Consider a right circular cone of base radius 4cm and height 10cm. A cylinder is to be placed inside the cone with one of the flat surfaces resting on the base of the cone. Find the largest possible total surface area of the cylinder
Answers
Given : a right circular cone of base radius 4cm and height 10cm. A cylinder is to be placed inside the cone with one of the flat surfaces resting on the base of the cone.
To Find : the largest possible total surface area of the cylinder
Solution:
Let say Cylinder radius = r and height = h
A
Height above cylinder in cone = 10 - h
total surface area of the cylinder A = 2πrh + 2πr²
r/4 = (10-h) /10
=> 2.5r = 10 - h
=> h = 10 - 2.5r
A = 2πrh + 2πr²
A = 2πr(10 - 2.5r) + 2πr²
A = 20πr - 5πr² + 2πr²
A = 20πr - 3πr²
dA/dr = 20π - 6πr
dA/dr = 0
=> 20π - 6πr = 0
=> r = 20//6 = 10/3
d²A/dr² = -6π < 0 hence Surface area id maximum when r = 10/3
r = 10/3
h = 10 - 2.5r = 5/3
A = 20πr - 3πr² = 20π(10/3) - 3π(10/3)²
= 200π/3 - 100π/3
= 100π/3 cm²
largest possible total surface area of the cylinder = 100π/3 cm²
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