Question:
A rectangle of perimeter 30 cm is revolved along one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.
A rectangle of perimeter 30 cm is revolved along one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.
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To maximize volume, the side being squared (the radius) should generally be the larger side.
Updated On: Sep 10, 2026
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Solution and Explanation
Concept:
• Volume of a cylinder is \( V = \pi r^2 h \).
• If a rectangle of sides \( x, y \) is revolved about side \( x \), the radius of the resulting cylinder is \( y \) and height is \( x \).
Step 1: Express dimensions in one variable
Perimeter \( P = 2(x + y) = 30 \implies x + y = 15 \implies y = 15 - x \).
Step 2: Formulate the volume function
Revolving about side \( x \): Radius \( r = y = 15 - x \), Height \( h = x \).
\[ V(x) = \pi(15 - x)^2 x = \pi(225x - 30x^2 + x^3) \]
Step 3: Maximize the volume
\[ V'(x) = \pi(225 - 60x + 3x^2) = 3\pi(x^2 - 20x + 75) \]
Set \( V'(x) = 0 \): \( (x-5)(x-15) = 0 \).
As \( x < 15 \), we have \( x = 5 \).
Checking second derivative: \( V''(x) = 3\pi(2x - 20) \).
At \( x = 5 \), \( V''(5) = 3\pi(10 - 20) = -30\pi < 0 \). Maxima at \( x = 5 \).
Dimensions are \( x = 5 \text{ cm} \) and \( y = 15 - 5 = 10 \text{ cm} \).
• Volume of a cylinder is \( V = \pi r^2 h \).
• If a rectangle of sides \( x, y \) is revolved about side \( x \), the radius of the resulting cylinder is \( y \) and height is \( x \).
Step 1: Express dimensions in one variable
Perimeter \( P = 2(x + y) = 30 \implies x + y = 15 \implies y = 15 - x \).
Step 2: Formulate the volume function
Revolving about side \( x \): Radius \( r = y = 15 - x \), Height \( h = x \).
\[ V(x) = \pi(15 - x)^2 x = \pi(225x - 30x^2 + x^3) \]
Step 3: Maximize the volume
\[ V'(x) = \pi(225 - 60x + 3x^2) = 3\pi(x^2 - 20x + 75) \]
Set \( V'(x) = 0 \): \( (x-5)(x-15) = 0 \).
As \( x < 15 \), we have \( x = 5 \).
Checking second derivative: \( V''(x) = 3\pi(2x - 20) \).
At \( x = 5 \), \( V''(5) = 3\pi(10 - 20) = -30\pi < 0 \). Maxima at \( x = 5 \).
Dimensions are \( x = 5 \text{ cm} \) and \( y = 15 - 5 = 10 \text{ cm} \).
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