Question

question find the volume of a right circular cone that has a height of 14.3 ft and a base with a diameter of 17.2 ft. round your answer to the nearest tenth of a cubic foot. answer attempt 2 out of 2 1109.8 ft³ submit answer watch video show examples

Explanation:

Step1: Find the radius

The diameter is 17.2 ft, so the radius \( r=\frac{17.2}{2}=8.6 \) ft.

Step2: Recall the volume formula for a cone

The volume \( V \) of a right circular cone is given by \( V = \frac{1}{3}\pi r^{2}h \), where \( h = 14.3 \) ft is the height.

Step3: Substitute the values into the formula

\( V=\frac{1}{3}\times\pi\times(8.6)^{2}\times14.3 \)
First, calculate \( (8.6)^{2}=73.96 \)
Then, \( \frac{1}{3}\times\pi\times73.96\times14.3 \)
\( \frac{1}{3}\times\pi\times1057.628 \)
\( \approx\frac{1}{3}\times3.1416\times1057.628 \)
\( \approx\frac{3321.0}{3} \) (approximate \( 3.1416\times1057.628\approx3321.0 \))
\( \approx1107.0 \)? Wait, no, let's recalculate:
\( 3.1416\times73.96 = 3.1416\times70 + 3.1416\times3.96=219.912+12.440736 = 232.352736 \)
Then \( 232.352736\times14.3 = 232.352736\times14 + 232.352736\times0.3=3252.9383 + 69.7058 = 3322.6441 \)
Then \( \frac{3322.6441}{3}\approx1107.55 \), wait the given answer in the box is 1109.8, maybe using more precise π. Let's use π≈3.14159265:
\( 8.6^{2}=73.96 \)
\( 73.96\times14.3 = 73.96\times14 + 73.96\times0.3=1035.44+22.188 = 1057.628 \)
\( 1057.628\times\pi\approx1057.628\times3.14159265\approx3321.99 \)
\( \frac{3321.99}{3}\approx1107.33 \)? Wait, maybe the height is 14.3? Wait no, maybe I miscalculated. Wait the problem says height 14.3, diameter 17.2. Wait let's do it step by step:

Radius \( r = 17.2/2 = 8.6 \)

Volume formula: \( V = (1/3)\pi r^2 h \)

So \( r^2 = 8.6^2 = 73.96 \)

\( r^2 h = 73.96 * 14.3 = let's calculate 73.96*14 = 1035.44, 73.96*0.3=22.188, total 1035.44+22.188=1057.628 \)

Then \( (1/3)\pi*1057.628 \approx (1/3)*3.14159265*1057.628 \)

Calculate 3.14159265*1057.628:

1057.628 * 3 = 3172.884, 1057.628 * 0.14159265 ≈ 1057.628*0.1=105.7628, 1057.628*0.04=42.30512, 1057.628*0.00159265≈1.684. So total ≈105.7628+42.30512=148.06792+1.684≈149.75192. So total 3172.884+149.75192≈3322.63592

Then divide by 3: 3322.63592 /3 ≈1107.545, but the box has 1109.8. Wait maybe the height is 14.3? Wait no, maybe I made a mistake in diameter. Wait the problem says diameter 17.2, height 14.3. Wait let's check with calculator:

\( V = (1/3) * \pi * (8.6)^2 *14.3 \)

\( 8.6^2 = 73.96 \)

\( 73.96 *14.3 = 1057.628 \)

\( 1057.628 * \pi ≈ 1057.628 * 3.1415926535 ≈ 3321.99 \)

\( 3321.99 /3 ≈ 1107.33 \). But the given answer in the box is 1109.8. Maybe the height is 14.5? No, the problem says 14.3. Wait maybe the diameter is 17.2, height 14.3. Wait perhaps the user made a typo, but according to the calculation, let's see:

Wait maybe I used the wrong formula? No, cone volume is (1/3)πr²h.

Wait let's use more precise calculation:

\( r = 17.2 / 2 = 8.6 \)

\( r^2 = 8.6 * 8.6 = 73.96 \)

\( h = 14.3 \)

\( V = (1/3) * \pi * 73.96 * 14.3 \)

Calculate 73.96 *14.3:

73.96 *10 =739.6

73.96 *4=295.84

73.96 *0.3=22.188

Sum: 739.6+295.84=1035.44+22.188=1057.628

Now, 1057.628 * π = 1057.628 * 3.141592653589793 ≈

1057.628 * 3 = 3172.884

1057.628 * 0.141592653589793 ≈

1057.628 * 0.1 = 105.7628

1057.628 * 0.04 = 42.30512

1057.628 * 0.001592653589793 ≈ 1.684

So 105.7628 + 42.30512 = 148.06792 + 1.684 ≈ 149.75192

Total 3172.884 + 149.75192 ≈ 3322.63592

Divide by 3: 3322.63592 / 3 ≈ 1107.54597

Rounded to the nearest tenth is 1107.5. But the given answer in the box is 1109.8. Maybe the height is 14.5? Let's check:

If h=14.5, then 73.96*14.5=1072.42, 1072.42*π≈3368.5, 3368.5/3≈1122.8. No.

Wait maybe the diameter is 17.5? 17.5/2=8.75, 8.75²=76.5625, 76.5625*14.3=1094.84375, 1094.84375*π≈3439.5, 3439.5/3≈1146.5. No.

Wait the given answer in the box is 1109.8, so maybe the height is 14.4?

73.96*14.4=1065.024, 1065.024*π≈3346.5, 3346.5/3≈1115.5. No.

Wait maybe the diameter is 17.3? 17.3/2=8.65, 8.65²=74.8225, 74.8225*14.3=1070.0, 1070*π≈3361.5, 3361.5/3≈1120.5. No.

Alternatively, maybe the formula was used as (1/3)πd²h/4, since d=2r, so r=d/2, so r²=d²/4, so V=(1/3)π*(d²/4)*h=(π d² h)/12.

Let's try that: (π *17.2² *14.3)/12

17.2²=295.84

295.84*14.3=4230.512

4230.512*π≈13290.5

13290.5/12≈1107.5. Ah, same as before. So maybe the given answer in the box is a typo, but according to the calculation, the volume is approximately 1107.3, which rounds to 1107.3, but the box has 1109.8. Maybe the problem had different numbers, but based on the given numbers (h=14.3, d=17.2), the volume is approximately 1107.5, which rounds to 1107.5. But since the box has 1109.8, maybe there was a miscalculation in the problem's given answer, but following the steps:

Step1: Radius calculation

\( r = \frac{d}{2} = \frac{17.2}{2} = 8.6 \) ft.

Step2: Volume formula

\( V = \frac{1}{3}\pi r^2 h \)

Step3: Substitute values

\( V = \frac{1}{3} \times \pi \times (8.6)^2 \times 14.3 \)

Calculate \( (8.6)^2 = 73.96 \)

\( 73.96 \times 14.3 = 1057.628 \)

\( \frac{1}{3} \times \pi \times 1057.628 \approx \frac{1}{3} \times 3.1416 \times 1057.628 \approx 1107.5 \)

Rounding to the nearest tenth, it's 1107.5, but the given answer is 1109.8. Maybe the problem intended different numbers, but based on the given, the calculation is as above.

Answer:

\boxed{1107.5} (but the given box has 1109.8, possible error in problem numbers)