Question

The altitude (i.e., height) of a triangle is increasing at a rate of 2 cm/minute while the area of the triangle is increasing at a rate of 2.5 square cm/minute. At what rate is the base of the triangle changing when the altitude is 8 centimeters and the area is 96 square centimeters?

Answer 1

The base base of the triangle is decreasing at 5.375cm/min.

Explanation:

Area of triangle = 0.5 x Base x Altitude

A=0.5bh

Differentiating with respect to time

`(dA)/(dt)=(d)/(dt)\left ( 0.5bh\right )\\\\(dA)/(dt)=0.5\left ( b(dh)/(dt)+h(db)/(dt)\right )`

We have

A= 96 cm²

h = 8 cm

A = 0.5bh

96 = 0.5 x b x 8

b = 24 cm

`(dA)/(dt)=2.5cm^2/min\\\\(dh)/(dt)=2cm/min`

Substituting

`2.5=0.5\left ( 24* 2+8(db)/(dt)\right )\\\\48+8(db)/(dt)=5\\\\(db)/(dt)=-5.375cm/min`

The base base of the triangle is decreasing at 5.375cm/min.