A car[enter will be building a stained glass jewelry box. The frame of the box is to be built from a 48 inch stripe of metal which will be cut into 12 pieces. The 12 pieces are then soldered to form a rectangular box whose base is twice as long as it is wide. How should the metal be cut to maximize the volume of the box.

6y

To maximize the volume of the box, we need to consider the dimensions of the rectangular box that can be formed with the metal strip. Let's go through the steps:

1. Define variables:
Let's assume the width of the box is "x" inches. Given that the length of the base is twice as long as it is wide, the length would be "2x" inches.

2. Perimeter constraint:
The 48-inch metal strip needs to be cut into 12 pieces, which means each piece would be 48/12 = 4 inches long. Considering the four sides of the rectangle, we can calculate the perimeter as follows:
Perimeter = 2x + 2(2x) = 6x

3. Volume calculation:
The volume of a rectangular box is calculated as Length x Width x Height. In our case, the height is not given, but in order to maximize the volume, we can use the complete length of the metal strip as the height. Therefore, the volume can be expressed as:
Volume = 2x * x * 48

4. Optimization:
To maximize the volume, we can express it as a function of a single variable, x, and then find the critical points. Since the volume is a quadratic function, we can use calculus to maximize it.

Taking the derivative of the volume with respect to x and setting it equal to zero, we can find the critical points. Differentiate the equation with respect to x:
dV/dx = 96x - 144x^2

Setting the derivative equal to zero and solving for x:
96x - 144x^2 = 0
Dividing both sides by 48x:
x - 2x^2 = 0
Simplifying:
x(1 - 2x) = 0

This equation has two possible solutions: x = 0 or 1/2.

5. Determine the maximum:
To determine if x = 0 is valid, we need to check if it satisfies the perimeter constraint. Plugging x = 0 into the equation, we find the perimeter to be zero, which is not valid for our case. Therefore, x = 1/2 is the only valid solution.

6. Final dimensions:
Given that x = 1/2, the width of the box is 1/2 inch, and the length is 2 times the width, so the length is 1 inch.

7. Final volume:
Using these dimensions, we can calculate the final volume:
Volume = 2(1/2) * (1/2) * 48 = 24 cubic inches.

So, to maximize the volume of the box, the metal should be cut into 12 pieces with a width of 1/2 inch and a length of 1 inch.