Related rates problem

Question

one leg of a right triangle is decreasing at a rate of 3 cm/sec and the otherleg is increasing at a rate of 4 cm/sec. At what rate is the area of thetriangle changing when the length of the decreasing leg is 8 cm and that ofthe increasing leg is 5 cm? Is the area of the triangle increasing or decreasingat that instant?

Answer 1

Let x = length (in cm) of the decreasing leg at time t seconds

 

     y = length (in cm) of the increasing leg at time t seconds

 

     A = area of triangle at time t seconds

 

Given:  dx/dt = -3   dy/dt = 4

 

Find:  dA/dt when x = 8 and y = 5

 

Since A = (1/2)xy, we have dA/dt = (1/2)[x(dy/dt) + y(dx/dt)]

 

                                                 = (1/2)[(8)(4) + (5)(-3)]

 

                                                 = 8.5 square cm/sec

 

When x = 8 cm and y = 5 cm, the area of the triangle is increasing at the rate of 8.5 square cm/sec.