The altitude (i. e., height) of a triangle is increasing at a rate of 1.5 cm/minute while the area of the triangle is increasing at a rate of 1.5 square cm/minute. At what rate is the base of the triangle changing when the altitude is 10 centimeters and the area is 82 square centimeters?

Answer: The base of the triangle is decreasing at the rate of 2.16cm per minute

Step-by-step explanation:

The area of a triangle is = 1/2 base * height.

We would make use of the product rule to find the rate of change of the base of this triangle.

Let us use A to represent the area, h to represent height, and b to represent base.

dA/dt = [ (1/2) * (dh/dt) * b] + [ (1/2) * (db/dt) * h]

Where:-

dh/dt = 1.5, dA/dt = 1.5, h = 10, db/dt = the unknown.

Before proceeding further, we must find the base of the triangle. Since we already know that the height = 10cm and the area = 82 square centimeters. We can find the base.

Formula for area of a triangle is 1/2 * b * h = A.

We substitute accordingly: 1/2 * b * 10 = 82.

10b/2 = 82

10b = 164

b = 16.4cm

Since the base is 16.4cm, we can calculate the rate of change of the base of the triangle (db/dt) using the product rule formula stated earlier on.

1.5 = [ (1/2) * (1.5) * (16.4) ] + [ (1/2) * (db/dt) * (10) ]

1.5 = (1.5 * 8.2) + (db/dt * 5)

1.5 = (12.3) + (db/dt * 5)

db/dt * 5 = 1.5 - 12.3

db/dt * 5 = - 10.8

db/dt = - 10.8/5

db/dt = - 2.16cm/min

Therefore, the base of the triangle is decreasing at the rate of 2.16cm per minute