Answer: the area of the lateral surface of the cone is approximately 429 square inches.
Explanation:
a. To sketch and label a diagram of the cone, start by drawing a triangle with a curved top. The triangle should have a vertical line in the middle, representing the height of the cone. The curved top should connect the two ends of the base of the triangle, which represents the circular base of the cone. Label the height as 14 in and the diameter as 17 in.
b. To determine the area of the lateral surface of the cone, we first need to find the slant height of the cone. The slant height can be found using the Pythagorean theorem. The slant height, l, is the hypotenuse of a right triangle formed by the height (h) and the radius (r) of the base.
Using the given diameter of 17 in, we can find the radius by dividing it by 2. So, the radius is 8.5 in.
Next, we can use the Pythagorean theorem: l^2 = r^2 + h^2. Plugging in the values, we get l^2 = 8.5^2 + 14^2.
Simplifying the equation, we have l^2 = 72.25 + 196.
Adding the two values, we get l^2 = 268.25.
To find the slant height, we take the square root of both sides: l = √268.25.
Rounding to the nearest whole number, the slant height is approximately 16 in.
Finally, we can calculate the area of the lateral surface of the cone. The lateral surface area is given by the formula A = πrl, where r is the radius of the base and l is the slant height.
Using the given radius of 8.5 in and the calculated slant height of 16 in, we can substitute these values into the formula: A = π(8.5)(16).
Evaluating the expression, we get A ≈ 429 square inches.