Chapter 1.2, Problem 86E

a.

To determine

Verify the geometry of the statement.

Answer to Problem 86E

The rectangle is drawn.

Explanation of Solution

Given information:

A soccer playing field of length $x$ and width $y$ has a perimeter of $360$ meters. Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle.

Calculation:

The a visual representation of the rectangle.

  

Hence the rectangle is drawn.

b.

To determine

Verify the geometry of the statement.

Answer to Problem 86E

The area and length of rectangle is verified.

Explanation of Solution

Given information:

A soccer playing field of length $x$ and width $y$ has a perimeter of $360$ meters. Show that the width of the rectangle is $y=180-x$ and its area is $A=x\left(180-x\right)$

Calculation:

The perimeter of the rectangle is given as $360$ meters.

Perimeter= $2x+2y$

  $P=2x+2y$

  $\begin{array}{l}360=2x+2y\\ 360=2\left(x+y\right)\\ 180=x+y\end{array}$

Subtract $x$ from both sides.

  $\begin{array}{l}180-x=x-x+y\\ y=180-x\end{array}$

Now verify the area,

  $\begin{array}{l}A=width\times length\\ =xy\\ A=x\left(180-x\right)\end{array}$

Thus length of the rectangle is $\boxed{y=180-x}$

The area of the rectangle is $\boxed{A=x\left(180-x\right)}$

Hence area and length of rectangle is verified.

c.

To determine

Verify the geometry of the statement.

Answer to Problem 86E

The graph of area is verified.

Explanation of Solution

Given information:

A soccer playing field of length $x$ and width $y$ has a perimeter of $360$ meters. Use a graphing utility to graph the area equation. Be sure to adjust your window settings.

Calculation:

Consider the expression $\boxed{A=x\left(180-x\right)}$

To graph the equation use graphic utility $TI-83$

Now press button $Y=$ and enter the equation.

  

Press Window button and set parameters,

  

Press GRAPH button,

  

Hence the graph of area is verified.

d.

To determine

Verify the geometry of the statement.

Answer to Problem 86E

  $\boxed{x=9,y=110}$ Yield maximum area.

Explanation of Solution

Given information:

A soccer playing field of length $x$ and width $y$ has a perimeter of $360$ meters

From the graph in part $\left(c\right)$ , estimate the dimensions of the rectangle that yield a maximum area.

Calculation:

To graph the equation use graphic utility $TI-83$

Now press button $2nd$ and CALC.

Select option $4$ .

  

Now select left and right bound where area is maximum.

  

Thus area is maximum when $x=9,y=110$ .

Hence the dimensions of rectangle $\boxed{x=9,y=110}$ yield maximum area.

e.

To determine

Verify the geometry of the statement.

Answer to Problem 86E

  $\boxed{Area={960}^{2}m}$

Explanation of Solution

Given information:

A soccer playing field of length $x$ and width $y$ has a perimeter of $360$ meters

Use your school’s library, the Internet, or some other reference source to find the actual dimensions and area of a regulation Major League Soccer field and compare your findings with the results of part $\left(d\right)$ .

Calculation:

To graph the equation use graphic utility $TI-83$

Now press button $2nd$ and CALC.

Select option $4$ .

  

Now select left and right bound where area is maximum.

  

Thus area is maximum when $x=9,y=110$ .

The length of the Major League Soccer field is $110-120$ meters.

Width is $70-80$

Area of Major League Soccer field is $960$ square meters.

Hence $\boxed{Area={960}^{2}m}$ .