A farmer wants to construct a pen with 100 feet of fencing. The pen will be divided into two areas as shown below. Find the maximum dimensions of the big pen.
All I have so far is that 3x+2y=100. I am not sure how to move forward.
the area is
2xy = 2x(100-3x)/2 = 100x-3x^2
Now use what you know about parabolas to find the vertex (maximum area).
You will find that the total fencing is divided equally among the lengths and widths.
"as shown below" does not work here, but I have seen that problem many many times.
ok so far as your 3x + 2y = 100, assuming you defined y as the length and x as each of the dividing widths.
from that y = (100-3x)/2 = 50 - (3/2)x
I object to your wording of "Find the maximum dimensions of the big pen."
It should say:
"Find the dimensions of the big pen that will yield a maximum area."
Area = xy
= x(50-(3/2)x)
= 50x - (3/2)x^2
I will also assume that you are not in Calculus, or else this would now be easy
You are looking at a parabola which opens downwards. The vertex will give you all you want to know.
For a parabola in standard form, the x of the vertex is -b/(2a)
x = -50/(-3) = 50/3
then y = 50 - (3/2)(50/3) = 25
so the width is 25 ft and the length is 50/3 ft
for a maximum area of 1250/3 ft
If you have been finding the vertex by competing the square, I will leave it up to you to find the vertex of
(-3/2)x^2 + 50x using that method.
To find the maximum dimensions of the big pen, we can use optimization techniques.
First, let's assign variables to the dimensions of the big pen. Let's say the length is represented by "x" and the width is represented by "y".
The perimeter of the big pen is 100 feet, so we can form an equation with the perimeter:
2x + y + 2y = 100
Simplifying this equation, we get:
2x + 3y = 100
Now, we need to express one variable in terms of the other so we can find the maximum value for the area. To do this, let's solve the equation for x:
2x = 100 - 3y
x = (100 - 3y) / 2
Now we have the expression for x in terms of y.
Next, we need to express the area of the big pen in terms of one variable. The area of a rectangle is given by the product of its length and width, so the area is:
Area = x * y
Substituting the expression for x from the earlier equation:
Area = ((100 - 3y) / 2) * y
Expanding this equation, we get:
Area = (100y - 3y^2) / 2
To find the maximum area, we need to take the derivative of the equation with respect to y, set it equal to zero, and solve for y. This will give us the critical points where the area is at a maximum.
d(Area) / dy = (100 - 6y) / 2
0 = (100 - 6y) / 2
Now, we can solve for y:
100 - 6y = 0
6y = 100
y = 100 / 6
y = 16.67 (approximately)
Since the length and width cannot be negative, we can conclude that the dimensions of the big pen that maximize its area are approximately x = 33.33 feet and y = 16.67 feet.