Propeller Size An ideal diameter d, in feet, of a ship’s propeller is given by the formula
d=50(h/r exponent 3) exponent 1/5.
Here h is the horsepower of the engine driving the propeller, and r is the (maximum) revolutions per minute of the propeller.

a. The Victory Ships produced in World War II had an engine that produced 6000 horsepower and rotated the propeller at 100 revolutions per minute. According to the formula, what should have been the diameter of their propellers? (The actual diameter was 18.2 feet.)

b. If a ship’s engine has a horsepower of 8500 turning a propeller with a diameter of 20 feet, what should be the (maximum) revolutions per minute of the propeller?
Solve b only

To solve part (b), we need to find the maximum revolutions per minute (denoted as \(r\)) of the propeller. The formula given is:

\[
d = 50 \left(\frac{h}{r^3}\right)^{1/5}
\]

Given:
- \( h = 8500 \) horsepower
- \( d = 20 \) feet

We need to solve for \(r\).

### Step-by-Step Solution:

1. Start with the formula:

   \[
   20 = 50 \left(\frac{8500}{r^3}\right)^{1/5}
   \]

2. Divide both sides by 50 to isolate the term involving \(r\):

   \[
   \frac{20}{50} = \left(\frac{8500}{r^3}\right)^{1/5}
   \]
   
   Simplifying:

   \[
   0.4 = \left(\frac{8500}{r^3}\right)^{1/5}
   \]

3. Raise both sides to the power of 5 to eliminate the 1/5 exponent:

   \[
   0.4^5 = \frac{8500}{r^3}
   \]

   Simplifying \(0.4^5\):

   \[
   0.01024 = \frac{8500}{r^3}
   \]

4. Multiply both sides by \(r^3\) and divide by 0.01024 to solve for \(r^3\):

   \[
   r^3 = \frac{8500}{0.01024}
   \]

5. Calculate:

   \[
   r^3 = 830078.125
   \]

6. Take the cube root of both sides to find \(r\):

   \[
   r = \sqrt[3]{830078.125}
   \]

   Calculating the cube root:

   \[
   r \approx 94.19 \text{ revolutions per minute (rpm)}
   \]

Thus, the maximum revolutions per minute of the propeller should be approximately **94.19 rpm**.

Would you like more details or have any questions?

### Related Questions:
1. How would the propeller size change if the engine produced 9000 horsepower instead of 8500?
2. How does the power of the engine affect the propeller's diameter if rpm is constant?
3. What is the relationship between propeller size and revolutions per minute if horsepower is fixed?
4. How would doubling the revolutions per minute affect the propeller size if the engine power is constant?
5. If the propeller's diameter is increased to 25 feet, what should the new maximum rpm be?

### Tip:
When solving equations involving exponents and roots, always isolate the term with the variable before applying powers or roots for a more straightforward solution!

Propeller Size: An ideal diameter d, in feet, of a ship’s propeller is given by the formula d=50(h/r^3)^(1/5). If a ship’s engine has a horsepower of 8500 turning a propeller with a diameter of 20 feet, what should be the (maximum) revolutions per minute of the propeller?

Learn how to solve for the maximum revolutions per minute (RPM) of a ship's propeller using the formula d = 50(h/r^3)^(1/5). This problem involves algebraic concepts such as exponents, roots, and equation solving. Given a 8500 horsepower engine and a propeller diameter of 20 feet, find the RPM in this detailed solution.

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