Which of the following must be true about the area A of the triangle?

[lude1]

Homework Statement

If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is decreasing at a rate of 3 inches per minute, which of the following must be true about the area A of the triangle?

a. A is always increasing
b. A is always decreasing
c. A is decreasing only when b < h
d. A is decreasing only when b > h
e. A remains constant

Correct answer is d. A is decreasing only when b > h

Homework Equations

The Attempt at a Solution

I don't understand why the answer is d. If the area of a triangle is (1/2)(base)(height) and the base increases by 3 while the height decreases by 3, wouldn't they just cancel out each other?

Moreover, I don't understand why the area would only be decreasing when the base is bigger than the height and not vice versa. But I guess I won't understand this part until I understand why the area can't be constant.

 

[Mark44]

Homework Statement

If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is decreasing at a rate of 3 inches per minute, which of the following must be true about the area A of the triangle?

a. A is always increasing
b. A is always decreasing
c. A is decreasing only when b < h
d. A is decreasing only when b > h
e. A remains constant

Correct answer is d. A is decreasing only when b > h

Homework Equations

The Attempt at a Solution

I don't understand why the answer is d. If the area of a triangle is (1/2)(base)(height) and the base increases by 3 while the height decreases by 3, wouldn't they just cancel out each other?

Moreover, I don't understand why the area would only be decreasing when the base is bigger than the height and not vice versa. But I guess I won't understand this part until I understand why the area can't be constant.

You need to find the dA/dt. Since A is a function of both h and b, and both can be assumed to be functions of t, dA/dt will involve the partial derivatives with respect to h and the partial derivative with respect to b.

 

[hgfalling]

To see why the area wouldn't be constant, take a triangle with height a and base b. Then its area would be ab/2, right? Now suppose that a minute passes. So now it has base b+3 and height a-3.

Find the area of the new triangle and figure out when it's larger or smaller than the other one.