# quadratic function application

• Dec 10th 2006, 06:08 PM
paul
I came here about a month ago asking a question and you guys were very helpful. Now, I need help again. These questions are part of a project, but the questions deal with quad. functions/optimization:

1) You want to fence off a rectangular garden plot by putting a short brick wall along one edge and wooden fencing along the other three edges. The brick wall will cost \$8 per linear foot, while the wooden fence costs \$2 per linear foot.

a) Find a function that gives the total cost of the material in terms of the variable l and w; ignore the thickness of the brick.

b) If you have only \$500 to spend on materials, what are the dimensions of the largest (biggest area) plot you can enclose?

And the second:

2) A box is made from a 16 inch by 28 inch piece of cardboard by cutting equal squares from each corner and folding up the sides.

a) Write a function expressing the volume of the box as a function of x.
b) Determine the domain of this function
c) Determine the size of the square that will produce a maximum volume for the box. Give a complete explanation of your solution.

I understand it's a lot, but I would appreciate any sort of help. I'm really struggling with this. Thanks.

- Paul
• Dec 10th 2006, 06:42 PM
Quick
Quote:

Originally Posted by paul
I came here about a month ago asking a question and you guys were very helpful. Now, I need help again. These questions are part of a project, but the questions deal with quad. functions/optimization:

1) You want to fence off a rectangular garden plot by putting a short brick wall along one edge and wooden fencing along the other three edges. The brick wall will cost \$8 per linear foot, while the wooden fence costs \$2 per linear foot.

a) Find a function that gives the total cost of the material in terms of the variable l and w; ignore the thickness of the brick.

Let's say the brick wall is one of the "length" sides.

The equation for perimeter is: P = l + l + w+ w

And since the perimeter is also the amount of fence, the cost of the fence is: C = \$2l + \$8l + \$2w + \$2w

Quote:

b) If you have only \$500 to spend on materials, what are the dimensions of the largest (biggest area) plot you can enclose?
Well then, you have the equation: 500 = \$2l + \$8l + \$2w + \$2w

Substitute: 500 = 2l + 8l + 2w + 2w

Add: 500 = 10l + 4w

Subtract 10l from both sides: 500 - 10l = 4w

Divide both sides by 4: 125 - 2.5l = w

Now you want to find the largest area, so the equation for Area is: A = lw

Substitute: A = l(125-2.5l)

So: A = 125l - 2.5l^2

Let's say that A = y and l = x: y = 125x - 2.5x^2

Now graph that and find what value of x gives the largest area.
• Dec 10th 2006, 06:57 PM
ThePerfectHacker
Quote:

Originally Posted by Quick

Now graph that and find what value of x gives the largest area.

Let me teach you a rule about quadradic functions that might help.

If a>0
Then,
y=ax^2+bx+c
Is a parabola that opens up.
Thus it has a minimum value.
Which is located at,
x=-b/(2a)
(Actually that is where derivative is zero).

If a<0
Then,
y=ax^2+bx+c
Is a parabola that opens down.
Thus it has a maximum value.
Which is located at,
[tex]x=-b/(2a)

Thus, to find a max/min for "a" non-zero
You need to compute,
-b/(2a)
And determine what the sign of 'a' is.
• Dec 11th 2006, 09:54 AM
earboth
Quote:

Originally Posted by paul
I came here about a month ago asking a question and you guys were very helpful. Now, I need help again. These questions are part of a project, but the questions deal with quad. functions/optimization:

1) You want to fence off a rectangular garden plot by putting a short brick wall along one edge and wooden fencing along the other three edges. The brick wall will cost \$8 per linear foot, while the wooden fence costs \$2 per linear foot.

a) Find a function that gives the total cost of the material in terms of the variable l and w; ignore the thickness of the brick.

b) If you have only \$500 to spend on materials, what are the dimensions of the largest (biggest area) plot you can enclose?

And the second:

2) A box is made from a 16 inch by 28 inch piece of cardboard by cutting equal squares from each corner and folding up the sides.

a) Write a function expressing the volume of the box as a function of x.
b) Determine the domain of this function
c) Determine the size of the square that will produce a maximum volume for the box. Give a complete explanation of your solution.

I understand it's a lot, but I would appreciate any sort of help. I'm really struggling with this. Thanks.

- Paul

Hello, Paul,

you have to cut off 4 equal squares which have the side x. Thus the greatest square to cut of has a side length of 8".

The base area of the box is:
A=(16-2x)*(28-2x)=4x^2-88x+448, 0 ≤ x ≤ 8

the volume of the box is:

V=x * A=4x^3-88x^2+448x

The volume will become an extreme value (Minimum or maximum) if the first derivative equals zero:

V'(x)=12x^2-176x+448. Now calculate:

0=12x^2-176x+448. Use the formula to solve a quadratic equation. I've got the results:

x=22/3-2/3*sqrt(37) ≈ 3.278...[/tex]
or
x=22/3+2/3*sqrt(37) ≈ 11.388...[/tex]

The 2nd value doesn't belong to the domain of the volume function.

Plug in the 1st value into the volume function and you'll get the greatest Volume: V = 663.851 cubic inches.

EB

PS.: I've got some troubles to use Latex, so I send you this text as plain text. I hope you can use it.
• Dec 12th 2006, 01:38 AM
earboth
Hi,

I've attached a sketch of your problem.

EB
• Dec 12th 2006, 02:38 AM
Quick
Quote:

Originally Posted by earboth
Hi,

I've attached a sketch of your problem.

EB

Here's a good sketch of the problem