The cost of polishing the total surface area of a solid cone at ₹ 0.50 per cm² is ₹ 5632, and the circumference of its base is 176 cm. What is the height of the cone ? (Use π=\(\frac{22}{7}\))

Given:

Cost of polishing the total surface area of a solid cone = ₹ 5632

Rate of polishing = ₹ 0.50 per cm²

Circumference of the base of the cone = 176 cm

Formula used:

Total Surface Area (TSA) = Total Cost / Rate per cm²

Circumference of the base of a cone (C) = 2πr (where r is the radius of the base)

Total surface area of a cone (TSA) = πr(r + l) (where l is the slant height)

Relationship between height (h), radius (r), and slant height (l): l² = r² + h² (Pythagorean theorem)

Calculation:

TSA = Total Cost / Rate per cm²

TSA = 5632 / 0.50

TSA = 5632 × 2

TSA = 11264 cm²

Circumference (C) = 2πr

176 = 2 × (22/7) × r

176 = (44/7) × r

r = (176 × 7) / 44

r = 4 × 7 = 28 cm

Total Surface Area (TSA) = πr(r + l)

11264 = (22/7) × 28 × (28 + l)

11264 = 22 × 4 × (28 + l)

11264 = 88 × (28 + l)

28 + l = 11264 / 88

28 + l = 128

l = 128 - 28 = 100 cm

Using the Pythagorean theorem: l² = r² + h²

100² = 28² + h²

10000 = 784 + h²

h² = 10000 - 784

h² = 9216

h = √9216 = 96

∴ The correct answer is option 4.