An open tank of square base is to be made from a square sheet of metal if sides 20m by cutting square pieces from each of the corners, lifting the resulting flaps and soldering the edges together. Find the maximum capacity of such tank
To find the maximum capacity of the tank, we need to determine the dimensions of the square base after the corners are cut.
Let x be the length of one side of the square cut from each corner.
The length of the base of the tank will be (20 - 2x) meters, and the height of the tank will also be (20 - 2x) meters.
The volume of a rectangular prism is given by the formula: V = length * width * height.
In this case, the volume of the tank is: V = (20 - 2x) * (20 - 2x) * x.
To find the maximum capacity of the tank, we need to maximize this volume.
Taking the derivative of the volume function with respect to x:
dV/dx = -8x^2 + 80x - 400.
Setting this derivative equal to zero and solving for x:
-8x^2 + 80x - 400 = 0.
Dividing by -8:
x^2 - 10x + 50 = 0.
Using the quadratic formula to solve for x:
x = (-(-10) ± sqrt((-10)^2 - 4(1)(50))) / (2(1)).
x = (10 ± sqrt(100 - 200)) / 2.
x = (10 ± sqrt(-100)) / 2.
Since the square root of a negative number is not a real number, there are no real solutions to this equation.
This means that the volume of the tank is always decreasing as x increases, which implies that the maximum capacity of the tank occurs when x = 0.
When x = 0, the tank is just a square box without any corners cut, so the dimensions of the tank are 20m x 20m x 20m.
Therefore, the maximum capacity of the tank is V = 20m * 20m * 20m = 8,000 cubic meters.
To find the maximum capacity of the tank, we need to determine the dimensions that will result in the largest volume.
Let's assume that the side length of the cut-out square is x meters. Since we are cutting square pieces from each corner of the metal sheet, the length and width of the resulting tank will be reduced by 2x.
Therefore, the length of the tank will be (20 - 2x) meters, and the width will also be (20 - 2x) meters.
The height of the tank can be represented by x, as it is the distance between the original metal sheet and the upper surface formed by folding the corner flaps.
The volume of a rectangular prism (tank) is given by the formula: V = length * width * height.
So, the volume of the tank can be expressed as V = (20 - 2x) * (20 - 2x) * x.
To find the maximum capacity, we need to find the value of x that maximizes this volume function.
Taking the derivative of V with respect to x and setting it equal to zero will give us the critical points. We can then determine which critical point corresponds to the maximum value.
Differentiating V with respect to x:
dV/dx = (20 - 2x)(20 - 2x) + (20 - 2x)(20 - 2x) - 2x(20 - 2x)(2) = 0
Simplifying the equation:
4(20 - 2x)(20 - 2x) - 4x(20 - 2x) = 0
(20 - 2x)(20 - 2x) - x(20 - 2x) = 0
Expanding and simplifying:
400 - 80x + 4x^2 - 20x + 4x^2 - 2x^2 = 0
6x^2 - 100x + 400 = 0
Dividing through by 2 for simplicity:
3x^2 - 50x + 200 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula.
Factoring this equation is not straightforward, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Substituting the values:
x = (-(-50) ± √((-50)^2 - 4(3)(200)))/(2(3))
x = (50 ± √(2500 - 2400))/6
x = (50 ± √100)/6
x = (50 ± 10)/6
So, x equals either (50 + 10)/6 = 13.33 meters or (50 - 10)/6 = 6.67 meters.
Since the dimensions of the tank cannot be in decimal values, we need to choose between these two options.
If we choose x = 6.67 meters, the resulting tank dimensions will be (20 - 2(6.67)) = 6.66 meters. However, this is not a feasible solution as it would involve cutting negative lengths from the corners.
Therefore, we choose x = 13.33 meters.
The dimensions of the tank will be (20 - 2(13.33)) = 20 - 26.66 = -6.66 meters. Again, this is not a feasible solution as negative dimensions are not possible.
Hence, it implies that the maximum capacity is obtained when x = 0, which means we do not cut anything from the corners.
Therefore, the maximum capacity of the tank is given by V = length * width * height = 20m * 20m * 0m = 0 cubic meters.