Geometry worksheet focusing on volume calculations for prisms and cylinders, with visual diagrams and problem statements.

Here are the step-by-step solutions for each problem on the worksheet.

1. Volume of a right prism (Parallelogram base)
* Step 1: Find the area of the base. The base is a parallelogram with a base length of 14 and a height of 10.
* Area = $14 \times 10 = 140$ square units.
* Step 2: Multiply by the height of the prism (36).
* Volume = $140 \times 36$.
* Calculation: $140 \times 36 = 5,040$.

2. Volume of a cylinder
* Step 1: Find the radius. The diameter is 18 inches, so the radius ($r$) is half of that.
* $r = 9$ inches.
* Step 2: Use the volume formula $V = \pi r^2 h$. The height ($h$) is 22.
* $V = \pi \times 9^2 \times 22$
* $V = \pi \times 81 \times 22$
* $V = 1,782\pi$ cubic inches.
* Approximate value: $1,782 \times 3.14159 \approx 5,598.32$.

3. Amount of water in the pool (Cylinder)
* Step 1: Identify dimensions. Diameter = 25 ft, so Radius ($r$) = 12.5 ft. Height ($h$) = 1.5 ft.
* Step 2: Calculate volume using $V = \pi r^2 h$.
* $V = \pi \times (12.5)^2 \times 1.5$
* $V = \pi \times 156.25 \times 1.5$
* $V = 234.375\pi$
* Calculation: $234.375 \times 3.14159 \approx 736.31$.
* Step 3: Round to the nearest cubic foot.
* Result: 736 cubic feet.

4. Volume of a right triangular prism
* Step 1: Find the area of the triangular base. Base of triangle = 15, Height of triangle = 12.
* Area = $\frac{1}{2} \times \text{base} \times \text{height}$
* Area = $\frac{1}{2} \times 15 \times 12 = 90$ square units.
* Step 2: Multiply by the length (height) of the prism, which is 20.
* Volume = $90 \times 20 = 1,800$ cubic units.

5. Total surface area of a cylinder
* Step 1: Find the radius. We know Volume ($V$) = $144\pi$ and Height ($h$) = 4.
* Formula: $V = \pi r^2 h$
* $144\pi = \pi \times r^2 \times 4$
* Divide both sides by $4\pi$: $36 = r^2$.
* So, radius ($r$) = 6 inches.
* Step 2: Calculate Total Surface Area ($SA = 2\pi r^2 + 2\pi rh$).
* Area of two bases: $2 \times \pi \times 6^2 = 72\pi$.
* Lateral Area (side): $2 \times \pi \times 6 \times 4 = 48\pi$.
* Total SA = $72\pi + 48\pi = 120\pi$ square inches.
* Approximate value: $120 \times 3.14159 \approx 376.99$.

6. Volume of a right hexagonal prism
* Step 1: Find the area of the hexagon base.
* A regular hexagon can be split into 6 equilateral triangles. The side length is 8.
* Area of one equilateral triangle = $\frac{\sqrt{3}}{4} \times s^2$.
* Area = $\frac{\sqrt{3}}{4} \times 8^2 = \frac{\sqrt{3}}{4} \times 64 = 16\sqrt{3}$.
* Total Base Area = $6 \times 16\sqrt{3} = 96\sqrt{3}$.
* Step 2: Multiply by the height of the prism (12).
* Volume = $96\sqrt{3} \times 12 = 1,152\sqrt{3}$.
* Calculation: $1,152 \times 1.732 \approx 1,995.3$.

7. Volume of a prism with an isosceles trapezoid base
* Step 1: Find the area of the trapezoid base.
* Top base ($b_1$) = 18. Bottom base ($b_2$) = 42.
* To find the height ($h_{trap}$), look at the slanted side (20). The difference between bases is $42 - 18 = 24$. Since it's isosceles, each side overhang is $24 / 2 = 12$.
* Use Pythagorean theorem on the side triangle: $12^2 + h_{trap}^2 = 20^2$.
* $144 + h_{trap}^2 = 400$.
* $h_{trap}^2 = 256$, so $h_{trap} = 16$.
* Area of Trapezoid = $\frac{18 + 42}{2} \times 16 = 30 \times 16 = 480$.
* Step 2: Multiply by the prism height (10).
* Volume = $480 \times 10 = 4,800$ cubic units.

8. Volume of soup in a cylindrical can
* Step 1: Identify dimensions. Height ($h$) = 8 inches. Diameter = 4 inches, so Radius ($r$) = 2 inches.
* Step 2: Calculate volume using $V = \pi r^2 h$.
* $V = \pi \times 2^2 \times 8$
* $V = \pi \times 4 \times 8$
* $V = 32\pi$ cubic inches.
* Approximate value: $32 \times 3.14159 \approx 100.53$.

Final Answer:
1. 5,040 cubic units
2. $1,782\pi$ cubic inches (approx. 5,598.32)
3. 736 cubic feet
4. 1,800 cubic units
5. $120\pi$ sq inches (approx. 377.0)
6. $1,152\sqrt{3}$ cubic feet (approx. 1,995.3)
7. 4,800 cubic units
8. $32\pi$ cubic inches (approx. 100.53)