QuestionHemisphere
7. Find the volume of the umbrella if its radius is 2.1 ft .
8. Find the volume of the umbrella if its diameter is 48 in .

STEP 1

1. The umbrella is in the shape of a hemisphere.
2. The formula for the volume of a hemisphere is $\frac{2}{3} \pi r^3$.
3. For problem 7, the radius is given as $2.1$ feet.
4. For problem 8, the diameter is given as $48$ inches, which needs to be converted to feet for consistency.
_HIGH_LEVEL_APPROACH_ for Problem 7:
1. Recall the formula for the volume of a hemisphere.
2. Substitute the given radius.
3. Calculate the volume.
_HIGH_LEVEL_APPROACH_ for Problem 8:
1. Convert the diameter to radius in feet.
2. Recall the formula for the volume of a hemisphere.
3. Substitute the calculated radius.
4. Calculate the volume.
**Problem 7:**

STEP 2

STEP 3

Recall the formula for the volume of a hemisphere:
$$V = \frac{2}{3} \pi r^3$$

STEP 4

Substitute the given radius into the formula:
$$V = \frac{2}{3} \pi (2.1)^3$$

STEP 5

Calculate the volume:
$$V = \frac{2}{3} \pi (2.1)^3$$ $$= \frac{2}{3} \pi \times 9.261$$ $$\approx 19.44 \text{ ft}^3$$
The volume of the umbrella is approximately:
$$\boxed{19.44 \text{ ft}^3}$$
**Problem 8:**
STEP_1: Convert the diameter to radius in feet:
$$\text{Diameter} = 48 \text{ in}$$ $$\text{Radius} = \frac{48}{2} = 24 \text{ in}$$ $$\text{Radius in feet} = \frac{24}{12} = 2 \text{ ft}$$
STEP_2: Recall the formula for the volume of a hemisphere:
$$V = \frac{2}{3} \pi r^3$$
STEP_3: Substitute the calculated radius into the formula:
$$V = \frac{2}{3} \pi (2)^3$$

STEP 6

Calculate the volume:
$$V = \frac{2}{3} \pi (2)^3$$ $$= \frac{2}{3} \pi \times 8$$ $$\approx 16.76 \text{ ft}^3$$
The volume of the umbrella is approximately:
$$\boxed{16.76 \text{ ft}^3}$$