11. expanding square the sides of a square increase in length at a rate of 2 m/s.\na. at what rate is the…

11. expanding square the sides of a square increase in length at a rate of 2 m/s.\na. at what rate is the area of the square changing when the sides are 10 m long?\nb. at what rate is the area of the square changing when the sides are 20 m long?

11. expanding square the sides of a square increase in length at a rate of 2 m/s.\na. at what rate is the area of the square changing when the sides are 10 m long?\nb. at what rate is the area of the square changing when the sides are 20 m long?

Answer

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)