the circle r = 3sinθ and outside the cardioid r = 1 + sinθ, you need to set up an integral in terms of θ. First, find the points of intersection by setting the equations equal to each other:

3sinθ = 1 + sinθ

Solve for θ to find the points of intersection:

2sinθ = 1
sinθ = 1/2
θ = π/6, 5π/6

Now, set up the integral for the area. The area of a polar curve is given by the formula:

A = 1/2 ∫(r² dθ)

So the integral for the area inside the circle and outside the cardioid is:

A = 1/2 ∫[(3sinθ)² - (1 + sinθ)²] dθ from θ = π/6 to θ = 5π/6

Do not evaluate the integral, as per the instructions. This expression represents the area of the region that lies inside the circle and outside the cardioid.

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The correct answer is not (B). The critical value or values of X2 based on the given information are not -14.573 and 14.573. The correct answer is (C) 16.151, 40.113.

To find the critical value or values of X2, we need to refer to the Chi-Square distribution table or use a calculator that can calculate Chi-Square probabilities.

Given information:

Hypothesis: H1: σ ≠9.3 (which means the population standard deviation is not equal to 9.3)

Sample size: n = 28

Significance level: α = 0.05 (which corresponds to a 95% confidence level)

We need to find the critical value or values of X2 at a significance level of 0.05 with 27 degrees of freedom (n - 1 = 28 - 1 = 27) because we are dealing with a sample size of 28.

Using a Chi-Square distribution table or a calculator, the critical value of X2 at a significance level of 0.05 with 27 degrees of freedom is found to be 40.113. Since X2 is always positive, we only need to consider the upper tail of the Chi-Square distribution. Therefore, the critical value or values of X2 based on the given information are 16.151 and 40.113.

Therefore, the correct answer is (C) 16.151, 40.113.

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The normal vector of the tangent plane to the surface defined by (b) <-44e168, -80e168, 0>.

The normal vector of the tangent plane to a surface at a given point is perpendicular to the tangent plane and points outward from the surface. In this case, the surface is defined by the equation z = ex² + 18xy + 2y², and the point of interest is (-4, -2, e⁹⁶).

To find the normal vector, we need to calculate the gradient of the surface at the given point, which involves finding the partial derivatives of the surface equation with respect to x, y, and z, and evaluating them at the point (-4, -2, e⁹⁶).

The resulting vector will be the normal vector of the tangent plane at that point.

Taking the partial derivatives of the surface equation, we get:

∂z/∂x = 2ex² + 18y

∂z/∂y = 18x + 4y

Evaluating these partial derivatives at (-4, -2, e⁹⁶), we get:

∂z/∂x at (-4, -2, e⁹⁶) = 2e(-4)^2 + 18(-2) = -44e¹⁶⁸

∂z/∂y at (-4, -2, e⁹⁶) = 18(-4) + 4(-2) = -80e¹⁶⁸

Hence , the normal vector of the tangent plane at the point (-4, -2, e⁹⁶) is <-44e¹⁶⁸, -80e¹⁶⁸, 0>, which corresponds to option (b) <-44e168, -80e168, 0>.

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The probability P(−1.60 ≤ Z ≤ 0) is 0.44500.

The probability P(−1.60 ≤ Z ≤ 0) can be found using a standard normal distribution table or calculator.

Using a standard normal distribution table, we can look up the area under the curve between z = −1.60 and z = 0, which is 0.44500. Therefore, the answer is 0.44500.

Alternatively, we can use a calculator that can calculate probabilities for a standard normal distribution. In this case, we would enter the following: P(−1.60 ≤ Z ≤ 0) = normdist(0, 1, 0, TRUE) − normdist(-1.60, 1, 0, TRUE), which also gives us 0.44500 as the answer.

Therefore, the probability P(−1.60 ≤ Z ≤ 0) is 0.44500.
To find the probability P(-1.60 ≤ Z ≤ 0), we'll use the standard normal distribution table or Z-table.

Step 1: Look up the Z-scores in the standard normal distribution table.
For Z = -1.60, the table value is 0.0548, which represents the probability P(Z ≤ -1.60).
For Z = 0, the table value is 0.5000, which represents the probability P(Z ≤ 0).

Step 2: Calculate the probability P(-1.60 ≤ Z ≤ 0).
Subtract the probability of Z ≤ -1.60 from the probability of Z ≤ 0.
P(-1.60 ≤ Z ≤ 0) = P(Z ≤ 0) - P(Z ≤ -1.60)
P(-1.60 ≤ Z ≤ 0) = 0.5000 - 0.0548

Step 3: Solve for the probability.
P(-1.60 ≤ Z ≤ 0) = 0.4452

Therefore, the probability P(-1.60 ≤ Z ≤ 0) is approximately 0.4450.

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The open intervals on which f(x) is concave up are (-1/√6, 1/√6) and the open intervals on which f(x) is concave down are (-∞, -1/√6) and (1/√6, ∞). The x-coordinates of the inflection points are x = ±1/√6.

To determine where f(x) is concave up or down, we need to find the

second derivative of f(x) and examine its sign. The second derivative of

f(x) is:

[tex]f''(x) = -18x^2 + 3[/tex]

To find the intervals where f(x) is concave up, we need to solve the

inequality:

f''(x) > 0

[tex]-18x^2 + 3 > 0[/tex]

Solving this inequality, we get:

[tex]x^2 < 1/6[/tex]

-1/√6 < x < 1/√6

Therefore, f(x) is concave up on the interval (-1/√6, 1/√6).

To find the intervals where f(x) is concave down, we need to solve the inequality:

f''(x) < 0

[tex]-18x^2 + 3 < 0[/tex]

Solving this inequality, we get:

[tex]x^2 > 1/6[/tex]

x < -1/√6 or x > 1/√6

Therefore, f(x) is concave down on the intervals (-∞, -1/√6) and (1/√6, ∞).

To find the inflection points, we need to find the x-coordinates where the

concavity changes, i.e., where f''(x) = 0 or is undefined.

From [tex]f''(x) = -18x^2 + 3[/tex], we see that f''(x) is undefined at x = 0. At x = ±1/

√6, f''(x) changes sign from positive to negative or vice versa, so these

are the inflection points.

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Step-by-step explanation:

The given line has slope , m = 4   parallel will be the same value m

  using the point (-1.6)   and slope m = 4

     y - 6 = 4(x -  -1)

(Which reduces to        y - 6 = 4 ( x+1) )

lim(x→0) sin(x) = 0

To show that lim(x→0) sin(x) = 0, we will use the squeeze theorem, which states that if a function g(x) is bounded between two other functions f(x) and h(x) such that lim(x→a) f(x) = lim(x→a) h(x) = L, then lim(x→a) g(x) = L.

Here, f(x) = -x, g(x) = sin(x), and h(x) = x. The hint given is that -x ≤ sin(x) ≤ x for all x ≥ 0.

As x approaches 0, both f(x) and h(x) also approach 0:

lim(x→0) -x = 0 and lim(x→0) x = 0

Now, we apply the squeeze theorem. Since -x ≤ sin(x) ≤ x and both f(x) and h(x) have a limit of 0 as x approaches 0, then:

lim(x→0) sin(x) = 0

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The data provided does indicate that due to the passing of the new law the number of reported crimes have dropped.

Based on the data provided for the six neighborhoods, we want to determine if the new law, which gave city police greater powers in apprehending suspected criminals, has led to a decrease in the number of reported crimes.

1. Neighborhood 1: The number of reported crimes increased from 18 to 21.

2. Neighborhood 2: The number of reported crimes decreased from 35 to 23.

3. Neighborhood 3: The number of reported crimes decreased from 44 to 30.

4. Neighborhood 4: The number of reported crimes decreased from 28 to 19.

5. Neighborhood 5: The number of reported crimes increased from 22 to 24.

6. Neighborhood 6: The number of reported crimes decreased from 37 to 29.

Out of the six neighborhoods, four experienced a decrease in the number of reported crimes, while two experienced an increase.

Based on this comparative analysis, it can be indicated that the number of reported crimes has generally dropped in the majority of the neighborhoods (4 out of 6) after the new law was implemented. However, it's important to consider additional factors and data to draw a more comprehensive conclusion about the law's overall effectiveness.

Note: The question is incomplete. The complete question probably is: A new law has been passed giving city police greater powers in apprehending suspected criminals. For six neighborhoods, the numbers of reported crimes one year before and one year after the new law are shown. Does this indicate that the number of reported crimes have dropped?

Neighborhood 1 2 3 4 5 6

Before 18 35 44 28 22 37

After 21 23 30 19 24 29

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The mean of the distribution of sample means (also known as the expected value of the sample mean) is equal to the population mean, which is u = 118.9.

The standard deviation of the distribution of sample means (also known as the standard error of the mean) is equal to the population standard deviation divided by the square root of the sample size. Therefore,

o/sqrt(n) = 22.3/sqrt(94) = 2.30 (rounded to 2 decimal places)

So the standard deviation of the distribution of sample means is 2.30.

For a population with parameters μ = 118.9 (mean) and σ = 22.3 (standard deviation), if you draw a random sample of size n = 94, the mean of the distribution of sample means (us) is equal to the population mean, which is:

us = μ = 118.9

The standard deviation of the distribution of sample means, also known as the standard error, can be calculated using the formula:

Standard Error (SE) = σ / √n

In this case:

SE = 22.3 / √94 ≈ 2.30

So, the standard deviation of the distribution of sample means is approximately 2.30 (accurate to 2 decimal places).

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A corporation is creating a sinking fund to replace machinery in 3 years. In order to have $340,000 in the fund, they need to calculate how much to place in the account at the end of each month. Assuming an annual interest rate that is compounded monthly, the answer is $61,025 rounded to the nearest cent.

To calculate how much interest they would earn over the life of the account, we would need to know the interest rate.

To determine the value of the fund after 2, 4, and 6 years, we would need to know the interest rate and the amount placed in the account each month.

To calculate how much interest was earned during the second month of the 4th year, we would need to know the interest rate and the amount in the fund at that time.

A corporation creates a sinking fund to have $340,000 in 3 years for machinery replacement. The account's annual interest rate is compounded monthly. To determine the monthly deposit amount, we can use the sinking fund formula:

FV = PMT * (((1 + r)^nt - 1) / r)

where FV is the future value of the account ($340,000), PMT is the monthly deposit amount, r is the monthly interest rate, n is the number of times the interest is compounded per year (12 for monthly), and t is the number of years (3 in this case).

We need to solve for PMT:

$340,000 = PMT * (((1 + r)^36 - 1) / r)

To find the monthly deposit amount, we need the annual interest rate (not provided in the question). Once we have the interest rate, we can find the PMT value and calculate the interest earned over the account's life, as well as the fund's value after 2, 4, and 6 years. Additionally, we can determine the interest earned during the second month of the 4th year.

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To find the total cost of producing 42 units, we will consider both the fixed costs and the variable costs. Since the marginal cost (MC) is given as $170.017 per unit, we can calculate the variable costs by multiplying MC by the number of units produced.

Variable costs = MC × Number of units = 170.017 × 42 = $7,140.714
Now, we'll add the fixed costs to the variable costs to get the total cost:
Total cost = Fixed costs + Variable costs = $3,350 + $7,140.714 = $10,490.714
Rounding to the nearest cent, the total cost of producing 42 units is $10,490.71.

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(a) The exponential equation to represent the amount of money in the account after t years is [tex]A(t) = 1600(1.0103)^{(4t)}[/tex].

(b) On solving the  exponential equation the amount of money that will be in the account after 7 years is $2,177.61.

What is an exponential function?

The formula for an exponential function is [tex]f(x) = a^x[/tex], where x is a variable and a is a constant that serves as the function's base and must be bigger than 0.

(a) The exponential function to represent the amount of money in the account after t years with quarterly compounding is -

[tex]A(t) = P(1 + \frac{r}{n})^{(nt)}[/tex]

where -

P = initial deposit = $1600

r = annual interest rate = 4.12%

n = number of compounding periods per year = 4 (since interest is compounded quarterly)

t = time in years

Substituting the given values, in the equation we get -

[tex]A(t) = 1600(1 + \frac{0.0412}{4})^{(4t)}[/tex]

Simplifying -

[tex]A(t) = 1600(1.0103)^{(4t)}[/tex]

Therefore, the equation is [tex]A(t) = 1600(1.0103)^{(4t)}[/tex].

(b) To find the amount of money in the account after 7 years, we need to substitute t = 7 in the equation -

[tex]A(7) = 1600(1.0103)^{(4\times7)}[/tex]

A(7) = 1600(1.3610)

A(7) = $2,177.61 (rounded to the nearest cent)

Therefore, the amount of money in the account after 7 years, assuming no additional deposits or withdrawals, will be $2,177.61.

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Answer:

A: A = P(1 + r/n)^nt

B: $2131.72

Step-by-step explanation:

A = P(1 + r/n)^nt

A = 1600(1 + .0412/4)^(4)(7)

A = 1600(1 + .0103)^(28)

A = 1600(1.0103)^(28)

A = $2131.72

The total amount accrued, principal plus interest, on a principal of $1600 at a rate of 4.12% per year compounded 4 times a year over 7 years is $2131.72.

The total weight of the container's contents is 11 pounds.

The container's volume can be estimated by multiplying the length, breadth, and height:

2 feet * 2 feet * 12.5 feet equals 50 cubic feet

Because the contents weigh 0.22 pounds per cubic foot, calculating the volume by the weight per cubic foot yields the total weight of the contents:

50 cubic feet * 0.22 pounds per cubic foot = 11 pounds

As a result, the total weight of the container's contents is 11 pounds.

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The area of the region enclosed by y = ln(x) is e - 1.

The area of the region enclosed by y = ln(x), the x-axis, the y-axis, and y = 1.

(A) Using the method of horizontal slices (dx), we can integrate with respect to x:

The limits of integration are x = 1 (where the curves intersect) and x = e (where y = 1).

The height of the slice is y = 1 - ln(x)

Therefore, the area is given by:

A = ∫[1,e] (1 - ln(x)) dx

= x - x ln(x) |[1,e]

= e - e ln(e) - 1 + 1 ln(1)

= e - 1

Therefore, the area of the region is e - 1 square units.

(B) Using the method of vertical slices (dy), we can integrate with respect to y:

The limits of integration are y = 0 (where the curve intersects the x-axis) and y = 1.

The width of the slice is x = [tex]e^y[/tex]

Therefore, the area is given by:

A = ∫[0,1] [tex]e^y[/tex] dy

= [tex]e^y[/tex] |[0,1]

= e - 1

Therefore, the area of the region is e - 1 square units, which is the same as the result obtained using horizontal slices.

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Sarina used 51/10 or 5.1 cups of brown sugar for her 8.5 batches of sugar cookies.

To find out how many cups of brown sugar Sarina used for 8.5 batches of sugar cookies.

We need to multiply the number of batches (8.5) by the amount of brown sugar per batch (3/5 cups).

We are doing the step by step calculation,

1. Write down the given values: 8.5 batches and 3/5 cups of brown sugar per batch.

2. Multiply the number of batches (8.5) by the amount of brown sugar per batch (3/5 cups): 8.5 × (3/5).

To perform the multiplication: (8.5) × (3/5) = (17/2) × (3/5) = (17×3) / (2×5) = 51/10

Hence, Sarina used 51/10 or 5.1 cups of brown sugar for her 8.5 batches of sugar cookies.

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The formula for f(t) is calculated to be f(t) = 15e^(-0.1714t)

Let's start with the given information: the rate of change of f(t) is proportional to the value of f(t) at any time. This means that we can write:

f'(t) = k*f(t)

where k is the proportionality constant. To solve for f(t), we need to find the value of k.

We know that 15 units of medicine were injected initially, and after 35 minutes, only 9 units remained. Let's use this information to find k.

We can write the following equation to represent the rate of change of f(t):

f'(t) = -r*f(t)

where r is the rate at which the medicine is leaving the dog's body. We know that after 35 minutes, 6 units of medicine were used, so:

r = (6 units) / (35 minutes) = 0.1714 units/minute

Now we can solve for k by using the given information that at t=0, f(0) = 15:

f'(t) = k*f(t)

f'(0) = kf(0) = -rf(0)

k = -r = -0.1714

So now we have k and we can solve for f(t) using the differential equation:

f'(t) = -0.1714*f(t)

Separating variables and integrating, we get:

ln(f(t)) = -0.1714*t + C

where C is the constant of integration. Solving for f(t), we get:

f(t) = e^(-0.1714*t + C)

To find the value of C, we use the initial condition f(0) = 15:

f(0) = e^(C) = 15

C = ln(15)

So the final formula for f(t) is:

f(t) = 15e^(-0.1714t)

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The equation of the line with value of a = 20 in the standard form ax + by  =c is equal to 20x - 5y = -10.

An equation of the line is,

ax + by = c

Equation is,

x + 4y = 19

Equation can be rearranged into the standard form,

⇒ x + 4y = 19

⇒4y = -x + 19

⇒y = (-1/4)x + (19/4)

Line is perpendicular to this line,

⇒ Slope is the negative reciprocal of (-1/4).

m = -1/m₁

   = -1/(-1/4)

   = 4

Since the line has y-intercept 2,

Use the point-slope form of the equation of a line

Then the equation of the line is,

y - y₁= m(x - x₁)

Substitute the values we have,

⇒ y - 2 = 4(x - 0)

⇒y - 2 = 4x

Rearranging this equation into the desired form ax + by = c, we get,

-4x + y =2

Multiplying both sides by -5 to ensure that a = 20

And there are no common factors between a, b, and c,

20x - 5y = -10

Therefore, the equation of the line in the desired form is 20x - 5y = -10.

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Answer:

8i + 34j

Step-by-step explanation:

[tex]4v - 2w \\ 4(5i + 4j) - 2(6i - 9j) \\ 20i + 16j - 12i + 18j \\ 20i - 12i + 16j + 18j \\ 8i + 34j[/tex]

An alternative method involves using the divisibility rule of 3, which states that a number is divisible by 3 if the sum of its digits is divisible by 3.

Since we are looking for multiples of 3, we can apply this rule to the digits of each number in the allowed set. We notice that the digits 1, 2, and 6 are already multiples of 3, so any combination of these digits will be a multiple of 3. Similarly, the digits 3 and 9 are also multiples of 3, but they are not in the allowed set. Therefore, any combination of the allowed digits that contains 3 or 9 will not be a multiple of 3.

Now, we consider the even numbers. An even number is a multiple of 2, so it must end in either 2 or 8.

Therefore, any combination of the allowed digits that ends in any other digit will not be even. Using this information, we can generate a list of all possible combinations of the allowed digits that end in 2 or 8.

Then, we check which ones have a sum of digits that is a multiple of 3. Finally, we eliminate any combination that repeats digits.

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Complete Question:

Ms. Hannabal assigned the students in her math class the task of finding all the multiples of 3 that even numbers and contain only the digits 1, 2, 3, 6, or 8. She told them not to repeat digits within a number, so a number like 222 would not be considered a solution

this was the result:

126, 128, 132, 136, 138, 162, 164, 168, 186, 212, 216, 218, 312, 316, 318, 362, 364, 368, 612, 614, 618, 812, 814, 816, 832, 834, 836

students identified 10 even three-digit multiples of 3, and these numbers are 126, 138, 162, 168, 186, 312, 318, 362, 368, and 618.

How could this be solved in a different way?