You have 300 square feet of wood you will use to construct a rectangular shed with a square base and top. What is the maximum volume of your shed? Optimization Question
To determine the maximum volume of the shed, we need to find the dimensions of the rectangular base and top that will allow us to use the entire 300 square feet of wood.
Let's denote the side length of the square base and top as "x". Since the base and top are square, both dimensions will be x. The height of the shed can be represented by "h".
The area of the base can be calculated as x * x = x^2, and the area of the top will also be x^2. Therefore, the total area of the base and top combined is 2 * (x^2) = 2x^2.
We are given that the total wood available is 300 square feet, so we can set up an equation:
2x^2 = 300
Now we can solve for x:
Divide both sides by 2:
x^2 = 150
Take the square root of both sides:
x = √150
Using a calculator or simplifying further, we find that x ≈ 12.25 (rounded to two decimal places).
Since the height of the shed can be any length, we can assume it is equal to x as well for simplicity. Therefore, the maximum volume of the shed is:
Volume = base area * height
Volume = x^2 * x
Volume = x^3
Plugging in the value of x we obtained earlier:
Volume ≈ (12.25)^3
Volume ≈ 1828.48 cubic feet
Hence, the maximum volume of the shed when using the entire 300 square feet of wood is approximately 1828.48 cubic feet.
base side x
height y
2x^2+4xy = 300 ==> y = (300-2x^2)/(4x)
v = x^2y = x^2(300-2x^2)/(4x) = 75x - x^3/2
dv/dx = 75 - 3x^2
dv/dx=0 at x=5
So, the shed is 5x5x12.5, with v=312.5 ft^3