find the lateral surface of the frustum of a cone 6inches high with upper and lower bases 11inches and 7inches in diameter respectively
To find the lateral surface area of a frustum of a cone, you need to calculate the area of the curved surface between the two bases. The formula for the lateral surface area of a frustum of a cone is given by:
Lateral Surface Area = π(R₁ + R₂)l
where R₁ and R₂ are the radii of the upper and lower bases respectively, and l is the slant height of the frustum.
In this case, the height of the frustum is given as 6 inches. The upper base has a diameter of 11 inches, so the radius of the upper base is 11/2 = 5.5 inches. The lower base has a diameter of 7 inches, so the radius of the lower base is 7/2 = 3.5 inches.
To find the slant height, we can use the Pythagorean theorem. The height of the original cone is given as 6 inches, and the difference in radii between the upper and lower bases is 5.5 - 3.5 = 2 inches. We can use these values to find the slant height as follows:
Slant height (l) = √(h² + (R₁ - R₂)²)
= √(6² + 2²)
= √(36 + 4)
= √40
= 2√10 inches
Now, we can substitute the values into the formula:
Lateral Surface Area = π(5.5 + 3.5)(2√10)
= π(9)(2√10)
= 18π√10 square inches
Therefore, the lateral surface area of the frustum of the cone is 18π√10 square inches.