https://file.aiquickdraw.com/mathbot/file/1729522850420lfoaw39w.jpg

This is a related rates problem. Here's how we can approach it step by step:

We are given:
- The rate of change of surface area: \(\frac{dA}{dt} = -7 \, \text{cm}^2/\text{min}\)
- The diameter \(D = 8 \, \text{cm}\)

We are asked to find:
- The rate at which the diameter is decreasing, i.e., \(\frac{dD}{dt}\).

### Step 1: Relate Surface Area and Diameter
The surface area \(A\) of a sphere is given by:
\[
A = 4\pi r^2
\]
where \(r\) is the radius of the snowball. The radius \(r\) is related to the diameter by \(r = \frac{D}{2}\), so the surface area in terms of diameter \(D\) is:
\[
A = 4\pi \left(\frac{D}{2}\right)^2 = \pi D^2
\]

### Step 2: Differentiate with Respect to Time
To find how the diameter changes with respect to time, differentiate both sides of the equation \(A = \pi D^2\) with respect to time \(t\):
\[
\frac{dA}{dt} = 2\pi D \frac{dD}{dt}
\]

### Step 3: Plug in the Given Values
We know:
- \(\frac{dA}{dt} = -7 \, \text{cm}^2/\text{min}\)
- \(D = 8 \, \text{cm}\)

Substitute these values into the differentiated equation:
\[
-7 = 2\pi(8) \frac{dD}{dt}
\]

### Step 4: Solve for \(\frac{dD}{dt}\)
Now solve for \(\frac{dD}{dt}\):
\[
\frac{dD}{dt} = \frac{-7}{16\pi}
\]

Using \(\pi \approx 3.1416\):
\[
\frac{dD}{dt} = \frac{-7}{16 \times 3.1416} \approx -0.139 \, \text{cm}/\text{min}
\]

Thus, the diameter is decreasing at a rate of approximately \(-0.139 \, \text{cm}/\text{min}\).

Would you like further details or have any questions about this problem?

---

Here are 5 related questions to expand your understanding:
1. How would the rate of change of volume compare to the rate of change of surface area in this scenario?
2. What happens to the rate at which the diameter changes as the snowball becomes smaller?
3. If the snowball had started with a larger diameter, how would that affect the rate at which the diameter decreases?
4. Can you derive a general formula for the rate of change of diameter in terms of surface area rate and initial diameter?
5. How does the radius of the sphere change as compared to the diameter when both are shrinking over time?

**Tip**: In related rates problems, always ensure you're differentiating with respect to time and correctly applying the chain rule for every variable involved.

If a snowball melts so that its surface area decreases at a rate of 7 cm²/min, find the rate (in cm/min) at which the diameter decreases when the diameter is 8 cm.

Find the rate at which the diameter of a melting snowball decreases, given its surface area decreases at a rate of 7 cm²/min and its diameter is 8 cm. This problem involves concepts of related rates, differentiation, and the geometry of spheres. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation