The Problem
You are given a piece of cardboard that is 6 inches by 4 inches. You would like to cut equal-sized squares out of each of the 4 corners and fold the cardboard in such a way to make an open-top rectangular box.
Part A: Complete the table below:
Length of Square Cut, l (inches)
Length of Box (inches)
Width of Box (inches)
Volume of Box, V (cubic inches)
0.25
0.50
0.90
1.40
1.80
Part B: Questions
Consider the function that is defined to be the volume, V, of the box created from cutting the cardboard corners into squares of length l.
1. What are the possible values of l?
2. What is the dependent and independent variable?
3. Using l and V,find a cubic function of best fit.
4. To maximize the volume of the box, what should be the side length of the corner squares?
1 mark for filling in the length and width column in the table
1 mark for filling in the volume column in the table
1 mark for writing the possible values of l
1 mark for identifying the dependent and independent variables
2 marks for the cubic function of best fit
2 marks for determining the side length of the corner squares that maximizes the volume
To solve the problem, let's work through each part step by step:
Part A:
To complete the table, we need to find the length and width of the box as well as the volume for each given length of square cut, l.
We know that the original cardboard dimensions are 6 inches by 4 inches.
To find the length of the box, we subtract twice the length of the cut square from the original length. So, the length of the box is 6 - 2l.
Similarly, to find the width of the box, we subtract twice the length of the cut square from the original width. Thus, the width of the box is 4 - 2l.
Finally, the volume of the box is given by multiplying the length, width, and height of the box. Since the height is equal to the length of the cut square, the volume of the box is (6 - 2l)(4 - 2l)l.
So, now let's fill in the table:
Length of Square Cut, l (inches) Length of Box (inches) Width of Box (inches) Volume of Box, V (cubic inches)
0.25 5.5 3.5 (5.5)(3.5)(0.25)
0.50 5 3 (5)(3)(0.50)
0.90 4.2 2.2 (4.2)(2.2)(0.90)
1.40 2.2 0.2 (2.2)(0.2)(1.40)
1.80 0.6 -0.2 (0.6)(-0.2)(1.80)
Note: Negative values for length or width are not possible in this context, so we can disregard the last row.
Part B:
1. The possible values of l are 0.25, 0.50, 0.90, and 1.40.
2. In this problem, the length of the cut square (l) is the independent variable, and the volume of the box (V) is the dependent variable.
3. To find a cubic function of best fit, we need to analyze the relationship between l and V. We can plot the values in a graph to identify the pattern and then determine the best cubic function that fits the data points.
4. To maximize the volume of the box, we need to find the value of l that corresponds to the highest volume in the table.