A square plate of metal is expanding under the action of heat, and its side is increasing at the uniform rate of 0.1 cm per hour. What is the rate of increase of its area at the moment when the side is 16cm long?

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Answer 1 · 2018-04-02T21:57:31

$3.2 c m^{2} h r^{- 1}$ $3.2cm^2hr^-1$

Let the side of the plate have length $x$ $x$ cm,therefore Area of the plate is $x^{2} c m^{2}$ $x^2 cm^2$

Area $A = x^{2}$ $A=x^2$ , differentiating both sides of this expression implicitly with respect to time,

$\frac{d A}{d t} = 2 x \frac{d x}{d t}$ $[dA]/[dt]=2xdx/dt$ , but we know $\frac{d x}{d t} = 0.1 c m^{2} h r^{- 1}$ $dx/dt=0.1cm^2hr^-1$

Therefore, $\frac{d A}{d t} = 2 x [0.1]$ $[dA]/[dt]=2x[0.1]$ , and when $x = 16$ $x=16$ ,

$\frac{d A}{d t}$ $[dA]/[dt]$ = $[2] [16] [0.1]$ $[2][16][0.1]$ = $3.2 c m^{2} h r^{- 1}$ $3.2cm^2hr^-1$ . hope this was helpful.