Question

A right triangle has legs of 27 inches and 36 inches whose sides are changing. The short leg is increasing by 10 in/sec and the long leg is shrinking at 9 in/sec. What is the rate of change of the area?

Answer 1

The rate of change of the area of a right triangle with sides changing at given rates is 18 square inches per second.

Step-by-step explanation:

The rate of change of the area of a right triangle when the lengths of the legs are changing can be determined using related rates. In this problem, we are given that one leg is increasing by 10 in/sec and the other leg is decreasing by 9 in/sec. The formula for the area of a right triangle is A = 0.5 * base * height. Let's denote the short leg (base) as 'b' and the long leg (height) as 'h'.

Using the related rates, we can find the rate of change of the area (dA/dt) with respect to time by differentiating the area formula with respect to time:

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

Plugging in the values, we get:
dA/dt = 0.5 * 10 * 36 + 0.5 * 27 * (-9)

After calculating, we find that the rate of change of the area is 18 square inches per second.

Answer 2

The rate of change of the area of the right triangle is decreasing at a rate of 135 in^2/sec.

Step-by-step explanation:

In order to find the rate of change of the area of the right triangle, we first need to determine its current area. Using the formula for the area of a right triangle (1/2 * base * height), we can calculate that the current area is 1/2 * 27 in * 36 in = 486 in^2.

Next, we can use the chain rule of differentiation to determine the rate of change of the area. Let A be the area of the triangle, x be the length of the short leg, and y be the length of the long leg. Then we have:

dA/dt = (dA/dx) * (dx/dt) + (dA/dy) * (dy/dt)

We know that dA/dx = 1/2 * y and dA/dy = 1/2 * x, so we can substitute these values in and simplify:

dA/dt = (1/2 * y) * (10 in/sec) + (1/2 * x) * (-9 in/sec)

Plugging in the given lengths of the legs, we get:

dA/dt = (1/2 * 36 in) * (10 in/sec) + (1/2 * 27 in) * (-9 in/sec) = 135 in^2/sec

Therefore, the rate of change of the area of the right triangle is decreasing at a rate of 135 in^2/sec.