. The area of the base of a cylinder is 150 square inches. If the volume of the cylinder is $$ 1,050 \mathrm{in}^{3} $$, what is the height of the cylinder? $$ \qquad $$ 9. Leann's cylindrical water tank in her backyard has a diameter of 3 meters and a height of 2 meters. What is the capacity of the water tank? Round to the nearest hundredth. 8. The volume of a cylinder is 942 cubic feet. If the height of the cylinder is 12 feet, what is the radius of the cylinder? Use 3.14 for pl . 10. Drew and Beau each brought a cylindrical water bottle to football practice. Drew's water bottle has a diameter of 3 inches and a height of 7 inches. Beau's water bottle has a radius of 2 inches and a height of 9 inches. Whose water bottle holds more water and how much more does it hold? Round to the nearest hundredth. 11. The height of a cylinder is 4 millimeters. The circumference of its base is $$ 10 \pi \mathrm{~mm} $$. a. What is the radius of the cylinder? Explain. b. Find the volume of the cylinder to the nearest tenth.

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Let's solve each problem step by step.

### Problem 1
**Given:**
- Area of the base of the cylinder, $$ A = 150 \, \text{in}^2 $$
- Volume of the cylinder, $$ V = 1,050 \, \text{in}^3 $$

**To find:** Height of the cylinder, $$ h $$.

**Formula for the volume of a cylinder:**
$$
V = A \cdot h
$$

**Step 1:** Rearranging the formula to find height:
$$
h = \frac{V}{A}
$$

**Step 2:** Substitute the known values:
$$
h = \frac{1,050}{150}
$$

Now, let's calculate $$ h $$:
$$
h = 7 \, \text{inches}
$$

### Problem 2
**Given:**
- Diameter of the tank, $$ d = 3 \, \text{m} $$
- Height of the tank, $$ h = 2 \, \text{m} $$

**To find:** Capacity of the water tank.

**Step 1:** Calculate the radius:
$$
r = \frac{d}{2} = \frac{3}{2} = 1.5 \, \text{m}
$$

**Step 2:** Use the volume formula for a cylinder:
$$
V = \pi r^2 h
$$

**Step 3:** Substitute the values (using $$ \pi \approx 3.14 $$):
$$
V = 3.14 \cdot (1.5)^2 \cdot 2
$$

Now, let's calculate $$ V $$:
$$
V = 3.14 \cdot 2.25 \cdot 2 = 3.14 \cdot 4.5 = 14.13 \, \text{m}^3
$$

### Problem 3
**Given:**
- Volume of the cylinder, $$ V = 942 \, \text{ft}^3 $$
- Height of the cylinder, $$ h = 12 \, \text{ft} $$

**To find:** Radius of the cylinder, $$ r $$.

**Step 1:** Rearranging the volume formula:
$$
V = \pi r^2 h \implies r^2 = \frac{V}{\pi h}
$$

**Step 2:** Substitute the known values (using $$ \pi \approx 3.14 $$):
$$
r^2 = \frac{942}{3.14 \cdot 12}
$$

Now, let's calculate $$ r^2 $$:
$$
r^2 = \frac{942}{37.68} \approx 25
$$

**Step 3:** Taking the square root to find $$ r $$:
$$
r = 5 \, \text{ft}
$$

### Problem 4
**Given:**
- Drew's water bottle: Diameter = 3 inches, Height = 7 inches
- Beau's water bottle: Radius = 2 inches, Height = 9 inches

**To find:** Compare the volumes of both bottles.

**Step 1:** Calculate Drew's volume:
$$
r_D = \frac{3}{2} = 1.5 \, \text{inches}
$$
$$
V_D = \pi r_D^2 h_D = 3.14 \cdot (1.5)^2 \cdot 7
$$
Calculating $$ V_D $$:
$$
V_D = 3.14 \cdot 2.25 \cdot 7 = 3.14 \cdot 15.75 \approx 49.50 \, \text{in}^3
$$

**Step 2:** Calculate Beau's volume:
$$
V_B = \pi r_B^2 h_B = 3.14 \cdot (2)^2 \cdot 9
$$
Calculating $$ V_B $$:
$$
V_B = 3.14 \cdot 4 \cdot 9 = 3.14 \cdot 36 \approx 113.04 \, \text{in}^3
$$

**Step 3:** Compare the volumes:
$$
\text{Difference} = V_B - V_D = 113.04 - 49.50 \approx 63.54 \, \text{in}^3
$$

### Problem 5
**Given:**
- Height of the cylinder, $$ h = 4 \, \text{mm} $$
- Circumference of the base, $$ C = 10\pi \, \text{mm} $$

**To find:**
a. Radius of the cylinder.
b. Volume of the cylinder.

**Step 1:** Find the radius from the circumference:
$$
C = 2\pi r \implies r = \frac{C}{2\pi} = \frac{10\pi}{2\pi} = 5 \, \text{mm}
$$

**Step 2:** Calculate the volume:
$$
V = \pi r^2 h
$$
Substituting the values:
$$
V = \pi (5)^2 (4) = \pi \cdot 25 \cdot 4 = 100\pi
$$

Calculating $$ V $$ using $$ \pi \approx 3.14 $$:
$$
V \approx 100 \cdot 3.14 = 314 \, \text{mm}^3
$$

### Summary of Results
1. Height of the cylinder: $$ 7 \, \text{inches} $$
2. Capacity of the water tank: $$ 14.13 \, \text{m}^3 $$
3. Radius of the cylinder: $$ 5 \, \text{ft} $$
4. Drew's water bottle holds $$ 49.50 \, \text{in}^3 $$ and Beau's holds $$ 113.04 \, \text{in}^3 $$; Beau's holds $$ 63.54 \, \text{in}^3 $$ more.
5. Radius of the cylinder: $$ 5 \, \text{mm} $$; Volume: $$ 314 \, \text{mm}^3 $$
1. The height of the cylinder is 7 inches. 2. The water tank's capacity is 14.13 cubic meters. 3. The radius of the cylinder is 5 feet. 4. Beau's water bottle holds more water by 63.54 cubic inches. 5. The radius of the cylinder is 5 millimeters, and its volume is 314 cubic millimeters.

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