Question

at a certain instant the base of a triangle is 5 inches and is increasing at the rate of 1 inch per minute. At the same instant, the height is 10 inches and is decreasing at the rate of 2.5 inches per minute. Is the area of the triangle increasing or decreasing? justify you answer.

Answer 1

a = bh/2
da/dt = h/2 db/dt + b/2 dh/dt
= 10/2 (1) + 5/2 (-2.5)
= -1.25 in^2/min

Answer 2

Ok. Just do it the brute-force way.

Initially, the area is 1/2 x 5 x 10 = 25
After one minute, the area is 1/2 x 6 x 7.5 = 22.5
After two minutes, the area is 1/2 x 7 x 5 = 17.5

You can see that the area is decreasing.

In terms of derivatives, the rate of change is exactly what a derivative is.

You can say +1 d/dx for the base and -2.5 d/dx for the height.

Overall that's a decrease in 1 - 2.5 = -1.5 inches every minute.

If you look at the equation for a triangle 1/2 x b x h, the overall value will be less because you're losing -1.5 inches overall.