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?

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

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