Chapter 11, Problem 44RP

To determine

To calculate: Thepercentage of honey comb which is made up of wax.

Answer to Problem 44RP

The honey comb which is made up of waxis $36\%$ .

Explanation of Solution

Given information:

Side of smaller hexagon= $4$

Side of longer hexagon = $5$

Formula used:

Apothem of polygon:

  $a=\frac{s}{2\tan \left(\frac{{180}^{\circ }}{n}\right)}$

s = side of polygon

n = number of sides

Area of regular polygon: $A=\frac{1}{2}\ast a\ast P$

  $\begin{array}{l}a=Apothem,\\ P=Perimeter.\end{array}$

Calculation:

  

Perimeter is sum of all sides.

Area of longer polygon:

n =6

Apothem of polygon:

  $a=\frac{5}{2\tan \left(\frac{{180}^{\circ }}{6}\right)}=\frac{5}{2\tan \left({30}^{\circ }\right)}=\frac{5}{2\ast \frac{1}{\sqrt{3}}}=\frac{5\sqrt{3}}{2}$

  $P=5+5+5+5+5+5=30$

  $\begin{array}{l}{A}_{LongerPolygon}=\frac{1}{2}\ast a\ast P\\ A_{LongerPolygon}=\frac{1}{2}\ast (\frac{5}{2}\sqrt{3})\ast (30)\\ A_{LongerPolygon}=\frac{75}{2}\sqrt{3}\end{array}$

Area of smallerpolygon:

n =6

Apothem of polygon:

  $a=\frac{4}{2\tan \left(\frac{{180}^{\circ }}{6}\right)}=\frac{4}{2\tan \left({30}^{\circ }\right)}=\frac{4}{2\ast \frac{1}{\sqrt{3}}}=2\sqrt{3}$

  $P=4+4+4+4+4+4=24$

  $\begin{array}{l}{A}_{SmallerPolygon}=\frac{1}{2}\ast a\ast P\\ A_{SmallerPolygon}=\frac{1}{2}\ast (2\sqrt{3})\ast (24)\\ A_{SmallerPolygon}=24\sqrt{3}\end{array}$

Area of wax = Area of longer polygon - Area of smaller polygon

  $A_{Wax}=\frac{75}{2}\sqrt{3}-24\sqrt{3}$

The percentage of wax is the ratio of area of wax and area of longer hexagon.

Honey comb which is made up of wax $=\frac{A_{Wax}}{A_{LongerHexagon}}=\frac{\frac{27}{2}\sqrt{3}}{\frac{75}{2}\sqrt{3}}=\frac{27}{75}=36\%$