Gina has 24 feet of fence, she wants to make the largest rectangular area possible for her rabbit to play in , what length should she make each side of the rabbit penn?
To determine the length of each side of the rabbit pen, Gina should aim for a rectangular shape since it provides the largest area given a fixed perimeter.
Let's break down the problem and find the solution step-by-step:
1. Start by understanding the given information. Gina has 24 feet of fence to use for the pen.
2. To find the largest rectangular area, we need to determine the dimensions that produce the maximum area while using the entire 24 feet of fence.
3. Let's assume the length of the pen is L feet, and the width is W feet.
4. The perimeter of a rectangle is given by the formula: 2L + 2W. In this case, the perimeter will be 24 feet. So we can write the equation as:
2L + 2W = 24.
5. Simplify the equation by dividing both sides by 2:
L + W = 12.
6. Now we need to express one variable in terms of the other. Let's solve for L:
L = 12 - W.
7. To maximize the area, we need to find the maximum value for L × W. Substituting the value of L from step 6 into this equation:
Area = (12 - W) × W.
8. Rearrange the equation to get it in a quadratic format:
Area = 12W - W^2.
9. The maximum area occurs when the quadratic equation is at its vertex. In this case, the vertex of the quadratic equation is given by the formula: x = -b / 2a. For our equation, a = -1, and b = 12. So the vertex occurs at:
W = -12 / (2 × -1) = 6.
10. By plugging the value of W into the equation from step 7, we can determine the length:
L = 12 - 6 = 6.
So, to create the largest rectangular area possible, Gina should make each side of the rabbit pen 6 feet long.
That is calculus question.
a = length
b = width
P = Perimeter
A = Area
P = 2 a + 2 b = 2 ( a + b )
24 = 2 ( a + b ) Divide both sides by 2
12 = a + b
a + b = 12 Subtract a to both sides
a + b - a = 12 - a
b = 12 - a
A = a * b = a * ( 12 - a )
A = 12 a - a ^ 2
The function has a minimum value if f ' ( A ) = 0
and f " ( A ) = a positive number.
The function has a maximum value if f '( A ) = 0
and f ''( A ) = a negative number.
f ´ ( A ) = 12 - 2 a
f " ( A ) = - 2
f´ ( A ) = 0
12 - 2 a = 0 Add 2 a to both sides
12 - 2 a + 2 a = 0 + 2 a
12 = 2 a Divide both sides by 2
6 = a
a = 6 m
f´ ( A ) = 0
f " ( A ) = - 2
That is a maximum value.
b = 12 - a = 12 - 6 = 6 m
Square 6 x 6 m