A veterinary clinic plans to build four identical dog kennels along the side of its building using 210 feet of fencing. (See the picture.) What should be the dimensions of each kennel to maximize the enclosed area? (Note: No fencing is needed along the side of the building.
26.25 feet and 21 feet :P
Well, if we want to maximize the enclosed area, we should make each kennel as big as possible. However, we also have to keep in mind that we only have 210 feet of fencing to go around.
Let's think about this... If we were to build one huge kennel, we would use all the fencing in one long rectangle shape, right? But since we want to build four identical kennels, we need to divide that fencing equally among all four.
So, let's start by thinking about how the fencing would be distributed. If we have four kennels next to each other, we only need two sides of fencing for each outer kennel, and only one side of fencing for each inner kennel. That means we have 2+2+1+1=6 sides of fencing for each kennel.
Now, if we take the total fencing of 210 feet and divide it by the number of sides per kennel (6), we get 210/6=35 feet for each side of the kennel.
Since we know all sides of each kennel are equal, we can calculate the dimensions by dividing 35 feet by 4 (to account for both the width and the length). So, each side of the kennel should be 35/4=8.75 feet.
Therefore, the dimensions of each kennel to maximize the enclosed area would be approximately 8.75 feet by 8.75 feet.
To maximize the enclosed area, each kennel should be in the shape of a square. Let's denote the length of each side of the square as 'x'.
Since there are four kennels, the total fencing required equals the perimeter of all four squares.
Each square has 4 equal sides, so the total fencing required is 4x for one square. Since there are four squares, the total fencing required is 4(4x) = 16x.
We are given that the total fencing available is 210 feet, so we can set up an equation:
16x = 210
To solve for x, divide both sides of the equation by 16:
x = 210/16 = 13.125
Therefore, each side of the square kennel should be approximately 13.125 feet long to maximize the enclosed area.
To find the dimensions of each kennel that will maximize the enclosed area, we can use the quadratic formula. Let's start by breaking down the problem into smaller steps:
Step 1: Define the variables.
Let's assume the width of each kennel is 'x' feet.
Step 2: Calculate the length of each kennel.
Since there are four kennels, we can divide the remaining fencing length (210 feet) by the number of kennels (4) to determine the length of each kennel's two sides. Therefore, the length of each kennel is (210/4) = 52.5 feet.
Step 3: Calculate the total area.
The total area of all four kennels can be found by multiplying the width and length of a single kennel by the number of kennels (4). The total area (A) is given by A = 4 * (width * length).
Step 4: Apply the quadratic formula.
Now, we want to maximize the enclosed area, so we need to find the maximum value of the area equation by finding the vertex of the quadratic equation.
Substituting the width (x) and length (52.5) into the area equation, we get A = 4 * (x * 52.5) = 210x.
To find the x-coordinate of the vertex, we can use the formula x = -b / (2a), where a = 0 (since the coefficient of x^2 is 0) and b = 210.
Hence, x = -210 / (2 * 0) = undefined
The result is undefined because the quadratic equation in our case only has a linear term (x), and there is no quadratic term (x^2). This means that the area increases linearly with the width of the kennels, and there is no maximum or minimum.
Therefore, there is no specific width that will maximize the enclosed area. As long as the width of each kennel is greater than zero, the area will continue to increase.
5x + 4y = 210 ... 4y = 210 - 5x
A = 4 x y = x (210 - 5x) = 210x - 5x^2
max A on axis of symmetry
... x = -210 / (2 * -5) = 21