4. A shipping crate has a square base with sides of length x feet, and it is half as tall as it is wide. If the material for the bottom and sides of the box costs $2.00 per square foot and the material for the top costs $1.50 per square foot, express the total cost of material for the box as a function of x.
c = 2(x^2 + 4*x(x/2)) + 1.50x^2 = 15/2 x^2
To find the total cost of the material for the box, you need to determine the area of each component and multiply it by the corresponding cost per square foot.
Let's break it down step-by-step:
1. Determine the area of the bottom:
The bottom of the box is a square with sides of length x feet. Therefore, the area of the bottom is given by x * x = x^2 square feet.
2. Determine the area of the four sides:
Since the box has a square base, all four sides are identical in shape and size. Each side is a rectangle with a width of x feet and a height of (1/2) * x feet. So, the area of each side is given by x * (1/2) * x = (1/2) * x^2 square feet. Since there are four sides, the total area of the four sides is 4 * (1/2) * x^2 = 2 * x^2 square feet.
3. Determine the area of the top:
The top of the box is also a square with sides of length x feet. Therefore, the area of the top is given by x * x = x^2 square feet.
4. Calculate the total cost of the material:
The cost of the material for the bottom and sides of the box is $2.00 per square foot, and the cost for the top is $1.50 per square foot. To find the total cost, we need to multiply the area of each component by its respective cost per square foot.
The cost for the bottom and sides is 2 * (x^2) = 2x^2 dollars.
The cost for the top is 1.50 * (x^2) = 1.5x^2 dollars.
Therefore, the total cost of the material for the box as a function of x is given by:
Total Cost = Cost of Bottom and Sides + Cost of Top
= 2x^2 + 1.5x^2
= 3.5x^2 dollars.
Thus, the total cost of material for the box is 3.5x^2 dollars.
To express the total cost of material for the box as a function of x, we need to determine the surface area of each component of the box.
Let's start by finding the surface area of the bottom and sides of the box. The base of the box is square with sides of length x feet, so the area of the base is x * x = x^2 square feet.
The box is also half as tall as it is wide, so its height is (1/2) * x = (1/2)x feet. Since the box has four identical sides, the combined area of the four sides is 4 * (x * (1/2)x) = 2x^2 square feet.
The total surface area of the bottom and sides of the box is the sum of the base area and the area of the four sides:
x^2 + 2x^2 = 3x^2 square feet.
Now, let's calculate the surface area of the top of the box. The top is also a square with sides of length x feet, so its area is x * x = x^2 square feet.
The total cost of material for the bottom and sides of the box is given as $2.00 per square foot, so the cost of the bottom and sides is 3x^2 * $2.00 = $6.00x^2.
The cost of material for the top of the box is given as $1.50 per square foot, so the cost of the top is x^2 * $1.50 = $1.50x^2.
The total cost of material for the box is the sum of the cost of the bottom and sides and the cost of the top:
$6.00x^2 + $1.50x^2 = $7.50x^2.
Therefore, the total cost of material for the box is a function of x and can be expressed as f(x) = $7.50x^2.