Chapter 1.2, Problem 83E

i.

To determine

To draw: A regulation NFL playing field of length $x$ and width $y$ has a perimeter of $346\frac{2}{3}$ or $\frac{1040}{3}$ yards. Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle.

Answer to Problem 83E

  

Explanation of Solution

A rectangle is a 2 dimensional figure having length and width. Here length is given as $y$

ii.

To determine

To show: A regulation NFL playing field of length $x$ and width $y$ has a perimeter of $346\frac{2}{3}$ or $\frac{1040}{3}$ yards. Show that the width of the rectangle is $y=\frac{520}{3}-x$ and its area $A=x\left(\frac{520}{3}-x\right)$ .

Explanation of Solution

Given information: length is $x$ and width is $y$ , perimeter is given as $\frac{1040}{3}$ yards

Calculation: Perimeter of a rectangle is sum of all 4 sides, so

Perimeter= length +length+breadth+breadth = 2(length + width)

  $\begin{array}{l}P=2(x+y)\\ \frac{1040}{3}=2\left(x+y\right)\end{array}$

Solve the above equation for $y$

  $\begin{array}{l}\frac{1040}{3\times 2}=\left(x+y\right)\\ \frac{520}{3}=\left(x+y\right)\\ y=\frac{520}{3}-x\end{array}$

Now area of rectangle is

  $\begin{array}{l}A=l\times w\\ A=x\times \left(\frac{520}{3}-x\right)\end{array}$

iii.

To determine

To graph: A regulation NFL playing field of length $x$ and width $y$ has a perimeter of $346\frac{2}{3}$ or $\frac{1040}{3}$ yards. Use a graphic utility to graph the area equation.

Explanation of Solution

Given information: $A=x\times \left(\frac{520}{3}-x\right)$

Plotting the graph for the equation: $A(x)=x\times \left(\frac{520}{3}-x\right)$

  

iv.

To determine

To calculate: A regulation NFL playing field of length $x$ and width $y$ has a perimeter of $346\frac{2}{3}$ or $\frac{1040}{3}$ yards. From the graph in part (c), estimate the dimensions of the rectangle that yields a maximum area.

Answer to Problem 83E

  $x=y=\frac{260}{3}$ , i.e. Length = Width = $\frac{260}{3}$

Explanation of Solution

Given information: $A=x\times \left(\frac{520}{3}-x\right)$

From this it can be considered that $y$ is maximum when $x=\frac{260}{3}$

Calculation: By putting value of $x$ in $x+y=\frac{520}{3}$ , we get

  $\begin{array}{l}\frac{260}{3}+y=\frac{520}{3}\\ y=\frac{520}{3}-\frac{260}{3}=\frac{260}{3}\end{array}$

Conclusion: $y=x=\frac{260}{3}$

v.

To determine

To calculate: A regulation NFL playing field of length $x$ and width $y$ has a perimeter of $346\frac{2}{3}$ or $\frac{1040}{3}$ yards. Use your school's library, the Internet, or some other reference source to find the actual dimensions and area of a regulation NFL playing field and compare your findings with the results of part (d).

Answer to Problem 83E

Actual dimension of NFL is 120 yards long and $53\frac{1}{3}$ yards wide.

Explanation of Solution

Given information: Actual dimension of NFL is 120 yards long and $53\frac{1}{3}$ yard wide

Conclusion: we got the shape as square but actual shape is not square, it is rectangle.