Chapter 3.4, Problem 3E

(a)

To determine

To calculate: The area of an equilateral triangle as a function of the side length s

Answer to Problem 3E

The area of an equilateral triangle is $\text{A}=\frac{\sqrt{3}}{4}{s}^2$.

Explanation of Solution

Given Information: The figure in discussion is an equilateral triangle.

Calculation:

The area of an equilateral triangle is,

  $\text{A}=\frac{\sqrt{3}}{4}{s}^2$ …… (1)

• Here, A is an equilateral triangle and r is the side of the equilateral triangle.

Conclusion:

The area of an equilateral triangle is $\text{A}=\frac{\sqrt{3}}{4}{s}^2$.

(b)

To determine

To calculate: The rate of change of area with respect to side s.

Answer to Problem 3E

The rate of change of area with respect to the side s is $\frac{\sqrt{3}}{2}s$.

Explanation of Solution

Given Information: The area of an equilateral triangle is $\text{A}=\frac{\sqrt{3}}{4}{s}^2$.

Calculation:

The area of an equilateral triangle is,

  $\text{A}=\frac{\sqrt{3}}{4}{s}^2$

The rate of change of area is equal to the derivative of area with respect to s ,

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

• Here, $\frac{dA}{ds}$ is the rate of change of area with respect to the side s .

Conclusion:

The rate of change of area with respect to the side s is $\frac{\sqrt{3}}{2}s$.

(c)

To determine

To calculate: The rate of change of area with respect to side s at $s=2\text{ and }s=10$

Answer to Problem 3E

The rate of change of area with respect to the area at $s=2\text{ is }\sqrt{3}$.

The rate of change of area with respect to the area at $s=10\text{ is }5\sqrt{3}$.

Explanation of Solution

Given Information: The rate of change of area with respect to the side is,

  $\frac{dA}{ds}=\frac{\sqrt{3}}{2}s$

Calculation:

The rate of change of area with respect to the side is,

  $\frac{dA}{ds}=\frac{\sqrt{3}}{2}s$

The rate of change of area with respect to the side s at $s=2$ is,

  $\begin{array}{c}{\left[\frac{dA}{ds}\right]}_{s=2}={\left[\frac{\sqrt{3}}{2}s\right]}_{s=2}\\ =\sqrt{3}\end{array}$

• Here, ${\left[\frac{dA}{ds}\right]}_{s=2}$ is the rate of change of area with respect to the circumference at $s=2$ .

The rate of change of area with respect to the side s at $s=10$ is,

  $\begin{array}{c}{\left[\frac{dA}{ds}\right]}_{s=10}={\left[\frac{\sqrt{3}}{2}s\right]}_{s=10}\\ =5\sqrt{3}\end{array}$

• Here, ${\left[\frac{dA}{ds}\right]}_{s=10}$ is the rate of change of area with respect to the circumference at $s=10$ .

Conclusion:

The rate of change of area with respect to the area at $s=2\text{ is }\sqrt{3}$.

The rate of change of area with respect to the area at $s=10\text{ is }5\sqrt{3}$.

(d)

To determine

To calculate: The appropriate units for $\frac{dA}{ds}$.

Answer to Problem 3E

The appropriate unit for $\frac{dA}{ds}=\text{in}$.

Explanation of Solution

Given Information: The rate of change of area with respect to the side.

Calculation: s is measured in inches and A is measured in square inches.

The rate of change of area with respect to the area is,

  $\frac{dA}{ds}$

Substitute the units in above equation,

  $\begin{array}{c}\frac{dA}{ds}=\frac{{\text{in}}^2}{\text{in}}\\ =\text{in}\end{array}$

Conclusion:

The appropriate unit for $\frac{dA}{ds}=\text{in}$.