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Question:
The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ?
Answer:

Let two equal sides of triangle  = a

Given base = b

Now area of triangle A = (1/2) * b*√{a2 - (b2 /4)}

Now differentiate with respect to x, we get

δA/δx = (b/2) * (1/2)* 2a/b*√{a2 - (b2 /4)} * δa/δx ............1

Given in the question

δa/δx = 3 and a = b

Put these values in equation 1, we get

δA/δx = (b/2) * (1/2)* [2b/[√{b2 - (b2 /4)}] *3

          = (3b2 / 2)* /[√{(4b2 - b2 )/4} 

          = 3b2 /√3b2

          = √3*√3b2 /b√3

          = √3b cm/second  

So when the two equal sides are equal to the base, then area decrease = √3b cm/second  

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