Question

the three - dimensional figure below is a cylinder with a hole in the shape of a rectangular prism going through the center of it. the radius is 10 feet. find the volume of the solid in cubic feet. use 3.14 for π.
(options: 4,935 cubic feet; 4,485 cubic feet; 4,710 cubic feet; 6,840 cubic feet)

Explanation:

Step1: Calculate Cylinder Volume

The formula for the volume of a cylinder is \( V_{\text{cylinder}} = \pi r^2 h \). Given \( r = 10 \) feet (wait, no, looking at the diagram: the radius of the cylinder is 5 ft? Wait, the diagram has a radius of 5 ft? Wait, the text says "the radius is 10 feet"? Wait, no, let's recheck. Wait, the diagram: the cylinder has radius 5 ft? Wait, the problem says "the radius is 10 feet"? Wait, no, the user's diagram: the cylinder has a radius of 5 ft? Wait, the problem statement: "The three - dimensional figure below is a cylinder with a hole in the shape of a rectangular prism going through the center of it. The radius is 10 feet." Wait, maybe I misread. Wait, the diagram: there's a 5 ft (radius?), 15 ft (height of cylinder?), and the hole is a triangular prism? Wait, no, the hole is a rectangular prism? Wait, the diagram shows a triangle with base 3 ft? Wait, maybe the hole is a triangular prism? Wait, no, the problem says "rectangular prism"? Wait, maybe the diagram has a triangular hole? Wait, let's re - examine the problem.

Wait, the correct approach: Volume of the solid = Volume of cylinder - Volume of the hole (prism).

First, cylinder volume: \( V_{\text{cylinder}}=\pi r^{2}h \). The radius \( r = 10 \) feet? Wait, no, the diagram: the radius of the cylinder is 5 ft? Wait, the problem says "the radius is 10 feet". Wait, maybe the diagram's 5 ft is something else. Wait, the problem says "Use 3.14 for \( \pi \)". Let's assume:

Cylinder: radius \( r = 10 \) ft, height \( h = 15 \) ft.

Volume of cylinder: \( V_{c}=\pi r^{2}h=3.14\times10^{2}\times15 = 3.14\times100\times15=4710 \) cubic feet.

Now, the hole: it's a rectangular prism? Wait, the diagram shows a triangle with base 3 ft? Wait, maybe the hole is a triangular prism? Wait, no, the problem says "rectangular prism"? Wait, maybe the hole has dimensions: let's see, the length of the hole is equal to the height of the cylinder, which is 15 ft. The cross - section of the hole: if it's a triangle with base 3 ft and height 5 ft? Wait, no, maybe the hole is a triangular prism with base area \( \frac{1}{2}\times3\times5 \) and length 15 ft.

Wait, no, let's re - read the problem. The problem says "a hole in the shape of a rectangular prism" – maybe that's a mistake, and it's a triangular prism. Wait, the diagram shows a triangle with base 3 ft and height 5 ft? Wait, let's calculate the volume of the hole. If the hole is a triangular prism, base area \( A=\frac{1}{2}\times3\times5 = 7.5 \) square feet, and length (height of cylinder) \( l = 15 \) ft. So volume of hole \( V_{h}=A\times l=7.5\times15 = 112.5 \) cubic feet? No, that can't be. Wait, maybe the hole is a rectangular prism with length 15 ft, width 3 ft, and height 5 ft? Then volume of hole \( V_{h}=3\times5\times15 = 225 \) cubic feet.

Wait, no, let's do it step by step.

Step 1: Calculate volume of cylinder.

Given \( r = 10 \) ft, \( h = 15 \) ft.

\( V_{c}=\pi r^{2}h=3.14\times10^{2}\times15=3.14\times100\times15 = 4710 \) cubic feet.

Step 2: Calculate volume of the hole.

If the hole is a triangular prism with base area \( \frac{1}{2}\times3\times5 \) (since the diagram has a triangle with base 3 ft and height 5 ft) and length 15 ft.

Base area of triangle: \( A=\frac{1}{2}\times3\times5 = 7.5 \) square feet.

Volume of triangular prism (hole): \( V_{h}=A\times h_{cylinder}=7.5\times15 = 112.5 \) cubic feet. Wait, no, that's too small.

Wait, maybe the radius of the cylinder is 5 ft? Let's try that.

If \( r = 5 \) ft, \( h = 15 \) ft.

\( V_{c}=\pi r^{2}h=3.14\times5^{2}\times15=3.14\times25\times15 = 1177.5 \) cubic feet. No, that's not matching the options.

Wait, the options include 4935, 4485, 4710, 6840.

Wait, let's re - evaluate. Maybe the cylinder has radius \( r = 10 \) ft, height \( h = 15 \) ft. Volume of cylinder: \( 3.14\times10^{2}\times15 = 4710 \). Now, the hole: let's say the hole is a triangular prism with base area \( \frac{1}{2}\times3\times5 = 7.5 \) and length 15. Volume of hole: \( 7.5\times15 = 112.5 \). No, 4710 - 112.5 = 4597.5, not in options.

Wait, maybe the hole is a rectangular prism with dimensions 3 ft (width), 5 ft (height), and 15 ft (length). Volume of hole: \( 3\times5\times15 = 225 \). Then volume of solid: \( 4710-225 = 4485 \) cubic feet. Ah, that's one of the options.

So:

Step1: Calculate Cylinder Volume

\( V_{c}=\pi r^{2}h \), \( r = 10 \), \( h = 15 \), \( \pi = 3.14 \)
\( V_{c}=3.14\times10^{2}\times15=3.14\times100\times15 = 4710 \) cubic feet.

Step2: Calculate Hole Volume

The hole is a rectangular prism with length \( l = 15 \) ft, width \( w = 3 \) ft, height \( h_{hole}=5 \) ft.
\( V_{h}=l\times w\times h_{hole}=15\times3\times5 = 225 \) cubic feet.

Step3: Calculate Solid Volume

\( V = V_{c}-V_{h}=4710 - 225=4485 \) cubic feet.

Answer:

4,485 cubic feet