Question

the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. 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 yards. find the volume of the solid in cubic yards, rounded to the nearest ten. use 3.14 for π.
diagram of a cylinder with a triangular - shaped figure on top labeled 4 yd, 4 yd, and the cylinders height is 6 yd
options: 1,788 cubic yards; 1,034 cubic yards; 1,884 cubic yards; 1,980 cubic yards

Explanation:

Answer:

To find the volume of the solid, we first calculate the volume of the cylinder and then subtract the volume of the rectangular prism (the hole).

Step 1: Volume of the Cylinder

The formula for the volume of a cylinder is \( V_{\text{cylinder}} = \pi r^2 h \), where \( r = 10 \) yards (radius) and \( h = 6 \) yards (height).

\[
V_{\text{cylinder}} = 3.14 \times (10)^2 \times 6 = 3.14 \times 100 \times 6 = 1884 \text{ cubic yards}
\]

Step 2: Volume of the Rectangular Prism (Hole)

The base of the rectangular prism is a parallelogram (or a rectangle, since the triangle and parallelogram suggest a rectangular cross - section). From the diagram, the length and width of the base of the prism are both 4 yards, and the height (same as the height of the cylinder) is 6 yards. The formula for the volume of a rectangular prism is \( V_{\text{prism}}=\text{length}\times\text{width}\times\text{height} \).

\[
V_{\text{prism}} = 4\times4\times6=96 \text{ cubic yards}
\]

Step 3: Volume of the Solid

Subtract the volume of the prism from the volume of the cylinder:

\[
V = V_{\text{cylinder}}-V_{\text{prism}}=1884 - 96=1788 \text{ cubic yards}
\]

Rounded to the nearest ten, 1788 is 1790? Wait, no, wait. Wait, maybe I misread the base of the prism. Wait, the diagram shows a triangle and a parallelogram. Wait, maybe the base of the prism is a rectangle with length equal to the diameter? No, wait, the radius is 10, diameter is 20. Wait, no, the diagram has a 4 yd and 4 yd. Wait, maybe the cross - section of the hole is a rectangle with length 8 yd (since the triangle has base 4 yd, maybe the rectangle has length 8 yd? Wait, no, let's re - examine.

Wait, maybe the base of the rectangular prism is a parallelogram with base 8 yd (since the radius is 10, and the diagram has a 4 yd segment, maybe the length of the prism's base is 8 yd? Wait, no, let's recalculate.

Wait, the volume of the cylinder is \( \pi r^{2}h=3.14\times10^{2}\times6 = 1884 \).

Now, the hole is a rectangular prism. From the diagram, the base of the prism is a parallelogram with base 8 yd (since the triangle has a base of 4 yd, and maybe the rectangle has a length of 8 yd? Wait, no, the original calculation was wrong. Wait, the diagram shows a 4 yd and 4 yd, but maybe the length of the prism is 8 yd (diameter of the cylinder is 20? No, radius is 10, diameter is 20. Wait, maybe the cross - section of the hole is a rectangle with length 8 yd and width 4 yd? Wait, no, let's check the answer options.

The answer options are 1788, 1034, 1884, 1980.

Wait, if we made a mistake in the volume of the prism. Let's assume that the base of the prism is a rectangle with length 8 yd (since the radius is 10, and maybe the length is 8 yd) and width 4 yd, height 6 yd. Then \( V_{\text{prism}}=8\times4\times6 = 192 \). Then \( V = 1884-192 = 1692 \), which is not an option.

Wait, maybe the cross - section of the hole is a triangle - based prism? Wait, the diagram has a triangle with base 4 yd and height 4 yd? No, the diagram shows a parallelogram and a triangle. Wait, maybe the base of the prism is a square with side 4 yd, and the length of the prism is 6 yd? No, that gives 96, and 1884 - 96 = 1788, which is one of the options.

So the volume of the solid is 1788 cubic yards.

1788 cubic yards (rounded to the nearest ten is 1790? But 1788 is one of the options, so the answer is 1788 cubic yards. So the correct option is 1,788 cubic yards.