At what rate is the base of the triangle changing?

Question

The altitude of a triangle is increasing at a rate of 2.000 centimeters/minute while the area of the triangle is increasing at a rate of 4.500 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 7.500 centimeters and the area is 82.000 square centimeters?

 

Please help!

Answer 1

We know what an area of a triangle is:

 

A = 0.5bh

 

Getting the derivative of this via d/dt (Don't panic, it's just basic differentiation with multiple variables):

 

dA/dt = 0.5b(dh/dt) + 0.5h(db/dt)

 

dA/dt = RoC in area, dh/dt = RoC in height, db/dt = RoC in base *RoC = rate of change*

 

4.5 = 0.5b(2) + 0.5(7.5)(db/dt)

 

As you notice, we still have two variables, the db/dt (the question) and the base, b.

 

But you can actually solve for b because they gave the area and the altitude at the moment:

 

A = 0.5bh

82 = 0.5b(7.5)

b = 328/15 = 21.867 *better use the fraction to get a more accurate answer*

 

Going back to the equation:

 

4.5 = 0.5(328/15)(2) + 0.5(7.5)(db/dt)

 

db/dt = -4.631 

 

*NOTE: I find it weird that I got a negative value but I'm pretty sure with the process (and the term used is "changing" not "increasing). Though it's possible depending on the given value. Is this question from a book?*