Question

the surface area of this cone is 734.76 square yards. what is the slant height of this cone? round your answer to the nearest hundredth. 9 yd s ≈ yards

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, \( s \) is the slant height. We know \( SA = 734.76\) square yards and \( r=9\) yards. First, calculate the area of the base \( \pi r^{2} \).
\( \pi r^{2}=\pi\times9^{2}=81\pi\approx 81\times 3.1416 = 254.4696 \) square yards.

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

The lateral surface area \( LSA=SA - \pi r^{2} \). So \( LSA=734.76 - 254.4696=480.2904 \) square yards.

Step3: Use the lateral surface area formula to solve for slant height

The lateral surface area of a cone is \( LSA = \pi r s \). We can solve for \( s \): \( s=\frac{LSA}{\pi r} \).
Substitute \( LSA = 480.2904 \), \( r = 9 \) and \( \pi\approx3.1416 \) into the formula:
\( s=\frac{480.2904}{3.1416\times9}=\frac{480.2904}{28.2744}\approx 17.0 \) (Wait, let's recalculate more accurately. Wait, maybe I made a mistake in step 2. Wait, let's do it again.

Wait, the surface area formula is \( SA=\pi r(r + s) \). So \( 734.76=\pi\times9\times(9 + s) \). Let's solve for \( (9 + s) \) first. \( 9 + s=\frac{734.76}{\pi\times9} \). \( \pi\approx3.14 \), so \( \pi\times9 = 28.26 \). Then \( \frac{734.76}{28.26}=26 \). So \( 9 + s=26 \), then \( s=26 - 9 = 17 \). Wait, but let's check with more precise \( \pi \). Let's use \( \pi = 3.14159265 \).

\( \pi\times9=28.27433385 \). Then \( \frac{734.76}{28.27433385}\approx25.986 \). Then \( 25.986-9 = 16.986\approx16.99 \)? Wait, no, wait the surface area formula: \( SA=\pi r^{2}+\pi r s=\pi r(r + s) \). So \( 734.76 = 3.14159265\times9\times(9 + s) \). Let's compute \( 3.14159265\times9 = 28.27433385 \). Then \( 734.76\div28.27433385\approx25.986 \). Then \( 25.986-9 = 16.986\approx16.99 \)? Wait, no, maybe my initial approach was wrong. Wait, let's start over.

Wait, the radius is 9 yards. The surface area is 734.76. The formula for the total surface area of a cone is \( SA = \pi r^2 + \pi r s \). So:

\( 734.76=\pi\times9^{2}+\pi\times9\times s \)

\( 734.76 = 81\pi+9\pi s \)

Factor out \( \pi \):

\( 734.76=\pi(81 + 9s) \)

Divide both sides by \( \pi \) (using \( \pi\approx3.14 \)):

\( \frac{734.76}{3.14}=81 + 9s \)

\( 234 = 81 + 9s \)

Subtract 81 from both sides:

\( 234 - 81=9s \)

\( 153 = 9s \)

\( s = 17 \). Oh! Because \( 734.76\div3.14 = 234 \) (since \( 3.14\times234 = 734.76 \)). Then \( 234-81 = 153 \), \( 153\div9 = 17 \). So the slant height is 17.00 yards.

Wait, let's verify: \( SA=\pi\times9^{2}+\pi\times9\times17=\pi\times81+\pi\times153=\pi\times(81 + 153)=\pi\times234\approx3.14\times234 = 734.76 \), which matches. So the slant height is 17.00 yards.

Answer:

\( 17.00 \)