In the driest part of an Outback ranch, each cow needs about 40 acres for grazing. Use an equation to find how many cows can graze on 720 acres of land.

The number of items that will maximize the profit is 2000. Thus, the correct answer is option c.

To calculate the maximum profit that can be earned we have to differentiate the equation and find the value of x

dP/dx = 1/dx (-3x² + 12x + 75)

= -6x + 12

Calculating dP/dx = 0

0 = -6x + 12

6x = 12

x = 2

Next, we calculate the next differential of the equation:

It comes out to be -6

Since it is smaller than zero, the value of x calculated is the maxima.

The maxima = 2

Thus, the item that will maximize the profit comes out to be 2000 as x is the number of items made in thousand.

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Thus, the height of cone for the given values of circumference an f volume is found as: 4 cm.

A tri shape that resembles a cone is what is known as a conical shape. A cone has a flat end that gradually taper towards a single point at the top known as the apex. Most commonly, a conical shape's flat end has an oval or circular shape. Conical shapes are on your mind when you imagine an ice cream cone with only a pointed end.

Volume of a cone = 1/3 * π *r²*h

Given that:

using circumference c = 6π

c = 2πr (for circular base)

6π  = 2πr

r = 3 cm

Now, using the volume;

Volume of a cone = 1/3 * π *r²*h

1/3 * π *3²*h = 12π

3h = 12

h = 4 cm

Thus, the height of the cone for the given values of circumference an f volume is found as: 4 cm.

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There are approximately 500 times more high school football players than professional football players in the United States.

To determine how many times more high school football players there are than professional football players in the United States, we need to divide the number of high school players by the number of professional players:

1 x 10⁶ / 2 x 10³ = 500

Therefore, there are approximately 500 times more high school football players than professional football players in the United States.

We can determine this by dividing the two numbers and finding the ratio of high school players to professional players. The result tells us how many times greater the number of high school players is than the number of professional players. In this case, the ratio is 500:1, which means that for every professional football player, there are 500 high school football players.

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The marked points are (-2,-4), (-2,-1), (0,-2), (2,-1), and (2,-4), under the condition that they are on the same coordinate plane having (A) y=x-2, (B) y=-x-2, (C) y=|x|-2.

In the given graph points on the coordinate plane, we have to plot the points (x,y)
Here
x = horizontal axis
y = vertical axis.

In the given point A, y=x-2, we can continue at the origin (0,0) and move 2 units go down on the y-axis and 2 units right on the x-axis to plot point A at (2,0).

In the given point B, y=-x-2, we can continue at the origin (0,0) and transfer 2 units down on the y-axis and 2 units left on the x-axis to plot point B at (-2,0).

In the given point C, y=|x|-2, we can continue plotting two points for this equation.
When x is considered negative, we can procees at the origin (0,0) and transfer 2 units down on the y-axis and 2 units left on the x-axis to plot point C at (-2,0).
When x is positive, we can start at the origin (0,0) and move 2 units down on the y-axis and 2 units right on the x-axis to plot point C at (2,0).

Then, all points (x,y) such that (A) y=x-2, (B) y=-x-2, (C) y=|x|-2 are (-2,-4), (-2,-1), (0,-2), (2,-1), and (2,-4).
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To use Lagrange multipliers, we need to define the Lagrangian function:

L(x, y, z, λ) = xyz + λ(x + y + z - 9)

Now, we need to find the partial derivatives of L with respect to x, y, z, and λ and set them equal to 0:

∂L/∂x = yz + λ = 0
∂L/∂y = xz + λ = 0
∂L/∂z = xy + λ = 0
∂L/∂λ = x + y + z - 9 = 0

From the first three equations, we can see that:

yz = -λ
xz = -λ
xy = -λ

Multiplying these equations together, we get:

(xyz)^2 = (-λ)^3

Substituting λ = -yz into the fourth equation, we get:

x + y + z - 9 = 0

Substituting λ = -yz into the first equation and solving for x, we get:

x = -λ/yz = (yz)^2/(-yz) = -y^2z^2

Similarly, we can solve for y and z:

y = -x^2z^2
z = -x^2y^2

Substituting these expressions into the constraint equation, we get:

(-y^2z^2) + (-x^2z^2) + (-x^2y^2) - 9 = 0

Simplifying and solving for xyz, we get:

xyz = sqrt(9/(x^2 + y^2 + z^2))

To maximize xyz, we need to minimize x^2 + y^2 + z^2. Therefore, we can set:

x^2 + y^2 + z^2 = 3

Substituting this into the expressions for x, y, and z, we get:

x = -y^2z^2
y = -x^2z^2
z = -x^2y^2

Substituting these expressions into xyz, we get:

xyz = sqrt(9/3) = 3

Therefore, the maximum value of f(x, y, z) = xyz subject to the constraint x + y + z - 9 = 0 is 3.
To solve this problem using Lagrange multipliers, we first set up the Lagrangian function L(x, y, z, λ) with the constraint function g(x, y, z) = x + y + z - 9.

L(x, y, z, λ) = f(x, y, z) - λ(g(x, y, z))

L(x, y, z, λ) = xyz - λ(x + y + z - 9)

Now we take the partial derivatives with respect to x, y, z, and λ, and set them equal to 0:

∂L/∂x = yz - λ = 0
∂L/∂y = xz - λ = 0
∂L/∂z = xy - λ = 0
∂L/∂λ = x + y + z - 9 = 0 (the constraint)

From the first three equations, we get:

yz = xz = xy

Since x, y, and z are positive, we can divide the first two equations:

y/z = x/z => y = x
x/z = y/z => x = y

So x = y = z. Now we can use the constraint equation:

x + x + x - 9 = 0 => 3x = 9 => x = 3

Thus, x = y = z = 3. Now we can find the maximum value of f(x, y, z):

f(3, 3, 3) = 3 * 3 * 3 = 27

So the maximum value of f(x, y, z) = xyz subject to the constraint x + y + z - 9 = 0 is 27, and this occurs at the point (3, 3, 3).

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The binomial conditions are not met as the cards are not being replaced, so the independence condition is not met. So, the correct answer is D).

The conditions for a binomial setting are

there are a fixed number of trials,

the trials are independent,

there are only two possible outcomes (success or failure),

the probability of success is constant for each trial.

In this scenario, the first two conditions are met as Lan is turning over the first four cards and they are independent events. The third condition is also met as the success is defined as observing an ace and the failure is observing any other card.

However, the fourth condition is not met as the probability of success changes for each trial. After the first card is turned over, the probability of observing an ace changes for the second trial. Therefore, the scenario does not meet all the conditions for a binomial setting. So, the correct option is D).

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The surface areas of the pyramids are listed below:

In this problem we find six cases of square pyramids, whose surface areas shall be found by means of the following formulas:

A = 4 · 0.5 · b · s + b²

Where:

Now we proceed to determine the surface area of the square pyramid:

Case 1:

A = 4 · 0.5 · (5 in) · (10 in) + (5 in)²

A = 125 in²

Case 2:

A = 4 · 0.5 · (8 ft) · (15 ft) + (8 ft)²

A = 304 ft²

Case 3:

A = 4 · 0.5 · (10 yd) · (16 yd) + (10 yd)²

A = 420 yd²

Case 4:

A = 4 · 0.5 · (3 yd) · (8 yd) + (3 yd)²

A = 57 yd²

Case 5:

A = 4 · 0.5 · (8 in) · (17 in) + (8 in)²

A = 336 in²

Case 6:

A = 4 · 0.5 · (4 ft) · (11 ft) + (4 ft)²

A = 104 ft²

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The value of M is a^4.

Given the information, we can express the given logarithms as follows:

1) log_a(MN) = 6
2) log_a(N/M) = 2
3) log_a(N^m) = 16

From equation (1), we can write:
MN = a^6

From equation (2), we can write:
N/M = a^2 → N = a^2 * M

Now, substitute N from equation (2) into equation (3):
log_a((a^2 * M)^m) = 16

Using the power rule of logarithms, we get:
m * log_a(a^2 * M) = 16

Since log_a(a^2 * M) = 2log_a(a) + log_a(M) = 2 + log_a(M), we have:
m * (2 + log_a(M)) = 16

We don't have enough information to determine the value of 'm', but we don't need it to find the value of 'M'.

Now, substitute N back into the equation MN = a^6:
M * a^2 * M = a^6

Divide both sides by M * a^2:
M = a^(6-2) = a^4

So, the value of M is a^4.

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the restaurant used 754,040 ounces of cream last year.

What is an Equations?

Equations are statements in mathematics that consist of two algebraic expressions separated by an equals (=) sign, indicating the equivalence between the expressions on either side. Equations can be solved to determine the value of a variable that represents an unknown quantity. A statement that does not have an "equal to" symbol is not considered an equation and is instead referred to as an expression.

If the restaurant used 50% less cream this year compared to last year, then it means that this year's usage is 50% of last year's usage.

Let x be the amount of cream used last year.

Then we can set up the following equation:

x * 50% = 377,020

To solve for x, we need to isolate it on one side of the equation.

x * 50% = 377,020

x = 377,020 / 50%

To convert 50% to a decimal, we divide it by 100:

x = 377,020 / 0.5

x = 754,040

Therefore, the restaurant used 754,040 ounces of cream last year.

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The calculated two double inequalities that define shaded region are 1 ≤ y < 5 and -3 < x ≤ 2

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

The graph

On the graph, we have the following properties

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

1 ≤ y < 5

-3 < x ≤ 2

Hence, the two double inequalities that define shaded region are 1 ≤ y < 5 and -3 < x ≤ 2

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Therefore, the correct answer is option (b) c + 0.92.

The expression that represents Evelyn's grocery bill after the 8% discount is:

c - 0.08c

This can be simplified as:

0.92c

Therefore, the correct answer is option (b) c + 0.92.

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The expression that represents Evelyn's grocery bill after the coupon is applied is 0.92c. Therefore, the correct option is D.

If Evelyn has a coupon that will reduce her grocery bill by 8% and c represents the cost of her groceries, the expression that represents Evelyn's grocery bill after using the coupon is 0.92c. It is determined as follows.

1. The coupon reduces the bill by 8%, which means Evelyn will pay 100% - 8% = 92% of the original cost.

2. Convert the percentage to a decimal: 92% = 0.92

3. Multiply the original cost (c) by the decimal: 0.92c

So, the correct answer is option D: 0.92c.

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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.

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Based on the above,  Frank will need to have about 533 water stations per kilometer for the 30-kilometer trail run.

If each runner is said to have about 250 milliliters (0.25 liters) of water per station and there are said to be 4000 liters of water available in total, we have to calculate the total number of water stations  by:

4000 liters of water ÷ 0.25 liters of water per station

= 16000 stations

we have 30-kilometer run, we have to divide the total number of stations by the distance and it will be:

16000 stations ÷ 30 kilometers

= 533.33 stations per kilometer

Therefore, about 533 water stations per kilometer for the 30-kilometer trail run is needed by Frank.

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see full question below

frank is designing a 30-kilometers trail run water that will be given to runners. if 4000 liters of water is available,  each runner was given about 250 milliliters (0.25 liters) of water per station.  how many water stations will there be 30-kilometer run.

The effect of the student presenting such a monologue would be B. The mean will decrease.

Order the data points:

1. 5, 1. 5, 2, 2. 5, 2. 5, 2. 5, 3, 3. 5, 3. 5, 4

Find the mean ;

= (1. 5 + 1. 5 + 2 + 2. 5 + 2. 5 + 2. 5 + 3 + 3. 5 + 3. 5 + 4) / 10

= 27 / 10

= 2. 7

Then find the new mean after the student presents the monologue:

= ( 0. 5 + 1. 5 + 1. 5 + 2 + 2. 5 + 2. 5 + 2. 5 + 3 + 3. 5 + 3. 5 + 4) / 11

= 2. 5

The mean therefore reduced.

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Answer:

b

Step-by-step explanation:

The equation of the tangent line is y = x - 1.

To find the equation of the tangent line of y=xlog(x) at the point (1,0), we will first need to find the derivative of the function y=xlog(x) with respect to x.

Step 1: Find the derivative of y=xlog(x) with respect to x.
Using the product rule, (uv)' = u'v + uv', where u=x and v=log(x).

u' = derivative of x with respect to x = 1
v' = derivative of log(x) with respect to x = 1/x

Now, apply the product rule:
y' = u'v + uv' = 1*log(x) + x*(1/x) = log(x) + 1

Step 2: Find the slope of the tangent line at the point (1,0).
Evaluate y' at x=1:
y'(1) = log(1) + 1 = 0 + 1 = 1

The slope of the tangent line at (1,0) is 1.

Step 3: Find the equation of the tangent line.
We will use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is the point (1,0) and m is the slope (1).

y - 0 = 1(x - 1)
y = x - 1

The equation of the tangent line of y=xlog(x) at the point (1,0) is y = x - 1.

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The value of x is 6 and the perimeter is 52 unints

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

The figure

If the lines that appear to be tangent are tangent, then we have the following equation

x + 2 = 8

Evaluate the like terms

x = 6

The perimeter is the sum of the side lengths

So, we have

Perimeter = x + 2 + 8 + 5 + 5 + 9 + 9 + 4 + 4

This gives

Perimeter = 6 + 2 + 8 + 5 + 5 + 9 + 9 + 4 + 4

Evaluate

Perimeter = 52

Hence, the perimeter is 52 unints

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Answer: Debnil has 2 Tablespoons of salt.

Step-by-step explanation:

3/1 is the ratio for teaspoons to tablespoons.

Substitute the 1 with the 6. What is six divided by three? 2.

You are an scalene triangle.

You are a scalene triangle.

A scalene triangle is a type of triangle where all three sides have different lengths, and no two angles are congruent. In this case, you mentioned that one of the angles has a measure of 100 degrees.

Here's a simple diagram of a scalene triangle to help illustrate:

    \

     \

      \

       \

        \

         \

In the diagram, the angles are not drawn to scale, but it represents a scalene triangle where one angle measures 100 degrees. The sides of the triangle would have different lengths, distinguishing it from an equilateral or isosceles triangle where at least two sides are congruent.

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The probability of the captain hitting a home run in his next two at-bats is 1/36, and the best simulations to model the situation are using a six-sided number cube or a random number generator between 1 and 60.

Determine the probability that the captain will hit a home run in his next two at-bats and find the best simulation to model the situation.
The probability of the captain hitting a home run in one at-bat is 1/6. To find the probability of hitting a home run in two consecutive at-bats, you can multiply the individual probabilities:

Probability = (1/6) * (1/6) = 1/36

Now let's evaluate the provided simulations:

1. Using a six-sided number cube: Yes, this can be used to model the situation because the probability of hitting a home run (1/6) and not hitting a home run (5/6) can be represented accurately by the numbers 1 and 2-6, respectively.
2. Using a three-sided number cube: No, this cannot be used to model the situation because the probability distribution is not accurately represented with only three sides.
3. Using a coin flip: No, this cannot be used to model the situation because the probability distribution is not accurately represented with only two outcomes (heads and tails).
4. Using a random number generator between 1 and 60: Yes, this can be used to model the situation because the probability of hitting a home run (1/6) and not hitting a home run (5/6) can be represented accurately by the numbers 1-10 and 11-60, respectively.

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In the equation , Felipe is correct.

What is equation?

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.

Here the given equation is (372) (675) =18k7.

We know that the factor is a number that divides the another number and leaves no reminder .

If we divide 18k7 by 372 the we get remainder 675. So 372 is not factor of 18k7.

But 372 is divides the number 18k7.

Hence Felipe is correct.

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Answer:

Step-by-step explanation:

To solve the equation (T/12) + 5 = (T/3) + (T/4), we need to simplify the right-hand side of the equation by finding a common denominator for T/3 and T/4.

The least common multiple of 3 and 4 is 12, so we can rewrite T/3 and T/4 as (4T/12) and (3T/12), respectively. Substituting these expressions into the equation, we get:

(T/12) + 5 = (4T/12) + (3T/12)

Simplifying the right-hand side, we get:

(T/12) + 5 = (7T/12)

Subtracting (T/12) from both sides, we get:

5 = (6T/12)

Simplifying the right-hand side, we get:

5 = (T/2)

Multiplying both sides by 2, we get:

T = 10

Therefore, the solution to the equation is T = 10.

[tex]\sf\longrightarrow \: \frac{t}{12} + 5 = \frac{t}{3} + \frac{t}{4} \\ [/tex]

[tex]\sf\longrightarrow \: \frac{t + 60}{12} = \frac{t}{3} + \frac{t}{4} \\ [/tex]

[tex]\sf\longrightarrow \: \frac{t + 60}{12} = \frac{4t + 3t}{12} \\ [/tex]

[tex]\sf\longrightarrow \: 12(t + 60) = 12(4t + 3t) \\ [/tex]

[tex]\sf\longrightarrow \: 12t + 720 = 48t + 36t \\ [/tex]

[tex]\sf\longrightarrow \: 12t + 720 = 84t \\ [/tex]

[tex]\sf\longrightarrow \: 720 = 84t - 12t\\ [/tex]

[tex]\sf\longrightarrow \: 720 =72t\\ [/tex]

[tex]\sf\longrightarrow \: 72t = 720\\ [/tex]

[tex]\sf\longrightarrow \: t = \frac{720}{72} \\ [/tex]

[tex]\sf\longrightarrow \: t = 10 \\ [/tex]

[tex]\longrightarrow { \underline{ \overline{ \boxed{ \sf{\: \: \: t = 10 \: \: \: }}}}} \: \: \bigstar\\ [/tex]

To calculate the interest earned by Luke and Alexa, we can use the following formulas:

For compound interest:

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

I = A - P

where:

A = the total amount

P = the principal amount

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the time period (in years)

I = the interest earned

For simple interest:

I = P*r*t

where:

P = the principal amount

r = the annual interest rate (as a decimal)

t = the time period (in years)

I = the interest earned

Using these formulas, we can calculate the interest earned by Luke and Alexa as follows:

For Luke:

P = £4000

r = 0.018 (1.8% as a decimal)

n = 1 (compounded annually)

t = 3 years

A = 4000(1 + 0.018/1)^(1*3) = £4316.83

I = 4316.83 - 4000 = £316.83

For Alexa:

P = £4000

r = 0.018 (1.8% as a decimal)

t = 3 years

I = 4000*0.018*3 = £216

Therefore, the total interest earned by Luke is £316.83 and the total interest earned by Alexa is £216. The difference between these two amounts is:

316.83 - 216 = £100.83

So Luke earned £100.83 more in interest than Alexa over the 3 years.

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The possible values of x that satisfy the equation |x+5| = c are x = c - 5 and x = -c - 5.

what is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.

Assuming you meant to write |x+5|= some value, I can give you a general method to solve equations involving absolute values.

If |a| = b, then either a = b or a = -b. Thus, to solve the equation |x+5| = c, where c is some given value, we can split it into two cases:

Case 1: x+5 = c

Solving for x, we get x = c - 5.

Case 2: -(x+5) = c

Solving for x, we get x = -c - 5.

So, the possible values of x that satisfy the equation |x+5| = c are x = c - 5 and x = -c - 5.

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Chris cannot build the triangular raised bed frame with the given lumber.

To determine if Chris can build her triangular raised bed frame, we need to check if the length of any one of the lumber pieces is greater than the sum of the other two. If this condition is not met, the pieces can be used to build the frame.

Let's check:

4 + 5 = 9 (no)

4 + 9 = 13 (no)

5 + 9 = 14 (yes)

Since the length of the 5-foot and 9-foot lumber pieces add up to be greater than the 4-foot piece, Chris can build her triangular raised bed frame. She can use the 4-foot and 5-foot pieces for the two shorter sides of the triangle and the 9-foot piece for the longer side.

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The value of g'(4) is -1/3 if f is a differential function such that f (2) = 4, f(4) = 6, f'(2) = -4, and f'(6) = -3.

First, let's use the information given to find the equation of the tangent line to f at x=2. We know that f(2) = 4 and f'(2) = -4, so the equation of the tangent line at x=2 is

y - 4 = -4(x - 2)

Simplifying, we get

y = -4x + 12

Now let's use the fact that g(x) = f-1(x) for all x. This means that g(f(x)) = x for all x. We want to find g'(4), which is the derivative of g at x=4.

Using the chain rule, we have

g'(4) = [g(f(4))]'

Since f(4) = 6 and g(f(4)) = g(6) (since g(x) = f-1(x)), we can rewrite this as

g'(4) = [g(6)]'

Now we can use the fact that g(x) = f-1(x) to rewrite g(6) as f-1(6)

g'(4) = [f-1(6)]'

Now we need to find the derivative of f-1(x) with respect to x. To do this, we can use the fact that f(f-1(x)) = x for all x. Differentiating both sides with respect to x using the chain rule, we get

f'(f-1(x)) * (f-1)'(x) = 1

Solving for (f-1)'(x), we get

(f-1)'(x) = 1 / f'(f-1(x))

Now we can plug in x=6 and use the information given to find f'(f-1(6)). Since f(4) = 6, we know that f-1(6) = 4. Therefore

f'(f-1(6)) = f'(4)

Using the tangent line equation we found earlier, we know that f(2) = 4 and f'(2) = -4. Therefore, the slope of the line connecting (2,4) and (4,6) is

(6 - 4) / (4 - 2) = 1

Since the line connecting (2,4) and (4,6) is the tangent line to f at x=2, we know that this slope is equal to f'(2). Therefore

f'(4) = f'(f-1(6)) = f'(4)

Now we can plug in x=6 and f'(4) into our expression for (f-1)'(x)

(f-1)'(6) = 1 / f'(4)

Substituting this into our expression for g'(4), we get

g'(4) = [f-1(6)]' = (f-1)'(6) = 1 / f'(4)

Plugging in f'(4) = f'(f-1(6)) = f'(4), we get

g'(4) = 1 / f'(4) = 1 / (-3) = -1/3

Therefore, g'(4) = -1/3.

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The volume of the cone with the same radius and height as the cylinder is 36 cm³.

To find the volume of a cone with the same radius and height as the cylinder, we first need to find the radius and height of the cylinder.

The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.

We are given that the volume of the cylinder is 108 cm^3.

So, 108 = πr^2h

To solve for r and h, we need more information. However, we can use the fact that the cone has the same radius and height as the cylinder to our advantage.

The formula for the volume of a cone is V = (1/3)πr^2h.

Since the cone has the same radius and height as the cylinder, we can substitute the values of r and h from the cylinder into the cone formula.

V = (1/3)π( r^2 )(h)

V = (1/3)π( r^2 )(108/π)

V = (1/3)( r^2 )(108)

V = 36( r^2 )

Therefore, the volume of the cone with the same radius and height as the cylinder is 36 cm³

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The radius of the cylinder with volume of 141.3and height of 7 cm is 2.53cm.

The formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height. Given that V = 141.3 cm³ and using π ≈ 3.14, we can solve for r.

Rearranging the formula, we get r² = V/(πh), and plugging in the given values, we get r² = 141.3/(3.14*7). Since we don't know the height of the cylinder, we cannot solve for r exactly.

However, we can say that the radius of the cylinder is proportional to the square root of the volume, the height is 7 cm, then r = √(141.3/3.14*7) ≈  2.53cm. If the height is different, the radius will change accordingly.

In summary, using the formula for the volume of a cylinder andheight of 7 cm, the radius of the cylinder with volume 141.3 cm³ and using π ≈ 3.14 is approximately 2.53 cm.

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Complete question:

The volume of the cylinder is 141. 3 cubiccentimeters. What is the radius of the cylinder given that height is 7cm? use  π ≈ 3.14

The unit tangent vector T is (1/√10)i + (3/√10)k

The principal unit normal vector N is j.

vector function r(t) = ti + at²j + 3tk.
Step 1: Find the derivative of r(t) with respect to t, which gives us the tangent vector.
r'(t) = (1)i + (2at)j + (3)k
Step 2: Evaluate r'(t) at t=0.
r'(0) = (1)i + (2a*0)j + (3)k = i + 3k
Step 3: Find the magnitude of r'(0).
|r'(0)| = √(1^2 + 3^2) = √10
Step 4: Normalize r'(0) to find the unit tangent vector T.
T = r'(0) / |r'(0)| = (1/√10)i + (3/√10)k
Step 5: Find the second derivative of r(t) with respect to t.
r''(t) = (0)i + (2a)j + (0)k
Step 6: Evaluate r''(t) at t=0.
r''(0) = (0)i + (2a)j + (0)k = 2aj
Step 7: Find the magnitude of r''(0).
|r''(0)| = √(2a)^2 = 2a
Step 8: Normalize r''(0) to find the principal unit normal vector N.
N = r''(0) / |r''(0)| = (2a/2a)j = j
So, at t=0, the unit tangent vector T is (1/√10)i + (3/√10)k, and the principal unit normal vector N is j.

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The rate at which the total number of bushels of corn currently increases per year depends on the value of "b", which represents the annual increase in yield per acre. If the yield per acre is not increasing (i.e., b = 0), then the rate of increase is a constant 1300 bushels per year.

Let's call the total number of acres the farmer is farming in corn in a given year as "a". We know that initially, a = 100 acres, and that it increases by 30 acres per year. So, in general:

a = 100 + 30t

where "t" is the number of years since the farmer started converting to corn.

Now, let's call the yield in bushels per acre in a given year as "y". We know that initially, y = 130 bushels per acre, and that it increases by "b" bushels per acre per year. So, in general:

y = 130 + bt

Finally, we can calculate the total number of bushels of corn produced in a given year by multiplying the number of acres by the yield per acre:

bushels per year = a * y

Substituting the expressions we have for "a" and "y", we get:

bushels per year = (100 + 30t) * (130 + bt)

Expanding this expression, we get:

bushels per year = 13000 + 1300t + 3900bt + 30tb

Now we can differentiate this expression with respect to time to find how rapidly the total number of bushels of corn currently increases per year:

d(bushels per year)/dt = 1300 + 3900b + 30b

Simplifying, we get:

d(bushels per year)/dt = 1300 + 3930b

So the rate at which the total number of bushels of corn currently increases per year depends on the value of "b", which represents the annual increase in yield per acre. If the yield per acre is not increasing (i.e., b = 0), then the rate of increase is a constant 1300 bushels per year. If the yield per acre is increasing, then the rate of increase will be greater than 1300 bushels per year, and the rate of increase will depend on the value of "b".

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