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Part 1
Scientists are drilling a hole in the ocean floor to learn more about the​ Earth's history. ​ Currently, the hole is in the shape of a cylinder whose volume is approximately 3700 cubic feet and whose height is 4.6 miles.  Find the radius of the hole to the nearest hundredth of a foot. ​ (Hint: Make sure the same units of measurement are​ used.)

To solve this problem, we need to find the radius of a cylindrical hole given its volume and height. The formula for the volume \( V \) of a cylinder is:

\[
V = \pi r^2 h
\]

Where:
- \( V \) is the volume of the cylinder.
- \( r \) is the radius of the cylinder.
- \( h \) is the height of the cylinder.

### Step 1: Convert the height from miles to feet.
Since the height is given in miles, we need to convert it to feet to use the same units. There are 5280 feet in a mile, so:

\[
h = 4.6 \text{ miles} \times 5280 \text{ feet/mile} = 24288 \text{ feet}
\]

### Step 2: Substitute the known values into the volume formula.
Given:
- \( V = 3700 \) cubic feet
- \( h = 24288 \) feet

Substitute these into the formula:

\[
3700 = \pi r^2 \times 24288
\]

### Step 3: Solve for the radius \( r \).
First, isolate \( r^2 \) by dividing both sides of the equation by \( \pi \times 24288 \):

\[
r^2 = \frac{3700}{\pi \times 24288}
\]

Now, calculate this value:

\[
r^2 = \frac{3700}{3.1416 \times 24288} \approx \frac{3700}{76341.97} \approx 0.04846
\]

Finally, take the square root of both sides to find \( r \):

\[
r = \sqrt{0.04846} \approx 0.2201 \text{ feet}
\]

### Answer:
The radius of the hole is approximately **0.22 feet** to the nearest hundredth of a foot.

Would you like more details on any step or have additional questions?

Here are 5 related questions to explore:
1. How would the radius change if the volume of the cylinder was doubled?
2. What is the relationship between height and radius if the volume remains constant?
3. How do you convert cubic feet to cubic meters?
4. How would you solve for the height if the radius and volume were given?
5. How does the volume of a cylinder change if both the radius and height are doubled?

**Tip:** When working with different units, always convert to the same unit system before performing calculations.

Learn how to calculate the radius of a cylindrical hole with a volume of 3700 cubic feet and a height of 4.6 miles. Detailed step-by-step solution included. Suitable for students in Grades 7-9.

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