A tank in the shape of a hemisphere has a diameter of 6 feet. If the liquid that fills the tank has a density of 91.4 pounds per cubic foot, what is the total weight of the liquid in the tank, to the nearest full pound?

To find the total weight of the liquid in the tank, we will follow these steps:

### Step 1: Determine the volume of the tank
The tank is in the shape of a hemisphere, and the volume \( V \) of a hemisphere is given by the formula:

\[
V = \frac{2}{3} \pi r^3
\]

where \( r \) is the radius of the hemisphere.

Since the diameter of the tank is 6 feet, the radius is:

\[
r = \frac{6}{2} = 3 \, \text{feet}
\]

Now, substitute \( r = 3 \) into the volume formula:

\[
V = \frac{2}{3} \pi (3)^3 = \frac{2}{3} \pi (27) = 18\pi \, \text{cubic feet}
\]

\[
V \approx 18 \times 3.1416 = 56.55 \, \text{cubic feet}
\]

### Step 2: Calculate the total weight of the liquid
The total weight \( W \) of the liquid is given by:

\[
W = \text{Density} \times \text{Volume}
\]

The density of the liquid is 91.4 pounds per cubic foot. Using the volume we calculated:

\[
W = 91.4 \times 56.55 = 5168.07 \, \text{pounds}
\]

### Step 3: Round to the nearest full pound
The total weight of the liquid in the tank is approximately:

\[
W \approx 5168 \, \text{pounds}
\]

Thus, the total weight of the liquid in the tank is **5168 pounds**.

Would you like further clarification on any part of this solution? Here are 5 related questions for practice:

1. What would the volume be if the tank's radius were doubled?
2. How would the total weight change if the liquid density were halved?
3. If only half the hemisphere were filled, what would be the weight of the liquid?
4. How would the weight change if the diameter was 10 feet instead of 6 feet?
5. What is the mass in kilograms if 1 pound is approximately 0.4536 kg?

**Tip:** The volume of a hemisphere is exactly half of the volume of a full sphere.

Learn how to calculate the total weight of a liquid filling a hemisphere tank with a diameter of 6 feet and a liquid density of 91.4 pounds per cubic foot. This problem covers key geometry concepts and weight calculation formulas, suitable for high school students in Grades 9-12.

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