How do I determine the rate of change in the area of a triangle if the height is 24 and the base is 7?

Question

How do I determine the rate of change in a triangle when the height is 24 and the base is 7? The ladder is falling away from the wall at 2 ft per second.

Answer 1

Let the area of the triangle be xy/2 and dA/dt becomes dA/dt= (xy' +yx')/2. To find y' use the Pythagorean theorem x² +y² =z² where z is constant. So xx' +yy' =0 and y'=-xx'/y and substituting this into dA/dt gives dA/dt =(-x²x'/y +x'y)/2 = x'(y² -x²)/2y. Using y=24ft and x=7ft and x'=2ft/sec gives dA/dt=2(576-49)/48 = 527/24 ft²/sec and thus dA/dt = 21.96 ft²/sec.

Answer 2

Your question is a little confusing. I will assume you mean that the top of the ladder is sliding down the wall at the rate of 2 feet/sec.

The ladder is 25 feet long (use Pythagorean theorem!) and at the outset, i.e. the top of the ladder is 24' high and the base of the ladder is 7 feet from the wall.

As the top of the ladder falls down the wall, the length of the base changes, i.e. both the base and the height of the triangle change with time.

The rate of change in area varies with time, i.e. you can only give a specific rate if you know at what point the ladder is when you want to know that rate of change...or you need to know how long after the ladder starts to fall to determine a rate of change at that point. Nevertheless, the equations you need are below.

base = sqrt(25-h²)

db/dt =[-h/sqrt(25-h²)] dh/dt

dh/dt = -2

A=bh/2

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