I need help on this math word problem!

Question

A rectangular field is to be enclosed by 1000 m of fencing. What is the maximum area that can be enclosed? What dimensions will give a maximum area? What dimensions will produce an area of 9600 m^2?

Answer 1

We can use the parabola model to find our maximum area.

 

Let x = length

Let y = width

 

 

Setting up the equations to represent perimeter and area,

 

2(x + y) = 1000

 

x + y = 500              eq1

 

Area = xy                 eq2

 

 

Substitute eq1 into eq2.  We will get eq2 in terms of x and obtain a parabolic formula for the area.

 

Area = x(500 - x)

 

Area = -x² + 500x

 

 

Because the coefficient of the leading term of this function is negative, it ensures that we will have a maximum.  To get the maximum, we need to find the vertex.

 

Vertex has coordinate (h, k).

 

h = -b / 2a

 

where:

a = -1

b = 500

 

Plug in these values to find h.  This will be the x coordinate of the maximum.

 

h = -500 / (2*-1)

h = 250

 

x = 250

 

 

Plug in this value of x into eq1 to solve for y.

 

y = 500 - 250

y = 250

 

 

The maximum area that is created by these dimensions is

 

Area = 250 cm * 250 cm

Area = 62500 cm²

 

 

We can also plug in x=250 (obtained from vertex) into the area function.  By doing so, you get 62500 cm².

 

 

The same solution that Daniel obtained.

 

 

The calculus method:

 

When finding maximum or minimum values, we set the derivative equal to zero.  This is because the derivative is the slope of the tangent line, and the line that is tangent to a maximum point ALWAYS has a slope of zero.

 

d/dx is the notation for derivative.

 

d/dx (Area) = 0

 

d/dx (-x² + 500x) = 0

 

-2x + 500 = 0

 

 

Solve for x.  The x value that we get is known as a critical point.  It is the x value where the maximum or minimum value is located.

 

-2x = -500

 

   x = 250

 

 

Does this number look familiar to you? 

 

Usually, when we have more than on critical point, we use test points to evaluate the derivative.  This determines if a critical point contains a maximum, or a minimum.  For example, if the derivative is positive before the critical point and negative after the critical point, then the critical point has a maximum point, otherwise it is minimum.

 

Then as a last step, you evaluate the function at that critical point to obtain your maximum or minimum value.  In this case, we will have a maximum value.

 

 

 

 

Answer 2

Hi Sun,

 

When you learn calculus, you'll find out that a square gives you the maximum area.  You can think of a square as a special type of rectangle.  The area would be 250 x 250 = 62500 square meters.

 

To calculate the dimensions for an area of 9600 square meters:

 

if the width is X, then the length is (1000 - 2X)/2 which equals 500 -X

 

So the area is X(500-X) which equals 9600

 

500X - X^2 = 9600

 

-X^2 + 500X -9600 = 0

 

Multiplying by -1--> X^2 - 500X + 9600 = 0

 

(X - 20) (X - 480) = 0

 

X = 20 and X = 480

 

Let's call the smaller value  (20) the width and then 480 has to be the length.

 

Dan