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Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%

Question:

Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%

Solution:

The surface area of a cube $(S)$ of side $x$ is given by $S=6 x^{2}$.

$\begin{aligned} \therefore \frac{d S}{d x} &=\left(\frac{d S}{d x}\right) \Delta x & & \\ &=(12 x) \Delta x & & \\ &=(12 x)(0.01 x) & &[\text { as } 1 \% \text { of } x \text { is } 0.01 x] \\ &=0.12 x^{2} & & \end{aligned}$

Hence, the approximate change in the surface area of the cube is $0.12 x^{2} \mathrm{~m}^{2}$.

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