A container with square base, vertical sides, and open top is to be made from 2700 square feet of material. Find the dimensions (length × width × heigth) of the container with greatest volume.
To find the dimensions of the container with the greatest volume, we need to optimize the volume function based on the given constraints.
Let's assume the dimensions of the square base are x (length) and x (width), and the height of the container is h.
Since the material for the container has an area of 2700 square feet, we can write the equation for the surface area as:
2(x^2) + 4xh = 2700
Now, we need to find a formula for the volume of the container. The volume (V) of a rectangular prism is given by V = length × width × height.
In this case, V = x × x × h = x^2h.
To optimize the volume function, we can express h in terms of x using the surface area equation.
Rearranging the surface area equation, we get:
2(x^2) + 4xh = 2700
Divide by 2:
x^2 + 2xh = 1350
Rearrange:
h = (1350 - x^2)/(2x)
Substituting this value of h into the volume equation, we have:
V = x^2 × [(1350 - x^2)/(2x)]
Simplifying:
V = (x^2 × (1350 - x^2))/(2x)
V = (x(1350 - x^2))/2
To find the dimensions that maximize the volume, we need to find the maximum value of the volume function.
To do this, we differentiate the volume equation with respect to x and set it equal to zero to find the critical points.
dV/dx = (1350 - 3x^2)/2
Setting dV/dx = 0, we have:
1350 - 3x^2 = 0
3x^2 = 1350
x^2 = 1350/3
x^2 = 450
x = √450
x ≈ 21.21 ft
Now we need to find the corresponding value of h:
h = (1350 - x^2)/(2x)
Substituting the value of x, we have:
h = (1350 - 450)/(2×21.21)
h = 900/42.42
h ≈ 21.21 ft
Thus, the dimensions of the container with the greatest volume are:
Length ≈ 21.21 ft
Width ≈ 21.21 ft
Height ≈ 42.42 ft
Therefore, the dimensions of the container with the greatest volume are approximately 21.21 ft × 21.21 ft × 42.42 ft.
To find the dimensions of the container with the greatest volume, we can use optimization techniques. Let's start by defining the variables:
Let x be the length and width of the square base.
Let h be the height of the container.
Since the container has a square base, the length and width are equal, so we can simplify the variables as follows:
x = length = width
h = height
The material used to create the container is given as 2700 square feet. We need to determine the dimensions (length, width, and height) that maximize the volume of the container.
To find the volume of the container, we need to multiply the area of the base (length × width) by the height:
Volume = Base Area × Height
= x * x * h
= x^2 * h
Now, we need to express the different quantities in terms of a single variable to create a function that represents the volume.
Since we know the total material used is 2700 square feet, we can express the surface area of the container as follows:
Surface Area = Base Area + 4 * Side Area
= x * x + 4 * (x * h)
= x^2 + 4xh
Since the surface area is given as 2700 square feet, we can write the equation:
x^2 + 4xh = 2700
Now, we can solve this equation for h in terms of x:
4xh = 2700 - x^2
h = (2700 - x^2) / (4x)
Substitute the value of h into the volume equation:
Volume = x^2 * [(2700 - x^2) / (4x)]
= (2700x - x^3) / 4
To maximize the volume, we need to find the value of x that maximizes this function.
To find the maximum, we can take the derivative of the volume function with respect to x and set it equal to zero:
dV/dx = (2700 - 3x^2) / 4
Set dV/dx = 0 and solve for x:
2700 - 3x^2 = 0
3x^2 = 2700
x^2 = 900
x = √900
x = 30
We get x = 30 as the value that maximizes the volume.
Now, substitute this value of x into the equation for h:
h = (2700 - x^2) / (4x)
h = (2700 - 30^2) / (4 * 30)
h = (2700 - 900) / 120
h = 1800 / 120
h = 15
The dimensions of the container with the greatest volume are:
Length = Width = 30 feet
Height = 15 feet
Therefore, the dimensions of the container with the greatest volume are 30 ft × 30 ft × 15 ft.
volume=s^2*h s is the base length.
but 2700=s^2+4sh
from that, sh=(2700-s^2)/4
then
volume= s(2700-s^2)/4
now, take the derivative of with respect to s...
dV/ds=0=2700/4 -3s^2/4
solve for s