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Chapter 6: Problem 1

What is the area of the curved surface of a right circular cone of radius 3 and height \(4 ?\)

Short Answer

Expert verified
Answer: The area of the curved surface of the right circular cone is \(15\pi\) square units.

Step by step solution

01

Find the slant height (l) of the cone

To find the slant height (l) of the cone, we can use the Pythagorean theorem, as l, r, and h form a right triangle. The Pythagorean theorem states that \(a^2+b^2=c^2\) for a right-angled triangle, where a and b are the legs and c is the hypotenuse. In this case, \(a=r=3\), \(b=h=4\), and \(c=l\). Solving for l: \(l^2 = r^2 + h^2\)
02

Calculate the slant height (l)

Plug in the values of r and h into the formula and calculate l: \(l^2 = (3)^2 + (4)^2\) \(l^2 = 9 + 16\) \(l^2 = 25\) \(l=5\)
03

Find the lateral surface area of the cone

Now that we have the slant height, we can use the formula for the lateral surface area (A) of a cone: \(A = \pi r l\)
04

Calculate the lateral surface area of the cone

Plug in the values of r and l into the formula and calculate the lateral surface area: \(A = \pi (3)(5)\) \(A = 15\pi\) So, the area of the curved surface of the given right circular cone is \(15\pi\) square units.

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