Chapter 2.4, Problem 3E

a.

To determine

To calculate: The area of the equilateral triangle in terms of side $s$

Answer to Problem 3E

The area of equilateral triangle is $\frac{\sqrt{3}}{4}{\left(s\right)}^2$ .

Explanation of Solution

Given information:

The side of the equilateral of the triangle is $s$ .

Concept used:

The area of the equilateral triangle is given by the formula $A=\frac{\sqrt{3}}{4}{\left(side\right)}^2$ .

Calculation:

Substitute $s$ for side in area of equilateral triangle $A=\frac{\sqrt{3}}{4}{\left(side\right)}^2$ .

  $\begin{array}{c}A=\frac{\sqrt{3}}{4}{\left(side\right)}^2\\ =\frac{\sqrt{3}}{4}{\left(s\right)}^2\end{array}$

Conclusion: The area of the equilateral triangle in terms of $A$ is $\frac{\sqrt{3}}{4}{\left(s\right)}^2$ .

b.

To determine

To calculate: The rate of change of the area of the equilateral triangle with respect to $s$ .

Answer to Problem 3E

The rate of change of area $A$ is $\frac{\sqrt{3}}{2}s$

Explanation of Solution

Given information:

The area of equilateral triangle is $\frac{\sqrt{3}}{4}{\left(s\right)}^2$ .

Concept used:

The formula used is $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$

Calculation:

The area of equilateral triangle in terms of $s$ is $A=\frac{\sqrt{3}}{4}{\left(s\right)}^2$ .

Differentiate the function with respect to $s$ ,

  $\begin{array}{c}\frac{d}{ds}\left(A\right)=\frac{d}{ds}\left(\frac{\sqrt{3}}{4}{\left(s\right)}^2\right)\\ =\frac{\sqrt{3}}{4}\left(2s\right)\\ =\frac{\sqrt{3}}{2}s\end{array}$

Conclusion: The rate of change $\frac{dA}{ds}$ is $\frac{dA}{ds}=\frac{\sqrt{3}}{2}s$ .

c.

To determine

To calculate: The rate of change at $s=2$ , $s=10$ .

Answer to Problem 3E

The rate of derivative at $s=2$ is $3$ and at $s=10$ is $75$ .

Explanation of Solution

Given information:

The rate of change is $\frac{dA}{ds}=\frac{\sqrt{3}}{2}s$ , $s=2$ , $s=10$ .

Calculation:

The rate of change is $\frac{dA}{ds}=\frac{\sqrt{3}}{2}s$ .

Substitute $2$ for $s$ in the rate of change $\frac{dA}{ds}=\frac{\sqrt{3}}{2}s$ .

  $\begin{array}{c}\frac{dA}{ds}=\frac{\sqrt{3}}{2}\times 2\\ =\sqrt{3}\end{array}$

Substitute $10$ for $s$ in the rate of change $\frac{dA}{ds}=\frac{\sqrt{3}}{2}s$ .

  $\begin{array}{c}\frac{dA}{ds}=\frac{\sqrt{3}}{2}\times 10\\ =5\sqrt{3}\end{array}$

Conclusion: The rate of change is $\sqrt{3}$ for $s=2$ and $5\sqrt{3}$ for $s=10$ .

d.

To determine

To calculate: The unit of rate of change $\frac{dA}{ds}$

Answer to Problem 3E

The unit of $\frac{dA}{ds}$ is $i{n}^2/in$

Explanation of Solution

Given information:

The rate of change is $\frac{dA}{ds}=\frac{\sqrt{3}}{2}s$ , unit of area is $\text{square inch}$ while that of side is $inch$ .

Calculation:

The rate of change is $\frac{dA}{ds}=\frac{\sqrt{3}}{2}s$ .

So the unit is $i{n}^2/in$