Question

A rectangle has one side of 8 cm. How fast is the diagonal of the rectangle changing at the instant when the other side is 6 cm and increasing at 3 cm per minute?

Answer 1

The diagonal is increasing at 1.8 cm/minute.

Step-by-step explanation:

Let l be one side,w be other side and s be diagonal.

We have

             s² = l² + w²

We have

          l = 8 cm

          w = 6 cm

           s² = l² + w²  

           s² = 8² + 6²

           s² = 100

           s = 10 cm

Taking derivative with respect to time in s² = l² + w²

           $s^2=l^2+w^2\\\\2s\frac{ds}{dt}=2l\frac{dl}{dt}+2w\frac{dw}{dt}\\\\s\frac{ds}{dt}=l\frac{dl}{dt}+w\frac{dw}{dt}\\\\\frac{dw}{dt}=3cm/min\\\\\frac{dl}{dt}=0cm/min\\\\10\frac{ds}{dt}=8\times 0+6\times 3\\\\\frac{ds}{dt}=1.8cm/min$

The diagonal is increasing at 1.8 cm/minute.