Chapter 1.1, Problem 55E

To determine

To find: The function of the area of an equilateral triangle in terms of the length of the side and the function domain.

Answer to Problem 55E

The formula for the function of area of an equilateral triangle in terms of the length of side is $A\left(x\right)=\frac{\sqrt{3}}{4}{x}^2$.

The domain of the function of the area (A) of an equilateral triangle in terms of length of the side is $x>0$.

Explanation of Solution

Formula used:

Area of the equilateral triangle, $A=\frac{1}{2}\times \text{base}\times \text{height}$.

Calculation:

Let the side length of the equilateral triangle be x.

Obtain the height of the equilateral triangle as follows:

Draw a line from any vertex to its opposite side so that it makes two equal right angled triangles.

Consider any one of the right angled triangle.

Since the side of the equilateral triangle is x, the base of the right angled triangle is $\frac{x}{2}$.

Let the height of the right angled triangle be h. Note that the height of the right angled triangle is the height of the equilateral triangle.

Apply Pythagoras theorem and obtain the value of height.

$\begin{array}{l}{\left(\frac{x}{2}\right)}^2+h^2=x^2\\ \frac{x^2}{4}+h^2=x^2\\ h^2=x^2-\frac{x^2}{4}\\ h=\frac{\sqrt{3}x}{2}\end{array}$

Therefore, the height of the equilateral triangle is $h=\frac{\sqrt{3}x}{2}$.

Thus, the area of the equilateral triangle is $A=\frac{\sqrt{3}{x}^2}{4}$.

Therefore, the function of the area of an equilateral triangle in terms of its length of a side is $A\left(x\right)=\frac{\sqrt{3}}{4}{x}^2$.

The domain of the function of the area (A) of an equilateral triangle in terms of length of side is $x>0$ as x cannot take the negative values.