Use the image to answer the question. please pick one of the answer choices.
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
863.5 square centimeters
863.5 square centimeters
2,383.26 square centimeters
2,383.26 square centimeters
1,354.125 square centimeters
1,354.125 square centimeters
3,689.5 square centimeters
3,689.5 square centimeters
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To find the surface area of the cone, we need to calculate the area of the base and the lateral surface area.
The formula for the area of the base of a cone is (pi * r^2), where r is the radius of the base. Since the diameter is given as 25 centimeters, the radius is half of that, which is 25/2 = 12.5 centimeters.
Therefore, the area of the base is (3.14 * 12.5^2) = 490.625 square centimeters.
The formula for the lateral surface area of a cone is (pi * r * l), where l is the slant height of the cone. The slant height can be found using the Pythagorean theorem, where the hypotenuse is given as 22 centimeters and one of the legs is the radius (12.5 centimeters).
Using the Pythagorean theorem, we can calculate the other leg as sqrt(hypotenuse^2 - radius^2) = sqrt(22^2 - 12.5^2) = sqrt(484 - 156.25) = sqrt(327.75) = 18.1 centimeters (approximately).
Therefore, the lateral surface area is (3.14 * 12.5 * 18.1) = 684.845 square centimeters (approximately).
Now, we can calculate the surface area by adding the area of the base and the lateral surface area:
Surface area = base area + lateral surface area = 490.625 + 684.845 = 1175.47 square centimeters (approximately).
Therefore, the correct answer is 1,175.47 square centimeters.
thats not even one of the answer choices
I apologize for the confusion. In that case, the closest answer choice to the calculated surface area of the cone would be 1,354.125 square centimeters.
A spyglass in the shape of a cone has a slant height of 18 centimeters and a radius of 6 centimeters. What is the surface area of the spyglass? Use 3.14 for pi.(1 point)
Responses
197.82 square centimeters
197.82 square centimeters
1,356.48 square centimeters
1,356.48 square centimeters
452.16 square centimeters
452.16 square centimeters
339.12 square centimeters
To calculate the surface area of the cone, we need to find the lateral surface area by using the formula (pi * r * l), where r is the radius and l is the slant height.
Given that the radius of the cone is 6 centimeters and the slant height is 18 centimeters, we can calculate the lateral surface area as (3.14 * 6 * 18) = 339.12 square centimeters.
Therefore, the surface area of the spyglass is 339.12 square centimeters.
The correct answer is 339.12 square centimeters.
To find the surface area of the cone, we need to calculate the lateral surface area and add it to the area of the base.
First, let's find the slant height of the cone. The slant height is the distance from the vertex of the cone to the edge of the base, which is the hypotenuse of the right triangle formed by the height and the radius of the base.
Given that the diameter of the base is 25 centimeters, the radius will be half of that, which is 12.5 centimeters. Now we can use the Pythagorean theorem to find the slant height.
Using the formula c^2 = a^2 + b^2, where c is the hypotenuse (slant height), and a and b are the other two sides of the triangle, we have:
22^2 = 12.5^2 + h^2
484 = 156.25 + h^2
h^2 = 327.75
Taking the square root of both sides, we get:
h ≈ 18.097
Now that we have the slant height, we can find the lateral surface area of the cone using the formula:
Lateral Surface Area = π × radius × slant height
Lateral Surface Area ≈ 3.14 × 12.5 × 18.097
Lateral Surface Area ≈ 687.16 square centimeters
The area of the base of the cone is given by the formula:
Base Area = π × radius^2
Base Area ≈ 3.14 × (12.5)^2
Base Area ≈ 3.14 × 156.25
Base Area ≈ 490.625 square centimeters
Finally, we add the lateral surface area and the base area to find the total surface area:
Total Surface Area = Lateral Surface Area + Base Area
Total Surface Area ≈ 687.16 + 490.625
Total Surface Area ≈ 1,177.785 square centimeters
Therefore, the correct answer is not among the given choices. The surface area of the cone is approximately 1,177.785 square centimeters.