Question

g The altitude of a triangle is increasing at a rate 2 inch/hour while the area of the triangle is decreasing at a rate of 0.5 square inch per hour. At what rate is the base of the triangle is changing when the altitude is 6 inch and the area is 24 square inch

Answer 1

Answer:

The base of the triangle is decreasing at a rate of 1.4167 inch/hour.

Step-by-step explanation:

Area of a triangle:

The area of a triangle of base b and height h is given by:

$A = bh$

In this question:

We have to derivate the equation of the area implicitly in function of time. So

$\frac{dA}{dt} = b\frac{dh}{dt} + h\frac{db}{dt}$

The altitude of a triangle is increasing at a rate 2 inch/hour while the area of the triangle is decreasing at a rate of 0.5 square inch per hour.

This means that:

$\frac{dh}{dt} = 2, \frac{dA}{dt} = -0.5$

At what rate is the base of the triangle is changing when the altitude is 6 inch and the area is 24 square inch?

This is $\frac{db}{dt}$ when $h = 6$

Area is 24, so the base is:

$A = bh$

$24 = 6b$

$b = \frac{24}{6} = 4$

Then

$\frac{dA}{dt} = b\frac{dh}{dt} + h\frac{db}{dt}$

$-0.5 = 4(2) + 6\frac{db}{dt}$

$6\frac{db}{dt} = -8.5$

$\frac{db}{dt} = -\frac{8.5}{6}$

$\frac{db}{dt} = -1.4167$

The base of the triangle is decreasing at a rate of 1.4167 inch/hour.