Use the image to answer the question.
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
3,689.5 square centimeters
3,689.5 square centimeters
2,383.26 square centimeters
2,383.26 square centimeters
863.5 square centimeters
To find the surface area of the cone, we need to find the slant height and the area of the base.
Given:
Diameter of the base = 25 cm
radius of the base = diameter / 2 = 25 cm / 2 = 12.5 cm
Perpendicular height from the base to the top vertex = 90 degrees
Hypotenuse (slant height) = 22 cm
To find the slant height, we can use the Pythagorean theorem:
(slant height)^2 = (perpendicular height)^2 + (radius)^2
(slant height)^2 = (25 cm)^2 + (12.5 cm)^2
(slant height)^2 = 625 cm^2 + 156.25 cm^2
(slant height)^2 = 781.25 cm^2
slant height = sqrt(781.25 cm^2)
slant height ≈ 27.92 cm
Now, let's find the area of the base:
Area of the base = (pi)(radius)^2
Area of the base = (3.14)(12.5 cm)^2
Area of the base ≈ 490.625 cm^2
Finally, let's find the surface area of the cone:
Surface area of the cone = area of the base + area of the lateral surface
Surface area of the cone = area of the base + (pi)(radius)(slant height)
Surface area of the cone ≈ 490.625 cm^2 + (3.14)(12.5 cm)(27.92 cm)
Surface area of the cone ≈ 490.625 cm^2 + 1089.5 cm^2
Surface area of the cone ≈ 1580.125 cm^2
Therefore, the surface area of the cone is approximately 1580.125 square centimeters.
However, none of the provided answer choices match this result.