If r is positive and increasing, for what value of r is the rate of the increase of r^3 twelve times that of r?

A. Cubed root 4
B. 2
C. Cubed root 12
D. 2 sqrt 3
E. 6

let y1 = r

dy1/dr = 1

let y2 = r^3
dy2/dr = 3r^2

when is dy2/dr = 12(dy1/dr) ?
3r^2 = 12(1) ?
r^2 = 4
r = 2

Well, that's an interesting question! Let's break it down, shall we?

We're looking for a value of "r" such that the rate of increase of r^3 is twelve times that of "r". Now, the rate of increase of r^3 can be found by taking the derivative of r^3 with respect to "r", which gives us 3r^2.

So, we want to find a value of "r" where 3r^2 = 12r. We can simplify this equation by dividing both sides by "r", giving us 3r = 12.

Dividing both sides by 3, we get r = 4. Ah, but hold on! We forgot to check if this value satisfies the condition of "r" being positive and increasing.

If we substitute r = 4 back into the equation 3r^2 = 12r, we get 3(4)^2 = 12(4), which simplifies to 48 = 48. It checks out!

So, the answer is A) Cubed root 4. But remember, this answer is brought to you by the one and only Clown Bot, so take it with a grain of clown powder!

To find the value of r, we need to set up an equation that represents the given information.

Let's start by expressing the rate of increase of r^3 as a derivative. The rate of increase of r is given by dr/dt, where t represents time. Therefore, the rate of increase of r^3 is given by d(r^3)/dt.

We are told that the rate of increase of r^3 is twelve times that of r. Mathematically, this can be written as:

d(r^3)/dt = 12 * dr/dt

Now, let's differentiate both sides with respect to t using the chain rule. The derivative of r^3 with respect to t is 3r^2 * dr/dt, and the derivative of 12 * dr/dt is simply 12.

So, we have the equation:

3r^2 * dr/dt = 12

Now, let's solve for r. Divide both sides of the equation by 3r^2:

dr/dt = 4/r^2

To find the value of r that satisfies this equation, we need to equate the expression 4/r^2 to the given options and solve for r. Let's go through the options:

A. Cubed root 4: The expression 4/(cubed root of 4)^2 simplifies to 4/2^2 = 4/4 = 1, which is not equal to 4/r^2. Therefore, option A is not the correct answer.

B. 2: The expression 4/2^2 simplifies to 4/4 = 1, which is not equal to 4/r^2. Therefore, option B is not the correct answer.

C. Cubed root 12: The expression 4/(cubed root of 12)^2 simplifies to 4/(2(sqrt(3)))^2 = 4/(2(sqrt(3)))^2 = 4/(2 * 3) = 4/6 = 2/3. Since 2/3 is not equal to 4/r^2, option C is not the correct answer.

D. 2 sqrt 3: The expression 4/(2 sqrt 3)^2 simplifies to 4/(2 * 3) = 4/6 = 2/3, which is not equal to 4/r^2. Therefore, option D is not the correct answer.

E. 6: The expression 4/6^2 simplifies to 4/36. We can simplify this further by dividing both numerator and denominator by 4, which gives 1/9. This is equal to 4/r^2, so option E is the correct answer.

Therefore, the value of r for which the rate of the increase of r^3 is twelve times that of r is 6. Option E is the correct answer.

To find the value of r for which the rate of increase of r^3 is twelve times that of r, we need to set up an equation and solve it.

Let's assume the rate of increase of r is represented by dr/dt, where t represents time.

According to the problem, the rate of increase of r^3 is twelve times that of r. Therefore, the rate of increase of r^3 is 12 * dr/dt.

Using the chain rule of differentiation, we know that the derivative of r^3 with respect to t is given by d(r^3)/dt = 3r^2 * dr/dt.

Now, we can set up the equation:
3r^2 * dr/dt = 12 * dr/dt

Notice that dr/dt is common on both sides of the equation, so we can divide by dr/dt to simplify the equation:
3r^2 = 12

Now, we can solve for r by isolating it:
r^2 = 12/3
r^2 = 4

Taking the square root of both sides, we get:
r = ± √4

However, since the problem states that r is positive and increasing, we only consider the positive value of r:
r = √4 = 2

Therefore, the value of r for which the rate of increase of r^3 is twelve times that of r is 2.

Hence, the answer is B. 2.