the sides of a square increase in length at a rate of 6 m/sec. a. at what rate is the area of the square…

the sides of a square increase in length at a rate of 6 m/sec. a. at what rate is the area of the square changing when the sides are 14 m long? b. at what rate is the area of the square changing when the sides are 21 m long? a. write an equation relating the area of a square, a, and the side length of the square, s.

the sides of a square increase in length at a rate of 6 m/sec. a. at what rate is the area of the square changing when the sides are 14 m long? b. at what rate is the area of the square changing when the sides are 21 m long? a. write an equation relating the area of a square, a, and the side length of the square, s.

Answer

Explanation:

Step1: Write area - side relationship

The area of a square $A$ with side length $s$ is given by the formula $A = s^{2}$.

Step2: Differentiate with respect to time $t$

Using the chain - rule, $\frac{dA}{dt}=2s\frac{ds}{dt}$. We know that $\frac{ds}{dt}=6$ m/sec.

Step3: Solve part (a)

When $s = 14$ m and $\frac{ds}{dt}=6$ m/sec, substitute into the derivative formula: $\frac{dA}{dt}=2\times14\times6=168$ m²/sec.

Step4: Solve part (b)

When $s = 21$ m and $\frac{ds}{dt}=6$ m/sec, substitute into the derivative formula: $\frac{dA}{dt}=2\times21\times6 = 252$ m²/sec.

Step5: Answer for the equation in part (a)

The equation relating the area of a square $A$ and the side length $s$ is $A=s^{2}$.

Answer:

a. 168 m²/sec b. 252 m²/sec a. $A = s^{2}$