Question

1. expanding square the sides of a square increase in length at a rate of 2 m/s.

a. at what rate is the area of the square changing when the sides are 10 m long?
b. at what rate is the area of the square changing when the sides are 20 m long?

Explanation:

Step1: Define variables and formula

Let the side length of the square be \(x\) and the area be \(A\). Then \(A = x^{2}\). Differentiate with respect to time \(t\) using the chain rule: \(\frac{dA}{dt}=2x\frac{dx}{dt}\). Given \(\frac{dx}{dt} = 2\space m/s\).

Step2: Solve part (a)

When \(x = 10\space m\), substitute into \(\frac{dA}{dt}=2x\frac{dx}{dt}\).
\(\frac{dA}{dt}=2\times10\times2\)
\(\frac{dA}{dt}=40\space m^{2}/s\)

Step3: Solve part (b)

When \(x = 20\space m\), substitute into \(\frac{dA}{dt}=2x\frac{dx}{dt}\).
\(\frac{dA}{dt}=2\times20\times2\)
\(\frac{dA}{dt}=80\space m^{2}/s\)

Answer:

a. \(40\space m^{2}/s\)
b. \(80\space m^{2}/s\)