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
2,383.26 square centimeters
2,383.26 square centimeters
863.5 square centimeters
863.5 square centimeters
3,689.5 square centimeters
3,689.5 square centimeters
1,354.125 square centimeters
To find the surface area of the cone, we need to calculate the lateral surface area and the base area separately.
The lateral surface area of a cone can be found using the formula: π * r * l, where r is the radius of the base and l is the slant height (in this case, the hypotenuse).
Given that the diameter of the base is 25 centimeters, the radius would be half of that: 25/2 = 12.5 cm.
Using the Pythagorean theorem, we can find the slant height (l):
h^2 + r^2 = l^2
Given that the height (h) is 22 centimeters, we can substitute the values:
22^2 + 12.5^2 = l^2
484 + 156.25 = l^2
640.25 = l^2
l ≈ 25.32 cm (rounded to 2 decimal places)
Now we can calculate the lateral surface area:
π * 12.5 * 25.32 ≈ 994.24 square centimeters (rounded to 2 decimal places)
To find the base area of the cone, we use the formula: π * r^2.
π * 12.5^2 ≈ 490.87 square centimeters (rounded to 2 decimal places)
Finally, we can add the lateral surface area to the base area to get the total surface area of the cone:
994.24 + 490.87 ≈ 1,485.11 square centimeters (rounded to 2 decimal places)
Therefore, the correct answer is not given in the options provided.
but choose one though
If I have to choose one of the given options, the closest surface area to the calculated result would be:
1,354.125 square centimeters.
However, please note that this is not the exact surface area of the cone based on the given dimensions.
To find the surface area of a cone, we need to calculate two parts: the area of the base and the lateral area.
1. Area of the base:
The base of the cone is a circle. The formula for the area of a circle is A = πr^2, where A is the area and r is the radius. In this case, we are given the diameter of the base, which is 25 centimeters. The radius is half the diameter, so it would be 25/2 = 12.5 centimeters. Using the formula, the area of the base is A = 3.14 * (12.5)^2.
2. Lateral area:
The lateral area of a cone is the curved surface area excluding the base. The formula for the lateral area of a cone is A = πrl, where A is the lateral area, r is the radius, and l is the slant height. The slant height can be found using the Pythagorean theorem, where the hypotenuse is the slant height and the other two sides are the radius and height of the cone. It is given that the hypotenuse is 22 centimeters, and the height is also given as the perpendicular height, which means it is just the radius. Using the Pythagorean theorem, we can find the slant height: l = √(22^2 - 12.5^2). Now we can use the formula for the lateral area to calculate it.
Finally, to find the surface area, we add the area of the base and the lateral area.
Compute the calculations to find the correct surface area.