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 bonding up the sides. find the formula that expresses the volume of the box as a function of x.
volume = x(12-2x)(10-2x)
expand if needed.
V=120x-44x2+4x3
To find the formula that expresses the volume of the box as a function of x, we need to follow a step-by-step approach.
Step 1: Calculate the dimensions of the box.
When squares of side x are removed from each corner, the length of the resulting box will be (12 - 2x) inches, and the width will be (10 - 2x) inches.
Step 2: Calculate the height of the box.
The height of the box will be equal to the side length of the square that was removed, which is x inches.
Step 3: Calculate the volume of the box.
The volume of a rectangular box is given by the formula:
Volume = Length x Width x Height
Substituting the values we calculated:
Volume = (12 - 2x) x (10 - 2x) x x
Simplifying this formula will give us the final expression for the volume of the box as a function of x.
To find the formula that expresses the volume of the box as a function of x, we need to understand the process of constructing the box.
Step 1: Understanding the dimensions
The rectangular piece of tin, initially measuring 12 inches long and 10 inches wide, will be used to construct an open box. We will cut x-inch squares from each corner of the tin sheet.
Step 2: Determining the dimensions of the box
When we cut out identical squares from each corner, the remaining tin sheet will form the base of the box. The height of the box will be equal to the length of the square cut out (x inches). The length and width of the box will be reduced by twice the value of x since we cut out x inches from each side.
The length of the box will be: 12 inches - 2x inches
The width of the box will be: 10 inches - 2x inches
The height of the box will be: x inches
Step 3: Calculating the volume of the box
The volume of a rectangular box is given by multiplying its length, width, and height. Therefore, the volume (V) of the box can be expressed as:
V = (length) * (width) * (height)
V = (12 - 2x) inches * (10 - 2x) inches * x inches
V = x(12 - 2x)(10 - 2x) cubic inches
So, the formula that expresses the volume of the box as a function of x is V = x(12 - 2x)(10 - 2x) cubic inches.