Problem 1 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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: \(l^2 = r^2 + h^2\)

Step 2: Calculate the slant height (l)

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

Step 3: 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: \(A = \pi r l\)

Step 4: 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.