Question

The altitude (i.e., height) of a triangle is increasing at a rate of 3 cm/minute while the area of the triangle is increasing at a rate of 4.5 square cm/minute. At what rate is the base of the triangle changing when the altitude is 7.5 centimeters and the area is 86 square centimeters?

Answer 1

The base of the triangle is changing at a rate of 3 cm/minute.

Step-by-step explanation:

To find the rate at which the base of the triangle is changing, we can use the formula for the area of a triangle:

Area = 1/2 * base * height.

We are given that the height is increasing at a rate of 3 cm/minute and the area is increasing at a rate of 4.5 square cm/minute.

We can differentiate the formula for the area of a triangle with respect to time to find the rate of change of the area:

d(Area)/dt = 1/2 * (d(base)/dt) * height

Now, we can substitute the given values into the formula:

4.5 = 1/2 * (d(base)/dt) * 7.5

Solving for d(base)/dt:

(d(base)/dt) = 2 * 4.5 / 7.5

= 3 cm/minute