Question

Let l be the length of a diagonal of a rectangle whose sides have lengths x and y, and assume that x and y vary with time. If x increases at a constant rate of 1/7 ft/s and y decreases at a constant rate of 1/8 ft/s, how fast is the size of the diagonal changing when x is ___?

a) Increasing
b) Decreasing
c) Constant
d) Cannot be determined

Answer 1

The size of the diagonal is changing when x is increasing.

Step-by-step explanation:

To find how fast the size of the diagonal is changing, we need to use the Pythagorean theorem, which states that in a rectangle, the square of the diagonal is equal to the sum of the squares of the two sides.

We have dx/dt = 1/7 ft/s and dy/dt = -1/8 ft/s. We are asked to find how fast the diagonal (l) is changing when x is increasing. Let's differentiate the Pythagorean theorem with respect to time:

d(l^2) / dt = d(x^2) / dt + d(y^2) / dt

2l * dl / dt = 2x * dx / dt + 2y * dy / dt

dl / dt = (x * dx / dt + y * dy / dt) / l

Substituting the given values, we get:

dl / dt = (x * (1/7) - y * (1/8)) / l

Since the question asks for how fast the diagonal is changing when x is increasing, the answer is positive, meaning the size of the diagonal is increasing.