Asked by In desperate need of help

A cone shows a base diameter of 25 centimeters, perpendicular height from the base to the top vertex at 90 degrees, and 22 centimeters as the hypotenuse or the side of the cone.
What is the surface area of the cone? Use 3.14 for pi.

(1 point)
Responses

1,354.125 square centimeters
1,354.125 square centimeters

2,383.26 square centimeters
2,383.26 square centimeters

3,689.5 square centimeters
3,689.5 square centimeters

863.5 square centimeters
Answered by In desperate need of help
Thanks your awesome
Answered by GPT-4o mini
To find the surface area of a cone, we calculate both the lateral surface area and the base area.

1. **Calculate the radius of the base:**

The diameter of the base is given as 25 centimeters.
So, the radius \(r\) is:
\[
r = \frac{\text{Diameter}}{2} = \frac{25}{2} = 12.5 \text{ centimeters}
\]

2. **Find the height of the cone:**

The height \(h\) is given as the perpendicular height from the base to the top vertex, which is:
\[
h = 22 \text{ centimeters}
\]

3. **Calculate the slant height \(l\):**

The hypotenuse (slant height) of the cone is given as 22 centimeters.
Thus:
\[
l = 22 \text{ centimeters}
\]

4. **Calculate the lateral surface area \(A_L\) of the cone:**

The formula for the lateral surface area of a cone is:
\[
A_L = \pi r l
\]

Substituting the values:
\[
A_L = 3.14 \times 12.5 \times 22
\]
\[
A_L = 3.14 \times 275 = 865.5 \text{ square centimeters}
\]

5. **Calculate the base area \(A_B\) of the cone:**

The formula for the area of the base of a cone (which is a circle) is:
\[
A_B = \pi r^2
\]

Substituting the radius:
\[
A_B = 3.14 \times (12.5)^2 = 3.14 \times 156.25 = 490.625 \text{ square centimeters}
\]

6. **Total surface area \(A_T\) of the cone:**

The total surface area is the sum of the lateral surface area and the base area:
\[
A_T = A_L + A_B = 865.5 + 490.625 = 1356.125 \text{ square centimeters}
\]

Rounding to two decimal places, the total surface area is approximately 1,356.13 square centimeters.

Among the given choices, the closest one appears to be:

**1,354.125 square centimeters**

So, the correct response is:
**1,354.125 square centimeters**
Answered by GPT-4o mini
You're very welcome! I'm glad I could help. If you have any more questions or need further assistance, feel free to ask. Have a great day!