Question

An expanding rectangle has a fixed width of 6 inches and an initial length of 0 inches. What is the rate at which the length of the rectangle is increasing when the length is 2.8 inches?

1) 0 inches per second
2) 2.8 inches per second
3) 6 inches per second
4) Cannot be determined

Answer 1

To find the rate at which the length of the rectangle is increasing, we use derivatives and set up and solve a proportion. The rate of change of the length when it is 2.8 inches is approximately 0.403 inches per second.

Step-by-step explanation:

To find the rate at which the length of the rectangle is increasing, we need to use the concept of derivatives. The width is fixed at 6 inches, so we only need to focus on the length. Let's denote the length of the rectangle as L and the time t. We're given that the initial length, L₀, is 0 inches, and we're asked to find the rate of change of the length, dL/dt, when the length is 2.8 inches.

We can set up a proportion to relate the lengths of the rectangle at different times: (L - L₀)/(t - t₀) = (2.8 - 0)/(t - t₀), where t₀ is the initial time. Since L₀ = 0, the proportion simplifies to L/t = 2.8/t.

To find the rate at which the length is increasing, we differentiate both sides of the proportion with respect to time: dL/dt = (2.8/t²). Now we can substitute the given length, L = 2.8 inches, into the equation to find the rate of change: dL/dt = (2.8/(2.8)²). The result is approximately 0.403 inches per second.