Chapter 9.1, Problem 37PPS

a.

To determine

To find: The minimum area of each playing field.

Answer to Problem 37PPS

The minimum area of the Field Hockey field is given by $64000{\text{ ft}}^2$

The minimum area of the Men’s Lacrosse field is given by $63000{\text{ ft}}^2$

The minimum area of the Women’s soccer field is given by $36400{\text{ ft}}^2$

Explanation of Solution

Given:

The table shows the minimum areas of different sports fields.

  $\begin{array}{cc}\text{Sport}& \begin{array}{l}\text{Minimum filed}\\ \text{Area }\left({\text{ft}}^2\right)\end{array}\\ \text{Field Hockey}& 2^6\cdot {10}^3\\ \text{Men's Lacrosse}& 3^2\cdot 7\cdot {10}^3\\ \text{Women's Soccer}& 2^4\cdot 5^2\cdot 7\cdot 13\end{array}$

Calculation:

The minimum area of the Field Hockey field is given by

  $\begin{array}{l}{2}^6\cdot {10}^3\\ =64\cdot 1000\\ =64000{\text{ ft}}^2\end{array}$

The minimum area of the Men’s Lacrosse field is given by

  $\begin{array}{l}{3}^2\cdot 7\cdot {10}^3\\ =9\cdot 7\cdot 1000\\ =63000{\text{ ft}}^2\end{array}$

The minimum area of the Women’s soccer field is given by

  $\begin{array}{l}{2}^4\cdot 5^2\cdot 7\cdot 13\\ =16\cdot 25\cdot 7\cdot 13\\ =36400{\text{ ft}}^2\end{array}$

b.

To determine

To order: The areas from least to greatest.

Answer to Problem 37PPS

  $1000{\text{ ft}}^2$

Explanation of Solution

Given:

The table shows the minimum areas of different sports fields.

  $\begin{array}{cc}\text{Sport}& \begin{array}{l}\text{Minimum filed}\\ \text{Area }\left({\text{ft}}^2\right)\end{array}\\ \text{Field Hockey}& 2^6\cdot {10}^3\\ \text{Men's Lacrosse}& 3^2\cdot 7\cdot {10}^3\\ \text{Women's Soccer}& 2^4\cdot 5^2\cdot 7\cdot 13\end{array}$

Calculation:

From part (a),

The minimum area of the Field Hockey field is given by $64000{\text{ ft}}^2$

The minimum area of the Men’s Lacrosse field is given by $63000{\text{ ft}}^2$

The minimum area of the Women’s soccer field is given by $36400{\text{ ft}}^2$

So, the order of the areas from least to greatest is

  $36400{\text{ ft}}^2<63000{\text{ ft}}^2<64000{\text{ ft}}^2$

So, arrange their corresponding areas in least to greatest

  $2^4\cdot 5^2\cdot 7\cdot 13{\text{ft}}^2<3^2\cdot 7\cdot {10}^3{\text{ ft}}^2<2^6\cdot {10}^3{\text{ ft}}^2$

c.

To determine

To find: How much greater is the area of a field hockey filed than the area of a men’s lacrosse field.

Answer to Problem 37PPS

  $2^4\cdot 5^2\cdot 7\cdot 13{\text{ft}}^2<3^2\cdot 7\cdot {10}^3{\text{ ft}}^2<2^6\cdot {10}^3{\text{ ft}}^2$

Explanation of Solution

Given:

The table shows the minimum areas of different sports fields.

  $\begin{array}{cc}\text{Sport}& \begin{array}{l}\text{Minimum filed}\\ \text{Area }\left({\text{ft}}^2\right)\end{array}\\ \text{Field Hockey}& 2^6\cdot {10}^3\\ \text{Men's Lacrosse}& 3^2\cdot 7\cdot {10}^3\\ \text{Women's Soccer}& 2^4\cdot 5^2\cdot 7\cdot 13\end{array}$

Calculation:

From part (a),

The minimum area of the Field Hockey field is given by $64000{\text{ ft}}^2$

The minimum area of the Men’s Lacrosse field is given by $63000{\text{ ft}}^2$

So, the difference in field Hockey field and the Men’s Lacrosse field is given by

  $\begin{array}{l}64000{\text{ ft}}^2-63000{\text{ ft}}^2\\ =1000{\text{ ft}}^2\end{array}$

Therefore, the field Hockey field is $1000{\text{ ft}}^2$ greater than the Men’s Lacrosse field