Two spheres that are tangent to each other have columes of `36pi` cubic centimeters and `972pi` cubic centimeters. What is the greatest possible distance, in centimeters, between a point on one sphere and a a second point on the other sphere?

Answer

To solve the problem of finding the greatest possible distance between a point on one sphere and a second point on the other sphere, we will follow these steps:

### Step 1: Understand the relationship between volume and radius of a sphere.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

### Step 2: Calculate the radius of the first sphere.
Given the volume of the first sphere is \( 36\pi \) cubic centimeters:
\[ \frac{4}{3} \pi r_1^3 = 36\pi \]
We can cancel \( \pi \) from both sides:
\[ \frac{4}{3} r_1^3 = 36 \]
Multiplying both sides by \( \frac{3}{4} \):
\[ r_1^3 = 36 \times \frac{3}{4} = 27 \]
Taking the cube root:
\[ r_1 = \sqrt[3]{27} = 3 \text{ cm} \]

### Step 3: Calculate the radius of the second sphere.
Given the volume of the second sphere is \( 972\pi \) cubic centimeters:
\[ \frac{4}{3} \pi r_2^3 = 972\pi \]
Again, we can cancel \( \pi \):
\[ \frac{4}{3} r_2^3 = 972 \]
Multiplying both sides by \( \frac{3}{4} \):
\[ r_2^3 = 972 \times \frac{3}{4} = 729 \]
Taking the cube root:
\[ r_2 = \sqrt[3]{729} = 9 \text{ cm} \]

### Step 4: Calculate the diameters of both spheres.
The diameter \( d_1 \) of the first sphere is:
\[ d_1 = 2r_1 = 2 \times 3 = 6 \text{ cm} \]
The diameter \( d_2 \) of the second sphere is:
\[ d_2 = 2r_2 = 2 \times 9 = 18 \text{ cm} \]

### Step 5: Calculate the greatest possible distance between points on the two spheres.
Since the spheres are tangent to each other, the greatest distance between a point on one sphere and a point on the other sphere is the sum of their diameters:
\[ \text{Greatest Distance} = d_1 + d_2 = 6 + 18 = 24 \text{ cm} \]

### Final Answer:
The greatest possible distance between a point on one sphere and a second point on the other sphere is \( \boxed{24} \) centimeters.

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