Question

65-70 Find a formula for the described function and state its domain.

67. Express the area of an equivalent triangle as a function of the length of a side.

Short answer

The formula for the described function is \(A\left( x \right) = \frac{{\sqrt 3 }}{4}{x^2}\), and the domain of \(A\) is \(x > 0\).

Step-by-step solution

Define the domain of the function

The domain of the function is the set of all inputs for which the formula produces a real number output.

Determine the formula for the described function and state its domain

Consider \(x\) as the length of a side of the equivalent triangle.

The figure of the equivalent triangle is shown below:

By the Pythagoras theorem,

\({y^2} + {\left( {\frac{1}{2}x} \right)^2} = {x^2}\)

\({y^2} = {x^2} - \frac{1}{4}{x^2}\)

\({y^2} = \frac{3}{4}{x^2}\)

\(y = \frac{{\sqrt 3 }}{2}x\)

Use the formula of area \(A\) of the triangle as shown below:

\(A = \frac{1}{2}b \times h\)

Express the area of the triangle as a function of the length of a side as shown below:

\(\begin{aligned}A\left( x \right) &= \frac{1}{2}\left( x \right)\left( {\frac{{\sqrt 3 }}{2}x} \right)\\ &= \frac{{\sqrt 3 }}{4}{x^2}\end{aligned}\)

Thus, the formula for the described function is \(A\left( x \right) = \frac{{\sqrt 3 }}{4}{x^2}\), and the domain of \(A\) is \(x > 0\).