Math Is Fun Forum

mathland

Member

Registered: 2021-03-25

Posts: 444

NFL Playing Field

A regulation NFL playing field of length
x and width y has a perimeter of (1040/3) yards.

(A) Show that the width of the rectangle is y = [(520/3) - x]
and its area is A = x[(520/3) - x)]

(B) From the graph, estimate the dimensions
of the rectangle that yield a maximum area.

1. Can someone set up part (A)? I will do the math. I know that
P = 2L + 2W is used here somehow, right?

2. For part (B), how do I use the graph of A = x[(520/3) - x)]
to estimate the dimensions of the rectangle that yield a maximum area.

Last edited by mathland (2021-04-26 14:34:19)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,580

Re: NFL Playing Field

Yes, that perimeter formula is the place to start.  Put in y and x and the known value for P. then it's just a case of making y the subject of the equation.

Then area = xy so you can use that equation for y to get area in terms of x only.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: NFL Playing Field

Yes, that perimeter formula is the place to start.  Put in y and x and the known value for P. then it's just a case of making y the subject of the equation.

Then area = xy so you can use that equation for y to get area in terms of x only.

Bob

What about part B?

Mathegocart

Member

Registered: 2012-04-29

Posts: 2,226

Re: NFL Playing Field

Yes, that perimeter formula is the place to start.  Put in y and x and the known value for P. then it's just a case of making y the subject of the equation.

Then area = xy so you can use that equation for y to get area in terms of x only.

Bob

What about part B?

This is a classic optimization problem - just graph A(x) and find its local maximum in a "reasonable" domain of x(i.e, from 0 to 520/3, as area can't be negative.)

---

The integral of hope is reality.
May bobbym have a wonderful time in the pearly gates of heaven.
He will be sorely missed.

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: NFL Playing Field

Yes, that perimeter formula is the place to start.  Put in y and x and the known value for P. then it's just a case of making y the subject of the equation.

Then area = xy so you can use that equation for y to get area in terms of x only.

Bob

What about part B?

This is a classic optimization problem - just graph A(x) and find its local maximum in a "reasonable" domain of x(i.e, from 0 to 520/3, as area can't be negative.)

I will use Desmos to graph the area function given to answer part B. I'll come back to this problem if need be.