Chapter 13.6, Problem 37PPSE

Solution

The graph of the function and the area of the placemat if the triangles are equilateral.

The graph of the function is drawn and the area of placemat is 259.8 in².

Given:

The function $y=5\tan \left(\theta \right)$

Concept used:

Equilateral triangle has all the sides equal and has $60°$ or $\frac{\pi }{3}$ as all the interior angles.

Area of the equilateral triangle is $A=\frac{1}{2}bh$

Where, b is the side and h is the height of the triangle.

Calculation:

The graph of the function $y=5\tan \left(\theta \right)$ on a graphing calculator is shown below:

  

The height can be found by substituting the value of $\theta$ as $\frac{\pi }{3}$ in the given function.

  

• Here, the height of the triangle is

  $\begin{array}{c}y=5\tan \left(\theta \right)\\ =5\tan \left(\frac{\pi }{3}\right)\\ =5\sqrt{3}\\ =5\times 1.732\\ y=8.66\end{array}$

  

• Area of an equilateral triangle is $A=\frac{1}{2}bh$

On substituting the values of b and h in the formula of area,

  $\begin{array}{c}A=\frac{1}{2}\times 10\times 8.66\\ A=43.3{\text{ in}}^{\text{2}}\end{array}$

Thus, the area of each triangle is 43.3 in².

• The last step of the solution will be to find out the area of the placemat which can be calculated by multiplying the area of triangle by 6 as it is a hexagonal placemat.

Thus, Area of placemat is

  $\begin{array}{c}{A}_{total}=\text{Area of triangle}\times 6\\ =43.3\times 6\\ A_{total}=259.8{\text{ in}}^2\end{array}$

Conclusion:

The graph of the function is drawn and the area of placemat is 259.8 in².