Chapter 12.8, Problem 35SR

To determine

To find: the ratio of the area of the larger flower bed to the smaller flower bed

Answer to Problem 35SR

The ratio of area of larger flower bed to smaller flower bed to smaller flower bed is, $\frac{25}{4}$

Explanation of Solution

Given:

The lengths of each side of the flower beds are $8feet$ and $20feet$

Calculation:

Area of equilateral triangle of side a unit is,

  $Area=\frac{\sqrt{3}}{4}{a}^2$

Area of larger equilateral triangle shaper flower bed is,

  $\begin{array}{l}{A}_I=\frac{\sqrt{3}}{4}{\left(20\right)}^2\\ =\frac{\sqrt{3}}{4}\times 400\\ =100\sqrt{3}\end{array}$

Area of smaller equilateral triangle shaper flower bed is,

  $\begin{array}{l}{A}_S=\frac{\sqrt{3}}{4}{\left(8\right)}^2\\ =\frac{\sqrt{3}}{4}\times 64\\ =16\sqrt{3}\end{array}$

Thus, the ratio of area of larger flower bed to smaller flower bed to smaller flower bed is,

  $\begin{array}{l}\frac{A_I}{A_S}=\frac{100\sqrt{3}}{16\sqrt{3}}\\ \Rightarrow \frac{A_I}{A_S}=\frac{25}{4}\end{array}$

Conclusion:

Thus, the ratio of area of larger flower bed to smaller flower bed to smaller flower bed is, $\frac{25}{4}$