Question

question 19
calculate the surface area. give the exact area or round to the nearest hundredth.
sa = m²
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Explanation:

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

The surface area \( SA \) of a cone is given by the sum of the lateral (or curved) surface area and the base area. The formula is \( SA=\pi r l+\pi r^{2} \), where \( r \) is the radius of the base, and \( l \) is the slant height of the cone.

First, we need to find the slant height \( l \). We know the radius \( r = 5\space m \) and the height \( h=12\space m \) of the cone. Using the Pythagorean theorem (since in a right triangle formed by the radius, height, and slant height of the cone, \( l^{2}=r^{2}+h^{2} \)), we calculate \( l \):

\( l=\sqrt{r^{2}+h^{2}}=\sqrt{5^{2}+ 12^{2}}=\sqrt{25 + 144}=\sqrt{169} = 13\space m \)

Step2: Calculate the lateral surface area and the base area

• Lateral surface area: \( \pi r l=\pi\times5\times13 = 65\pi \)
• Base area: \( \pi r^{2}=\pi\times5^{2}=25\pi \)

Step3: Calculate the total surface area

Now, add the lateral surface area and the base area to get the total surface area:

\( SA=65\pi+25\pi=90\pi \)

If we want to get the numerical value (rounding to the nearest hundredth), we know that \( \pi\approx3.14159 \), so:

\( SA = 90\times3.14159\approx282.74\space m^{2} \)

Answer:

If we take the exact value, the surface area is \( 90\pi\space m^{2}\approx282.74\space m^{2} \) (rounded to the nearest hundredth). So the answer is \( 90\pi \) (or approximately \( 282.74 \)) \( m^{2} \).