A hot water tank with a capacity of 85.0 gallons of water is being designed to have the shape of a right circular cylinder with diameter a 1.8 feet. Assuming that there are a 7.48 gallons in 1 cubic foot, how high in feet will the tank need to be?

Options

A. `4.50`
B. `4.75`
C. `5.00`
D. `5.25`

Answer

To find the height of the hot water tank shaped like a right circular cylinder, we can follow these steps:

### Step 1: Determine the radius of the cylinder
The diameter of the cylinder is given as 1.8 feet. The radius (r) is half of the diameter.

\[ r = \frac{\text{diameter}}{2} = \frac{1.8 \text{ feet}}{2} = 0.9 \text{ feet} \]

**Hint:** Remember that the radius is always half of the diameter.

### Step 2: Convert the volume from gallons to cubic feet
We know that 1 cubic foot holds 7.48 gallons. To find the volume in cubic feet that corresponds to 85 gallons, we can use the following formula:

\[ \text{Volume in cubic feet} = \frac{\text{Volume in gallons}}{\text{gallons per cubic foot}} = \frac{85 \text{ gallons}}{7.48 \text{ gallons/cubic foot}} \approx 11.4 \text{ cubic feet} \]

**Hint:** Use the conversion factor to change gallons to cubic feet.

### Step 3: Use the formula for the volume of a cylinder
The volume (V) of a right circular cylinder is given by the formula:

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

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

We already have the volume (11.4 cubic feet) and the radius (0.9 feet). We can rearrange the formula to solve for height (h):

\[ h = \frac{V}{\pi r^2} \]

**Hint:** Make sure to rearrange the formula correctly to isolate the height.

### Step 4: Substitute the values into the formula
Now we can substitute the values we have into the rearranged formula:

\[ h = \frac{11.4}{\pi (0.9)^2} \]

Calculating \( (0.9)^2 \):

\[ (0.9)^2 = 0.81 \]

Now substituting back into the height formula:

\[ h = \frac{11.4}{\pi \times 0.81} \]

### Step 5: Calculate the height
Using the approximate value of \( \pi \approx 3.14 \):

\[ h = \frac{11.4}{3.14 \times 0.81} \approx \frac{11.4}{2.54} \approx 4.49 \text{ feet} \]

Rounding to two decimal places, we get:

\[ h \approx 4.50 \text{ feet} \]

### Final Answer
The height of the tank needs to be approximately **4.50 feet**.

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