Question

Slant height of a right circular cone is 13 cm and its total surface area is 90\( \pi \) cm\(^2\). Find the diameter of its base.

Hint:

When solving a quadratic equation in a geometry problem, always check the physical feasibility of the solutions. Lengths, areas, and volumes cannot be negative, so discard any negative roots.

Answer 1

Step 1: Understanding the Concept:
The total surface area (TSA) of a cone is the sum of the area of its circular base and its curved surface area. We are given the TSA and the slant height, and we need to find the radius to calculate the diameter.
Step 2: Key Formula or Approach:
The formula for the TSA of a cone with radius \(r\) and slant height \(l\) is:
\[ \text{TSA} = \pi r^2 + \pi r l = \pi r(r+l) \] Step 3: Detailed Explanation:
Given values:
Slant height, \( l = 13 \) cm.
Total Surface Area, TSA = \( 90\pi \) cm\(^2\).
Substitute these values into the TSA formula:
\[ 90\pi = \pi r(r + 13) \] Divide both sides by \( \pi \):
\[ 90 = r(r + 13) \] \[ 90 = r^2 + 13r \] Rearrange into a standard quadratic equation:
\[ r^2 + 13r - 90 = 0 \] We can solve this by factoring. We need two numbers that multiply to -90 and add up to 13. These numbers are 18 and -5.
\[ (r + 18)(r - 5) = 0 \] This gives two possible values for the radius: \(r = -18\) or \(r = 5\).
Since the radius cannot be negative, we take \( r = 5 \) cm.
The question asks for the diameter of the base.
\[ \text{Diameter} = 2 \times r = 2 \times 5 = 10 \text{ cm} \] Step 4: Final Answer:
The diameter of the base of the cone is 10 cm.