Four corners are cut from a rectangular piece of cardboard that measures 5 ft by 3 ft. The cuts are x feet from the corners, as shown in the figure below. After the cuts are made, the sides of the rectangle are folded to form an open box. The area of the bottom of the box is 12 ft^2.

Question

Four corners are cut from a rectangular piece of cardboard that measures 5 ft by 3 ft. The cuts are x feet from the corners, as shown in the figure below. After the cuts are made, the sides of the rectangle are folded to form an open box. The area of the bottom of the box is 12 ft^2.

Answer 1

What is your question? I'm guessing.

(5 - 2x)(3 - 2x) = 12

Solve for x.

Answer 2

To find the value of x and determine the dimensions of the box, we need to use the given information and apply some geometry principles.

Let's start by visualizing the scenario. We have a rectangular piece of cardboard measuring 5 ft by 3 ft. We then cut four corners and fold the sides to form an open box. The cuts are x feet from the corners.

Let's denote the length of the cut as x and the width of the cut as y.

From the given information, we know that the dimensions of the rectangular piece of cardboard are 5 ft by 3 ft. We also know that the area of the bottom of the box is 12 ft^2.

The formula for the area of a rectangle is A = length x width. Therefore, the area of the bottom of the box is x * y.

We can set up an equation using the given information: x * y = 12.

Next, we need to express the dimensions of the box in terms of x. Since we cut x feet from each corner, the length of the box will be 5 - 2x (subtracting the length of two corners) and the width of the box will be 3 - 2x (subtracting the width of two corners).

Now we can substitute the dimensions of the box into the area equation:

(5 - 2x) * (3 - 2x) = 12.

Simplifying this equation will give us a quadratic equation:

(5 - 2x) * (3 - 2x) = 12
15 - 10x - 6x + 4x^2 = 12
4x^2 - 16x + 3 = 0.

We can solve this quadratic equation to find the value(s) of x. We can use factoring, completing the square, or the quadratic formula to solve for x.

Once we find the value of x, we can substitute it back into the expressions for the dimensions of the box (5 - 2x) and (3 - 2x) to determine the actual dimensions of the box.

I hope this explanation helps you understand the process to find the value of x and the dimensions of the box.