It looks like you've covered several parts of the problem! Let's break down each point.

1. Equation to Solve for Radius:

You're given the formula for the volume of a sphere:

$V = \frac{4}{3} \pi r^3$

For a sphere with a volume of 87 cubic inches, we can set $V = 87$:

$87 = \frac{4}{3} \pi r^3$

To solve for $r$, follow these steps:

• Multiply both sides of the equation by $\frac{3}{4}$ to isolate $\pi r^3$ on one side:

$\frac{3}{4} \times 87 = \pi r^3$

$65.25 = \pi r^3$

• Now, divide both sides by $\pi$ to get $r^3$:

$r^3 = \frac{65.25}{\pi}$

$r^3 \approx \frac{65.25}{3.1416} \approx 20.78$

• Finally, take the cube root of both sides to find $r$:

$r \approx \sqrt[3]{20.78} \approx 2.73 \text{ inches}$

So, the radius of the sphere is approximately 2.73 inches.

2. Volume when $r = 5.3$ inches:

You correctly calculated the volume when $r = 5.3$ inches using the formula for volume:

$V = \frac{4}{3} \pi r^3$

Substituting $r = 5.3$:

$V = \frac{4}{3} \pi (5.3)^3$

First, cube 5.3:

$5.3^3 = 148.877$

Then calculate the volume:

$V = \frac{4}{3} \pi (148.877) \approx \frac{4}{3} \times 3.1416 \times 148.877 \approx 623.77 \text{ cubic inches}$

So, your answer of 623.77 cubic inches is correct.

3. Volume Growth with Radius:

As the radius increases from 1 to 6 inches, the volume of the sphere increases exponentially, because the volume is proportional to the cube of the radius. So, as the radius increases, the volume grows rapidly.

In this case:

• When $r = 1$, the volume is $V = \frac{4}{3} \pi (1)^3 = 4.1887$ cubic inches.
• When $r = 6$, the volume is $V = \frac{4}{3} \pi (6)^3 = 904.78$ cubic inches.

So, the volume increases from 4.19 cubic inches to 904.78 cubic inches as the radius increases from 1 to 6 inches.

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Would you like further clarification or more details on any of these steps?

Here are 5 follow-up questions that might be helpful:

1. How does the volume change if the radius is doubled?
2. Can you derive a general formula for the volume of a sphere in terms of $r$ and $\pi$?
3. What happens to the volume if the radius decreases from 5 inches to 3 inches?
4. How would you solve for the radius if the volume is unknown and only the surface area is given?
5. What are the differences between the volume and surface area formulas for spheres?

Tip: When working with the volume of a sphere, always remember that the relationship is cubic, so a small change in radius can lead to a large change in volume!