Question

a cone has a height of 7 meters and a diameter of 14 meters. what is its volume?
round your answer to the nearest hundredth.
179.51
89.75
359.01
538.52
a cone has a height of 12 millimeters and a diameter of 22 millimeters. what is its volume?
round your answer to the nearest hundredth.
3039.52
1,519.76
379.94
759.88

Explanation:

Step1: Recall the volume formula for a cone

The volume \( V \) of a cone is given by the formula \( V=\frac{1}{3}\pi r^{2}h \), where \( r \) is the radius and \( h \) is the height. The radius \( r \) is half of the diameter \( d \), so \( r = \frac{d}{2} \).

Step2: Solve the first cone problem (height = 7 m, diameter = 14 m)

First, find the radius: \( r=\frac{14}{2}=7 \) meters.
Then substitute \( r = 7 \), \( h = 7 \) into the volume formula:
\( V=\frac{1}{3}\pi(7)^{2}(7)=\frac{1}{3}\pi\times49\times7=\frac{343\pi}{3}\approx\frac{343\times3.1416}{3}\approx\frac{1077.5688}{3}\approx359.19 \)? Wait, no, wait, let's recalculate. Wait, maybe I made a mistake. Wait, \( \pi\approx3.1416 \), \( 7^2 = 49 \), \( 49\times7=343 \), \( 343\times3.1416 = 343\times3 + 343\times0.1416=1029+48.5688 = 1077.5688 \), then divide by 3: \( 1077.5688\div3\approx359.19 \). But the options are 179.51, 89.75, 359.01, 538.52. Wait, maybe I used the wrong formula? Wait, no, wait, maybe the first problem: let's check again. Wait, height is 7, diameter 14, radius 7. So \( V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\times\pi\times7^{2}\times7=\frac{343\pi}{3}\approx359.01 \) (using \( \pi\approx3.14159 \), \( 343\times3.14159 = 343\times3 + 343\times0.14159=1029+48.56537 = 1077.56537 \), divided by 3 is \( 1077.56537\div3\approx359.19 \), but the option has 359.01, which is close (maybe using \( \pi\approx3.14 \)). Let's use \( \pi = 3.14 \): \( 343\times3.14 = 343\times3 + 343\times0.14 = 1029+48.02 = 1077.02 \), divided by 3: \( 1077.02\div3\approx359.01 \). So the first problem's answer is 359.01 (option C: 359.01).

Step3: Solve the second cone problem (height = 12 mm, diameter = 22 mm)

Radius \( r=\frac{22}{2}=11 \) mm.
Substitute into the volume formula: \( V=\frac{1}{3}\pi(11)^{2}(12)=\frac{1}{3}\pi\times121\times12 = 4\times121\pi=484\pi\approx484\times3.1416\approx1519.76 \)? Wait, no, wait, \( 11^2 = 121 \), \( 121\times12 = 1452 \), \( \frac{1}{3}\times1452 = 484 \), so \( V = 484\pi\approx484\times3.1416\approx1519.76 \)? But the selected option is 3039.52. Wait, that's a contradiction. Wait, no, wait, maybe I messed up the formula. Wait, no, the volume of a cone is \( \frac{1}{3}\pi r^{2}h \). Wait, height is 12, radius 11. So \( r = 11 \), \( h = 12 \). So \( V=\frac{1}{3}\pi\times11^{2}\times12=\frac{1}{3}\pi\times121\times12 = 4\times121\pi = 484\pi\approx1519.76 \). But the selected option is 3039.52, which is double that. Wait, maybe the formula was for a cylinder? No, the problem says cone. Wait, maybe the user made a mistake in the selected option, but let's focus on the first problem.

Wait, the first problem: height 7 m, diameter 14 m (radius 7 m). So \( V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\times\pi\times7^{2}\times7=\frac{343\pi}{3}\approx359.01 \) (using \( \pi\approx3.14 \), \( 343\times3.14 = 1077.02 \), divided by 3 is 359.01). So the answer for the first problem is 359.01 (option C: 359.01).

For the second problem, if we use the cone formula, the correct volume should be approximately 1519.76, which is option B: 1,519.76. But the selected option in the image is 3039.52, which is wrong (maybe a mistake, maybe it's a cylinder? If it's a cylinder, volume is \( \pi r^{2}h= \pi\times11^{2}\times12 = 1452\pi\approx4560.76 \), no. Wait, no, maybe the height is 24? No, the problem says 12. Wait, maybe I made a mistake. Wait, let's recalculate the second problem:

\( r = 22\div2 = 11 \) mm, \( h = 12 \) mm.

\( V=\frac{1}{3}\times\pi\times11^{2}\times12=\frac{1}{3}\times\pi\times121\times12 = 4\times121\times\pi = 484\pi\approx484\times3.1416 = 1519.76 \). So the correct answer for the second problem is 1,519.76 (option B: 1,519.76), but the image shows the selected option as 3039.52, which is incorrect.

But the first problem: let's confirm again. \( r = 7 \), \( h = 7 \).

\( V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\times3.1416\times49\times7=\frac{1}{3}\times3.1416\times343=\frac{1077.5688}{3}\approx359.19 \), which is approximately 359.01 (using \( \pi\approx3.14 \), \( 343\times3.14 = 1077.02 \), \( 1077.02\div3 = 359.0067\approx359.01 \)). So the first problem's answer is 359.01.

Answer:

First cone problem: C. 359.01
Second cone problem: B. 1,519.76