You want to cover a trapezoidal box with paper. You have a sheet of paper that is 1m by 500cm. What is the size box that can be made with minimal wastage?
To determine the size of the trapezoidal box that can be made with minimal wastage using a sheet of paper that is 1m by 500cm, we need to find the dimensions of the box that maximize the utilization of the paper.
First, let's convert the dimensions of the sheet of paper to a consistent unit. Since 1m equals 100cm, the sheet of paper is 100cm by 500cm.
Now, let's consider the dimensions of the trapezoidal box. A trapezoidal box has a top base, bottom base, and height. Let's assume the top and bottom bases have lengths x and y, respectively, and the height is h.
The total length of the paper that will be used to cover the trapezoidal box is equal to the sum of the lengths of all four sides of the box.
The lengths of the sides of the box are as follows:
- Top base: x cm
- Bottom base: y cm
- Two vertical sides: x cm and y cm
Thus, the total length of paper used is given by: x + y + x + y = 2x + 2y.
To minimize the wastage, we need to maximize the utilization of the paper. Therefore, we want to find the values of x and y that maximize the sum 2x + 2y, while adhering to the constraint imposed by the dimensions of the paper (100cm by 500cm).
Since the dimensions of the paper are fixed, we can express y in terms of x to get rid of one variable. One way to do this is by noting that the difference between the lengths of the two bases of a trapezoid, divided by two, is equal to the height (h) of the trapezoid.
So, we have the equation: (y - x) / 2 = h
To find the maximum value of 2x + 2y, we need to find the values of x and y that maximize y - x, as the coefficient 2 is constant for both variables.
The maximum value of y - x occurs when y = 500cm and x = 100cm, since these values maximize the width and minimize the length of the trapezoidal box within the given dimensions of the paper.
Substituting these values into our equation for the total length of paper used (2x + 2y), we get:
2(100) + 2(500) = 200 + 1000 = 1200 cm
Thus, the size of the trapezoidal box that can be made with minimal wastage using the given sheet of paper is 100cm by 500cm, resulting in a total length of paper used of 1200 cm.