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प्रश्न
If each side of an equilateral triangle increases at the rate of `(sqrt(2)"cm")/sec`, find the rate of increase of its area when its side of length 3 cm.
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उत्तर
If x cm is the side of the equilateral triangle and A is its area, then
`"A" = sqrt(3)/(4)x^2`
Differentiating w.r.t. t, we get
`"dA"/"dt" = sqrt(3)/(4) xx 2xdx/dt = sqrt(3)/(2).xdx/dt` ...(1)
Now, `dx/dt = (sqrt(2)"cm")/sec` aand x = 3 cm
∴ (1) gives, `"dA"/"dt" = sqrt(3)/(2) xx 3 xx sqrt(2)`
= `(3sqrt(6))/(2) "cm"^2/sec`
Hence, rate of increase of the area of equilateral triangle
= `(3sqrt(6))/(2) "cm"^2/sec`.
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