Let r(t) be a differentiable function that is positive and increasing. The rate of increase of r^3 is equal to 12 times the rate of increase of r when r(t)=?

the increasing rate or r^3 is 12 time of increasing rate of r

d/dt(r^3)=12dr/dt
3r^2dr/dt=12dr/dt
divde through by 3
r^2dr/dt=4dr/dt
r^2dr/dt-4dr/dt=0
dr/dt(r^2-4)=0
r^2-4=0
(r-2)(r+2)=0
r=2 or r=-2
that went awesome
now just have it in mind that any constant at all we always work

Oh, I love playing with rates and equations! Now, let's have some fun with math.

We start by saying that the rate of increase of a function r(t) is given by its derivative, which we can denote as r'(t). So, the rate of increase of r^3 is the same as the derivative of r^3.

Now, let's calculate the derivative of r^3 using the chain rule. The chain rule tells us that if we have a function raised to a constant power, we multiply by that constant power and then take the derivative of the function itself. In this case, let's call the constant power 3.

So, the derivative of r^3, which we'll call (r^3)' can be calculated as:

(r^3)' = 3r^2 * r'

Now, we're also given that this rate of increase of r^3 is equal to 12 times the rate of increase of r. So, we can equate the two expressions:

3r^2 * r' = 12 * r'

Now, we can simplify this equation a bit:

3r^2 * r' - 12 * r' = 0

Factoring out r' gives us:

(r' * (3r^2 - 12)) = 0

Now, either r' = 0, or (3r^2 - 12) = 0.

Let's solve the second equation:

3r^2 - 12 = 0

Dividing by 3 gives us:

r^2 - 4 = 0

Solving for r, we find:

r = ±2

So, when r(t) is equal to either -2 or 2, the rate of increase of r^3 is equal to 12 times the rate of increase of r.

There you have it! Math can be quite amusing, don't you think?

To find the value of r(t) when the rate of increase of r³ is equal to 12 times the rate of increase of r, we need to set up the equation.

Let's denote the rate of increase of r as dr/dt. According to the problem, the rate of increase of r³ is given by d(r³)/dt.

Using the power rule of differentiation, we can find the derivative of r³ as follows:

d(r³)/dt = 3r² * dr/dt

Now, we can set up the equation:

3r² * dr/dt = 12 * dr/dt

Since dr/dt is common to both sides of the equation, we can divide both sides by dr/dt:

3r² = 12

Dividing both sides by 3:

r² = 4

Taking the square root of both sides:

r = ±2

Since r is defined as a positive and increasing function, the value of r(t) would be r = 2 when the rate of increase of r³ is equal to 12 times the rate of increase of r.

To find the value of r(t) when the rate of increase of r^3 is equal to 12 times the rate of increase of r, we can start by using the chain rule of differentiation.

We are given that r(t) is a differentiable function that is positive and increasing. Let's denote the derivative of r(t) as dr/dt.

Now, we want to find the rate of increase of r^3. Let's call this derivative d(r^3)/dt. Using the chain rule:

d(r^3)/dt = 3r^2 * dr/dt

Next, we are given that the rate of increase of r^3 is equal to 12 times the rate of increase of r. This can be written as:

3r^2 * dr/dt = 12 * dr/dt

We can simplify this equation:

3r^2 * dr/dt - 12 * dr/dt = 0

Factor out dr/dt:

(3r^2 - 12) * dr/dt = 0

Now, for this equation to be true, either (3r^2 - 12) = 0 or dr/dt = 0.

First, let's solve (3r^2 - 12) = 0:

3r^2 - 12 = 0
3r^2 = 12
r^2 = 4
r = ±2

Taking into account that r(t) is positive and increasing, we can discard the negative value of r, leaving us with r = 2.

Therefore, when r(t) = 2, the rate of increase of r^3 is equal to 12 times the rate of increase of r.