The total surface area of a cone of slant height 1cm and base radius rcm is 224\(\pi\) cm\(^2\). If r : 1 = 2.5, find: (a) correct to one decimal place, the...

The total surface area of a cone of slant height 1cm and base radius rcm is 224\(\pi\) cm\(^2\). If r : 1 = 2.5, find: 

(a) correct to one decimal place, the value of r

(b) correct to the nearest whole number, the volume of the cone [Take \(\pi\) = \(\frac{22}{7}\)]

Answer Details

Given: Total surface area of a cone = 224π cm², slant height = 1 cm, and r : 1 = 2.5

(a) To find the value of r, we can use the formula for the total surface area of a cone:

Total surface area = πrℓ + πr², where ℓ is the slant height of the cone.

Substituting the given values, we get:

224π = πr√(1 + r²) + πr²

Simplifying and rearranging the terms, we get:

√(1 + r²) = 224/r - r

Squaring both sides, we get:

1 + r² = 50176/r² - 448 + r²

Simplifying and rearranging, we get:

r⁴ - 448r² - 50175 = 0

This is a quadratic equation in terms of r², so we can use the quadratic formula:

r² = (448 ± √(448² + 4(1)(50175)))/2

Simplifying, we get:

r² = 225 or r² = 22400

Since r : 1 = 2.5, we can choose the positive value of r², which is 225. Therefore, r = 15 (correct to one decimal place).

(b) To find the volume of the cone, we can use the formula:

Volume = (1/3)πr²ℓ

Substituting the given values, we get:

Volume = (1/3)(22/7)(15)²(1)

Simplifying, we get:

Volume = 350 cm³ (correct to the nearest whole number)

Therefore, the volume of the cone is approximately 350 cm³.