Suppose that a box has a base with a width of x, a length of x+3, and a height of 1 inch. It is cut from a square sheet of material with an area of 130 in2. Find the dimensions of the box.
Assuming the box is closed, it has 6 faces, so
2x(x+3) + 2x(1) + 2(x+3)(1) = 130
x^2+5x+3 = 65
x = 5.76
Hmmm. If the box is open, with no top, then
x(x+3) + 2x(1) + 2(x+3)(1) = 130
x^2+7x+6 = 130
x = 8.17
I guess x is not an integer.
To find the dimensions of the box, we need to solve for the value of x that satisfies two conditions:
1. The area of the square base of the box should be equal to the area of the material, which is given as 130 square inches.
2. The height of the box is given as 1 inch.
Let's break down the problem step by step to find the dimensions of the box.
Step 1: Determine the area of the base:
The area of the base can be calculated by multiplying the length and the width of the base. In this case, the length is x+3 and the width is x. So, the area of the base is (x+3) * x.
Step 2: Set up the equation:
Since the area of the base should be equal to 130 square inches, we can set up the following equation:
(x+3) * x = 130
Step 3: Solve the equation:
Expand the equation: x^2 + 3x = 130.
Rearrange the equation: x^2 + 3x - 130 = 0.
Now, we can solve the quadratic equation using factoring, completing the square, or the quadratic formula.
We find that the factors of -130 that sum to 3 are 13 and -10:
(x + 13)(x - 10) = 0
So, the possible values of x are x = -13 or x = 10.
Since we are talking about dimensions, x cannot be negative. Therefore, x = 10.
Step 4: Calculate the dimensions:
Now that we have found the value of x, we can calculate the dimensions of the box.
Width = x = 10 inches
Length = x + 3 = 10 + 3 = 13 inches
Height = 1 inch
Therefore, the dimensions of the box are:
Width = 10 inches
Length = 13 inches
Height = 1 inch.