Eli making a model castle out of clay. One of the roof peaks is in the shape of a cone with a diameter of 14 inches in a slant height of 20 inches. What is the surface area of the cone peak and your answer to the nearest hundredth. use 3.14 for pi. Why is the answer 593.46 and NOT 595.14
To find the surface area of the cone peak, we first need to calculate the slant height.
The slant height, denoted as 'l', can be found using the Pythagorean theorem in a right triangle formed by the height (h), the slant height (l), and half the diameter (d/2).
Using the Pythagorean theorem:
l² = h² + (d/2)²
Substituting the known values:
l² = 20² + (14/2)²
l² = 400 + 7²
l² = 400 + 49
l² = 449
Taking the square root of both sides:
l ≈ √449
l ≈ 21.18
Now that we have the slant height, we can calculate the surface area of the cone peak.
The surface area of the cone peak, denoted as 'A', includes the area of the base (B) and the lateral area (L).
The base of the cone peak is a circle, so its area can be calculated using the formula:
B = πr²
The radius of the base is half the diameter, so:
r = 14/2
r = 7
Substituting the known values and approximating π as 3.14:
B ≈ 3.14(7)²
B ≈ 3.14(49)
B ≈ 153.86
The lateral area of the cone peak can be calculated using the formula:
L = πrl
Substituting the known values:
L ≈ 3.14(7)(21.18)
L ≈ 3.14(147.26)
L ≈ 462.47
Finally, we can calculate the surface area by adding the base and lateral areas:
A = B + L
A ≈ 153.86 + 462.47
A ≈ 616.33
Rounded to the nearest hundredth, the surface area of the cone peak is approximately 616.33 square inches.
Therefore, the correct answer is 616.33, not 595.14.