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Mathematics · 2026 · 3 marks
CBSE 2026 · Region 5 · Set 3 · Q26
The volume of a wooden block in the shape of a cube increases at a constant rate as the air becomes moist during the rainy season. Show that the rate…
The volume of a wooden block in the shape of a cube increases at a constant rate as the air becomes moist during the rainy season. Show that the rate of change of its surface area varies inversely as the length of edge of the cube.
Marking-scheme solution
Let x be the length of edge of the cube
Volume $\displaystyle V=x^{3}$
$\displaystyle \dfrac{dV}{dt}=3 x^{2} \dfrac{dx}{dt}=k$ (constant)
Surface area, $\displaystyle S=6 x^{2}$
$\displaystyle \dfrac{dS}{dt}=12 x \dfrac{dx}{dt}=12 x \cdot \dfrac{k}{3 x^{2}}$ using (i)
$\displaystyle =\dfrac{4 k}{x}$
Thus, rate of change of surface area varies inversely as the length of edge of the cube
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