The altitude of a triangle is increasing at a rate of 2500 centimeters/minute while the area of the triangle is increasing at a rate of 4500 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 9000 centimeters and the area is 87000 square centimeters?

Note: The "altitude" is the "height" of the triangle in the formula "Area=(1/2)*base*height". Draw yourself a general "representative" triangle and label the base one variable and the altitude (height) another variable. Note that to solve this problem you don't need to know how big nor what shape the triangle really is.

A=(1/2)*b*h, and if h=9000 and A=87000 then b=19.(3)

A'=(1/2)*b'*h+(1/2)*b*h'
2*A'-b*h'=b'*h
2*4500-19.(3)*2500=b'*9000
b'=?

This definitely helped me find the answer to my question. However, why do we add a "2" to A'?

Thank you!

To solve this problem, we need to use the formula for the area of a triangle, which is given by the formula Area = (1/2) * base * height. We are given the rates of change of the altitude and the area, and we need to find the rate of change of the base.

Let's assign variables to the quantities given:
- Let b represent the base of the triangle.
- Let a represent the altitude (height) of the triangle.
- Let A represent the area of the triangle.

From the information given, we have the following rates of change:
da/dt = 2500 cm/min (altitude rate of change)
dA/dt = 4500 cm²/min (area rate of change)

We are asked to find db/dt, the rate at which the base is changing when altitude is 9000 cm and the area is 87000 cm².

To begin, let's differentiate the area formula with respect to time (t) using implicit differentiation:
dA/dt = (1/2) * (db/dt) * a + (1/2) * b * da/dt

We can rearrange this equation to solve for db/dt:
db/dt = (2 * dA/dt - b * da/dt) / a

Substituting the given values into the equation:
da/dt = 2500 cm/min
dA/dt = 4500 cm²/min
a = 9000 cm
A = 87000 cm²

Now we can calculate db/dt:
db/dt = (2 * 4500 cm²/min - b * 2500 cm/min) / 9000 cm

Simplifying the equation:
db/dt = (9000 cm²/min - 2.5bm²/min) / 9000 cm

Therefore, the rate at which the base is changing when the altitude is 9000 cm and the area is 87000 cm² is given by db/dt = (9000 cm²/min - 2.5bm²/min) / 9000 cm.