MHT CET · Maths · Application of Derivatives
The side of a square sheet of metal is increasing at the rate of \(3 \mathrm{~cm} / \mathrm{min}\). At what rate is the area increasing when the length of the side is \(6 \mathrm{~cm}\) long?
- A \(36 \mathrm{~cm}^2 / \mathrm{min}\)
- B \(12 \mathrm{~cm}^2 / \mathrm{min}\)
- C \(18 \mathrm{~cm}^2 / \mathrm{min}\)
- D \(9 \mathrm{~cm}^2 / \mathrm{min}\)
Answer & Solution
Correct Answer
(A) \(36 \mathrm{~cm}^2 / \mathrm{min}\)
Step-by-step Solution
Detailed explanation
\(\mathrm{A}=\mathrm{a}^2\) and \(\frac{\mathrm{da}}{\mathrm{dt}}=3 \mathrm{~cm} / \mathrm{min}\)
Now \(\frac{\mathrm{dA}}{\mathrm{dt}}=2 \mathrm{a} \frac{\mathrm{dA}}{\mathrm{dt}}\)
\(\Rightarrow \frac{\mathrm{dA}}{\mathrm{dt}}=2 \times 6 \times 3 \mathrm{~cm}^2 / \min =36 \mathrm{~cm}^2 / \min\)
Now \(\frac{\mathrm{dA}}{\mathrm{dt}}=2 \mathrm{a} \frac{\mathrm{dA}}{\mathrm{dt}}\)
\(\Rightarrow \frac{\mathrm{dA}}{\mathrm{dt}}=2 \times 6 \times 3 \mathrm{~cm}^2 / \min =36 \mathrm{~cm}^2 / \min\)
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