Question

the surface area of this cone is 621.72 square meters. what is the slant height of this cone? use π ≈ 3.14 and round your answer to the nearest hundredth. image of cone with radius 9 m s ≈ □ meters submit

Explanation:

Step1: Recall the formula for the surface area of a cone

The surface area \( SA \) of a cone is given by \( SA=\pi r^{2}+\pi r s \), where \( r \) is the radius of the base and \( s \) is the slant height. We know that \( SA = 621.72\) square meters, \( r = 9\) meters, and \( \pi\approx3.14\). First, we calculate the area of the base \( \pi r^{2} \).
\( \pi r^{2}=3.14\times9^{2}=3.14\times81 = 254.34 \)

Step2: Subtract the base area from the total surface area to get the lateral surface area

The lateral surface area \( LSA=\pi r s \), and \( LSA=SA - \pi r^{2} \). Substituting the known values:
\( LSA=621.72 - 254.34=367.38 \)

Step3: Solve for the slant height \( s \)

We know that \( LSA=\pi r s \), so we can solve for \( s \) by rearranging the formula: \( s=\frac{LSA}{\pi r} \). Substituting \( LSA = 367.38 \), \( \pi = 3.14 \), and \( r = 9 \):
\( s=\frac{367.38}{3.14\times9}=\frac{367.38}{28.26}=13.0 \) (Wait, let's check the calculation again. Wait, \( 3.14\times9 = 28.26 \), and \( 367.38\div28.26 = 13.0 \)? Wait, no, wait \( 28.26\times13=367.38 \), yes. Wait, but let's re - check the steps.

Wait, the surface area formula: \( SA=\pi r^{2}+\pi r s=\pi r(r + s) \). Let's use this formula directly.

\( 621.72=3.14\times9\times(9 + s) \)

First, divide both sides by \( 3.14\times9 \):

\( \frac{621.72}{3.14\times9}=9 + s \)

Calculate \( \frac{621.72}{28.26}=22 \) (Wait, I made a mistake earlier. \( 3.14\times9 = 28.26 \), \( 621.72\div28.26 = 22 \)). Then:

\( 22=9 + s \)

Subtract 9 from both sides:

\( s=22 - 9=13.00 \)? Wait, no, wait \( 28.26\times22 = 621.72 \), yes. Then \( 9 + s=22 \), so \( s = 13.00 \)? Wait, let's do the calculation step by step.

First, calculate \( \pi r^{2}=3.14\times9^{2}=3.14\times81 = 254.34 \)

Then, the lateral surface area is \( 621.72-254.34 = 367.38 \)

The formula for lateral surface area is \( \pi r s \), so \( s=\frac{\text{Lateral Surface Area}}{\pi r}=\frac{367.38}{3.14\times9}=\frac{367.38}{28.26}=13.00 \)

Wait, but when we use the formula \( SA=\pi r^{2}+\pi r s \), and substitute \( SA = 621.72 \), \( r = 9 \), \( \pi=3.14 \):

\( 621.72=3.14\times9^{2}+3.14\times9\times s \)

\( 621.72 = 254.34+28.26s \)

Subtract 254.34 from both sides:

\( 621.72 - 254.34=28.26s \)

\( 367.38 = 28.26s \)

Divide both sides by 28.26:

\( s=\frac{367.38}{28.26}=13.00 \)

Answer:

\( 13.00 \)