a box with an open top is constructed from a rectangular piece of cardboard with dimensions 14 inches by 18 inches by cutting out and discarding equal squares of side x at each corner and then folding up the sides as in the figure. The cost to create such a box is 2.50 per square foot for the bottom and 1.50 per square foot for the sides
a. Express the cost of the box as a function of x.
b. Graph c=c(x). For what value of x is C smallest?
To solve this problem, we need to break it down into steps:
Step 1: Determine the dimensions of the box
When we cut out squares of side x at each corner, the dimensions of the base of the box become (14-2x) inches by (18-2x) inches. The height of the box will be x inches.
Step 2: Calculate the area of each component
The cost of the box is based on the area. We need to calculate the area of the bottom and the area of the four sides.
The area of the bottom is (14-2x) inches multiplied by (18-2x) inches, which gives us (14-2x)(18-2x) square inches.
The area of the sides can be calculated by multiplying the perimeter of the base by the height. The perimeter is 2 times the sum of the length and width, which gives us 2(14-2x) + 2(18-2x) inches. multiplying the perimeter by the height x inches gives us x(2(14-2x) + 2(18-2x)) square inches.
Step 3: Convert the area to square feet
The cost is given per square foot, so we need to convert the area to square feet. There are 144 square inches in 1 square foot, so we divide the area by 144 to get the area in square feet.
For the bottom area, the cost is 2.50 per square foot, so the cost of the bottom is (2.50)( [(14-2x)(18-2x)] / 144 ) dollars.
For the side area, the cost is 1.50 per square foot, so the cost of the sides is (1.50)( [x(2(14-2x) + 2(18-2x))] / 144 ) dollars.
Step 4: Express the total cost as a function of x
To get the total cost, we add the cost of the bottom and the cost of the sides together. Let's call the total cost C(x).
C(x) = (2.50)( [(14-2x)(18-2x)] / 144 ) + (1.50)( [x(2(14-2x) + 2(18-2x))] / 144 )
Step 5: Find the value of x that minimizes C(x)
To find the value of x that minimizes the cost (C), we can either graph C(x) or use calculus. Let's start with graphing C(x).
To graph C(x), assign a range of possible values for x, such as 0 to 7 (since we can't have negative values or values larger than half the dimensions of the base which is 9 inches).
Plot the graph of C(x) on the y-axis and x on the x-axis. The lowest point on the graph corresponds to the value of x that minimizes C.
Once you have the graph, find the x-coordinate of the lowest point on the graph to determine the value of x that minimizes C.
And that's how you can express the cost of the box as a function of x and find the value of x that minimizes the cost.
To calculate the cost of the box, we need to determine the area of the bottom and the area of the sides. Let's break down the problem step by step.
Step 1: Find the dimensions of the box.
When we cut out squares of side x from each corner, the dimensions of the resulting box base will be (14-2x) inches by (18-2x) inches. The height of the box will be x inches.
Step 2: Calculate the area of the bottom.
The area of the bottom of the box is the product of the two dimensions: (14-2x) inches * (18-2x) inches.
Step 3: Calculate the area of the sides.
There are four sides to the box, each with the same width of x inches and a length equal to the perimeter of the base, which is 2(14-2x) + 2(18-2x) inches. Thus, the total area of the sides is 4x * [2(14-2x) + 2(18-2x)] square inches.
Step 4: Convert the areas to square feet.
Since the cost is given per square foot, we need to convert the areas from square inches to square feet. There are 144 square inches in 1 square foot.
Step 5: Calculate the cost.
The cost of the bottom is given as $2.50 per square foot, so the cost of the bottom is (2.5/144) * (14-2x) * (18-2x). The cost of the sides is given as $1.50 per square foot, so the cost of the sides is (1.5/144) * [4x * (2(14-2x) + 2(18-2x))].
a. Express the cost of the box as a function of x.
The total cost, C(x), is the sum of the cost of the bottom and the cost of the sides:
C(x) = (2.5/144) * (14-2x) * (18-2x) + (1.5/144) * [4x * (2(14-2x) + 2(18-2x))]
b. Graph C=C(x). For what value of x is C smallest?
To find the value of x at which C(x) is smallest, we need to graph the function C(x) and locate its minimum point. However, graphing the function may not be feasible in this format. To find the value of x, we can differentiate C(x), set it equal to zero, and solve for x.