Question

what is the surface area of this cone? use π ≈ 3.14 and round your answer to the nearest hundredth. cone image with 9 m (slant height) and 10 m (diameter) square meters submit

Explanation:

Step1: Find the radius of the base

The diameter of the base is 10 m, so the radius \( r = \frac{10}{2} = 5 \) m.

Step2: Recall the formula for the surface area of a cone

The surface area of a cone is \( SA = \pi r (r + l) \), where \( r \) is the radius and \( l \) is the slant height. Here, \( r = 5 \) m and \( l = 9 \) m.

Step3: Substitute the values into the formula

\( SA = 3.14 \times 5 \times (5 + 9) \)
First, calculate the sum inside the parentheses: \( 5 + 9 = 14 \)
Then, multiply: \( 3.14 \times 5 \times 14 = 3.14 \times 70 = 219.8 \)
Wait, no, wait. Wait, the formula for the surface area of a cone is also \( SA=\pi r^{2}+\pi rl \), where \( \pi r^{2} \) is the area of the base and \( \pi rl \) is the lateral (or curved) surface area. Let's recalculate using this correct breakdown.

Step1 (revised): Calculate the base area

The radius \( r = 5 \) m, so the base area \( A_{base}=\pi r^{2}=3.14\times5^{2}=3.14\times25 = 78.5 \) square meters.

Step2 (revised): Calculate the lateral surface area

The lateral surface area \( A_{lateral}=\pi rl \), where \( r = 5 \) m and \( l = 9 \) m. So \( A_{lateral}=3.14\times5\times9 = 3.14\times45 = 141.3 \) square meters.

Step3 (revised): Calculate the total surface area

Add the base area and the lateral surface area: \( SA = A_{base}+A_{lateral}=78.5 + 141.3 = 219.8 \) square meters. Wait, but let's check the formula again. Wait, the slant height is given as 9 m? Wait, the diagram shows the slant height as 9 m? Wait, no, wait, the diameter is 10 m, so radius 5 m, slant height 9 m. Wait, but let's confirm the formula for the surface area of a cone. The total surface area of a cone is \( \pi r (r + l) \), which is the same as \( \pi r^{2}+\pi rl \). So substituting \( r = 5 \), \( l = 9 \), \( \pi = 3.14 \):
\( SA = 3.14\times5\times(5 + 9)=3.14\times5\times14 = 3.14\times70 = 219.8 \). Wait, but that seems straightforward. Wait, maybe I made a mistake earlier. Wait, let's check with the two - part formula:
Base area: \( \pi r^{2}=3.14\times25 = 78.5 \)
Lateral surface area: \( \pi rl = 3.14\times5\times9 = 141.3 \)
Total surface area: \( 78.5+141.3 = 219.8 \) square meters.

Answer:

219.80