If 2100 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
Volume = cubic centimeters.
There are a few math errors in my answer.
x = sqrt700 = 26.45 cm
y = 525/x -x/4 = 13.24 cm
max volume = y*x^2 = 9263 cm^3
THANK YOU
i got it wrong
You did or I did?
You need to specify if all material gets used. If not, and you are just cutting square corners off the original piece, folding up tabs and throwing away the squares, then you need to say if the original material is a square or not.
Well, if the box has a square base, let's call the side of the base "x". So the area of the base would be x^2 (since it's a square), and the perimeter of the base would be 4x.
Now, to make the box, we need a base and four sides. Since the sides are identical, we can divide 2100 square centimeters by 5 (one base and four sides) to find the average area of each part.
2100 / 5 = 420.
So, the average area of each part is 420 square centimeters. To maximize the volume, we would want the base to have the largest area possible.
Now, since the perimeter of the base is 4x, we can divide the perimeter by 4 to find the side of the base.
4x / 4 = x.
And we know that the area of the base is x^2. So, the average area of each part is also x^2 = 420.
Now, we can solve for x:
x^2 = 420.
Taking the square root of both sides, we get:
x ≈ 20.5.
So, the side of the base (x) is approximately 20.5 centimeters. That means the largest possible volume of the box is:
Volume = x^2 * x = 20.5^2 * 20.5 = 8553.25 cubic centimeters.
Therefore, the largest possible volume of the box is approximately 8553.25 cubic centimeters with an x value of approximately 20.5 centimeters.
But hey, don't put all your boxes in one basket!
To find the largest possible volume of the box, we need to determine the dimensions of the box that would maximize the volume while using the given amount of material.
Let's assume that the side length of the square base is 'x' centimeters. The height of the box would also be 'x' centimeters to maintain the square shape.
To calculate the surface area of the box, we need to find the area of the square base and the area of the four sides. The area of the square base is given by (x * x) = x^2. The area of the four sides is given by (4 * x * x) = 4x^2.
The total surface area (including the base) can be found by adding the area of the base and the four sides: x^2 + 4x^2 = 5x^2.
Since we have 2100 square centimeters of material available, we can set up an equation: 5x^2 = 2100.
Dividing both sides of the equation by 5 gives us: x^2 = 420.
To find the value of 'x', we can take the square root of both sides: x = √420 ≈ 20.49.
Since we are dealing with measurements, we will round 'x' to the nearest whole number: x ≈ 20.
Now that we have the dimensions of the box (20cm by 20cm by 20cm), we can calculate the volume using the formula: volume = x^3 = 20^3 = 8000 cubic centimeters.
Therefore, the largest possible volume of the box is 8000 cubic centimeters when the base has a side length of 20 centimeters.
Let x = side width and y = height.
Area = x^2 + 4xy = 2100
y = (2100 -x^2)/(4x) = (525/x) - (x/4)
Volume = y*x^2 = 525x -(x^3/4)
dV/dx = 0 = 525 - (3/4)x^2
x = sqrt 733.3 = 26.45 cm
y = 13.24 cm
max volume = 14,410 cm^3