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) = ?

We are given that \( r(t) \) is a differentiable function that is positive and increasing. The problem states that the rate of increase of \( r^3 \) is equal to 12 times the rate of increase of \( r \). We need to find the value of \( r(t) \) when this is true.

### Step 1: Define the rates of increase
The rate of increase of \( r^3 \) is given by the derivative of \( r^3 \) with respect to \( t \), which is:
\[
\frac{d}{dt} (r^3) = 3r^2 \cdot \frac{dr}{dt}
\]
The rate of increase of \( r \) is simply:
\[
\frac{dr}{dt}
\]

### Step 2: Set up the equation
According to the problem, the rate of increase of \( r^3 \) is 12 times the rate of increase of \( r \). Therefore, we set up the equation:
\[
3r^2 \cdot \frac{dr}{dt} = 12 \cdot \frac{dr}{dt}
\]

### Step 3: Solve for \( r(t) \)
Assuming \( \frac{dr}{dt} \neq 0 \) (since \( r(t) \) is increasing), we can divide both sides of the equation by \( \frac{dr}{dt} \):
\[
3r^2 = 12
\]

Next, solve for \( r \):
\[
r^2 = \frac{12}{3} = 4
\]
\[
r = \sqrt{4} = 2
\]

### Final Answer:
The value of \( r(t) \) when the given condition holds is \( r(t) = 2 \).

Would you like more details on any part of this solution, or do you have any questions?

---

Here are 5 related questions:
1. How does the behavior of \( r(t) \) change if the rate of increase of \( r^3 \) is 6 times the rate of increase of \( r \)?
2. What is the general formula for the rate of change of a power of a function \( r(t) \)?
3. If \( r(t) \) is exponential, how would the condition affect the value of \( r(t) \)?
4. What happens if \( r(t) \) is decreasing instead of increasing? How would the answer change?
5. How can this problem be extended to higher powers, like \( r^n \)?

**Tip:** Always verify your assumptions, like \( \frac{dr}{dt} \neq 0 \), as it can change the outcome of your solution.

Explore how 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. This problem involves calculus concepts such as derivatives and rates of change. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation