The base of a right triangle is decreasing at a rate of 4 meters per minute and the height is decreasing at a rate of 2 meters per minute.

Question

When the base is 10 meters and the height is 23 meters, then how fast is the AREA changing?

The rate of change of AREA is _____ square meters per minute.

(Enter your answer as an integer or as a decimal number rounded to 2 places.)

Answer 1

Area of a right triangle = ba/2 = half the base times altitude

(ignore pretty much everything posted here, except the last 3 lines below. Although the first part gives you an idea of related rates too, as well as what's going on with the hypotenuse, should that show up in a similar problem)

the hypotenuse squared = base squared + altitude squared. Use the Pythagorean Theorem, a special case of the Law of Cosines

h^2 = b^2 + a^2. Take the derivative

2hh' = 2bb' + 2aa' where h', b' and a' are dh/dt, db/dt and da/dt, b' =-4 and a=-2, h^2 = 100+529, h= about 25.1

divide by 2

hh' = bb' + aa'

25.1h'= 10(-4) +23(-2)

h = -86/25.1 = -3.426 m/sec

Area = A = ba/2 = (1/2)ba

A' = dA/dt = (1/2)(ba' + ab') = (1/2)[10(-2)+ 23(-4)] = (1/2)(-20-92) = (1/2)(-112) = - 56 ft^2/sec

The area is decreasing at a rate of 56 square feet per second. IF you're really into 2 decimal places, call it - 56.00 ft^2/sec.