He altitude of a triangle is increasing at a rate of 3000 centimeters/minute while the area of the triangle is increasing at a rate of 3500 square centimeters/minute. at what rate is the base of the triangle changing when the altitude is 10000 centimeters and the area is 89000 square centimeters?

By definition, the area of a triangle is given by:

A = (1/2) * (b) * (h)

Where,

b: base of the triangle

h: height of the triangle

Deriving the area we have:

A ' = (1/2) * ((b') * (h) + (b) * (h '))

Clearing b' we have:

b ' = (2A' - (b) * (h ')) / (h)

We must look for the value of the base:

A = (1/2) * (b) * (h)

Substituting values:

89000 = (1/2) * (b) * (10000)

b = (89000) / (5000)

b = 17.8 cm

Then, the speed at which the base changes is:

b ' = (2 * (3500) - (17.8) * (3000)) / (10000)

b ' = - 4.64 cm / min

Answer:

The base of the triangle is decreasing at:

4.64 cm / min