Let's solve both parts of the question one by one.

Part a:

We are given the formula for the ideal diameter $d$ of a ship’s propeller:

$d = \frac{550 \cdot h}{r^{3/5}}$

Where:

• $h$ is the horsepower of the engine.
• $r$ is the revolutions per minute (RPM) of the propeller.

For the Victory Ships:

• $h = 6000$ horsepower.
• $r = 100$ revolutions per minute.

We need to find the ideal diameter $d$.

Step 1: Substitute the known values into the formula.

$d = \frac{550 \cdot 6000}{100^{3/5}}$

Step 2: Simplify the exponent.

We first calculate $100^{3/5}$:

$100^{3/5} = (10^2)^{3/5} = 10^{6/5} \approx 15.8489$

Step 3: Plug this value into the formula.

$d = \frac{550 \cdot 6000}{15.8489}$ $d = \frac{3300000}{15.8489} \approx 208,479.7 \text{ feet}$

So, according to the formula, the ideal diameter of the propeller should be approximately 208.48 feet. This is obviously much larger than the actual diameter of 18.2 feet, but this might suggest that either the formula needs adjustment or other factors affect the actual size.

Part b:

We are given:

• $h = 8500$ horsepower.
• $d = 20$ feet. We need to find the revolutions per minute $r$.

Rearrange the formula to solve for $r$:

$d = \frac{550 \cdot h}{r^{3/5}}$

$r^{3/5} = \frac{550 \cdot h}{d}$

Now substitute the known values:

$r^{3/5} = \frac{550 \cdot 8500}{20}$ $r^{3/5} = \frac{4675000}{20} = 233750$

Now, raise both sides to the power of $5/3$ to solve for $r$:

$r = (233750)^{5/3}$

Let’s compute this:

$r \approx 233750^{5/3} \approx 437.14 \text{ revolutions per minute}$

Thus, the maximum revolutions per minute of the propeller should be approximately 437 RPM.

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Would you like a deeper breakdown of any part of the calculation or have any further questions? Here are some related questions for practice:

1. How does the relationship between horsepower and revolutions per minute affect the propeller diameter?
2. What are the real-world factors that might cause the actual diameter of a propeller to differ from the calculated value?
3. How do you solve an equation involving fractional exponents, as seen in the problem?
4. How could the formula be adjusted if we included factors like drag or water resistance?
5. What impact would increasing the horsepower have on the diameter or RPM of a propeller?

Tip: When working with exponents, always remember to handle fractional exponents as roots or powers. For example, $a^{m/n} = \sqrt[n]{a^m}$.