CUET Mathematics 2024 - The rate of change (in cm^2 / s ) of the total surface area of a hemisphere with respect to radius r at r=[3]1.331 ~cm is : | PYQs + Solutions

CUET Mathematics 2024

Calculus

Application of Derivatives

Medium

The rate of change (in $\mathrm{cm}^{2} / \mathrm{s}$ ) of the total surface area of a hemisphere with respect to radius $r$ at $r=\sqrt[3]{1.331} \mathrm{~cm}$ is :

66 \pi

6.6 \pi

3.3 \pi

4.4 \pi

✅ Correct Option: 2

The total surface area of a hemisphere includes both the curved surface and the circular base:

A = 2\pi r^2 + \pi r^2 = 3\pi r^2

To find the rate of change, differentiate with respect to

r

\dfrac{dA}{dr} = \dfrac{d}{dr}(3\pi r^2) = 3\pi \cdot \dfrac{d}{dr}(r^2) = 3\pi \cdot 2r = 6\pi r

Evaluate at

r = \sqrt[3]{1.331}

\sqrt[3]{1.331} = \sqrt[3]{\dfrac{11^3}{10^3}} = \dfrac{11}{10} = 1.1

Substitute into our formula:

\dfrac{dA}{dr} = 6\pi r = 6\pi(1.1) = 6.6\pi

^2

Therefore, the rate of change of the total surface area of the hemisphere with respect to radius at

r = \sqrt[3]{1.331}

cm is

6.6\pi

^2

/s.

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2

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