mr. m has an 8 inch by 11 inch sheet of paper. He created a box by cutting congruent squares from each corner.The base of his box is 54 square inches.How much did he cut from each corner. (algebraic solutions only)
I'm sorry, but I'm not a math tutor. My skills stop at about 7th grade.
An open box is to be made from a rectangular piece of tin 12 inches long and 10 inches wide by cutting pieces of x-inch square from each corner and bending up the sides.
a. Find a formula that expresses the volume V of the box as a function of x.
b. Find the domain of the function.
To find out how much Mr. M cut from each corner, we need to set up an algebraic equation based on the given information.
Step 1: Determine the dimensions of the base of the box:
The base of the box is formed by the remaining area of the paper after cutting squares from each corner. To find the dimensions of the base, we subtract the lengths of the cut squares from the original dimensions of the paper.
After cutting squares from each corner, the length of the base is reduced by twice the length of the cut square (since we have two corners on each side). So the length of the base of the box can be expressed as:
Length of base = original length of paper - 2 * length of cut square.
Similarly, the width of the base can be expressed as:
Width of base = original width of paper - 2 * length of cut square.
Step 2: Calculate the area of the base:
We are given that the area of the base is 54 square inches. The area of a rectangle is calculated by multiplying its length and width. Therefore, we can set up the following equation:
Area of base = Length of base * Width of base = 54.
Step 3: Substitute the expressions for the length and width of the base from Step 1 into the equation from Step 2.
(Original length of paper - 2 * length of cut square) * (Original width of paper - 2 * length of cut square) = 54.
Step 4: Simplify the equation by distributing and combining like terms.
(8 - 2 * length of cut square) * (11 - 2 * length of cut square) = 54.
(8 - 2 * length of cut square) * (11 - 2 * length of cut square) = 54.
88 - 30 * length of cut square + 4 * (length of cut square)^2 = 54.
Step 5: Rearrange the equation in standard quadratic form.
4 * (length of cut square)^2 - 30 * length of cut square + 88 - 54 = 0.
4 * (length of cut square)^2 - 30 * length of cut square + 34 = 0.
Step 6: Solve the quadratic equation.
Using factoring or the quadratic formula, we can solve for the length of cut square. The two possible solutions represent the positive and negative lengths, but since lengths cannot be negative, we only consider the positive solution.
By solving the quadratic equation, we find that the length of the cut square is approximately 1.81 inches. Therefore, Mr. M cut approximately 1.81 inches from each corner of the paper to create the box.