To solve this word problem, we can break it down into several steps:
Step 1: Define the variables
Let's define the variables:
- Let's say the width of the base is 'x' feet.
- Since the length of the base is twice the width, the length would be '2x' feet.
- Let's assume the height of the container is 'h' feet.
Step 2: Write the equation for the volume
We know that the volume of a rectangular container is given by the formula: volume = length × width × height.
In this case, the volume is given as 48 ft³, so we can write the equation: 48 = 2x × x × h
Step 3: Express one variable in terms of the other
To make the equation easier to deal with, we need to express one variable in terms of the other two. Let's express the height 'h' in terms of 'x':
48 = 2x × x × h
48 = 2x²h
h = 48 / (2x²)
h = 24 / x²
Step 4: Write the equation for the cost of materials
The cost of materials for the base is given as $8 per square foot, and the cost for the sides is given as $6 per square foot.
To find the cost of materials, we need to calculate the area of each component and multiply it by the corresponding cost per square foot.
- The area of the base is length × width = (2x) × x = 2x²
- The area of each side is length × height = (2x) × (24 / x²) = 48 / x
So, the total cost of materials can be calculated as:
Cost = (Cost of base material) + (Cost of side material)
Cost = (2x²) × ($8 per square foot) + (4 × 48 / x) × ($6 per square foot)
Step 5: Simplify and find the cost of materials
To minimize the cost, we can differentiate the cost function with respect to 'x' and find its critical points.
Differentiating the cost function:
Cost'(x) = (16x) - (288 / x²)
Setting the derivative equal to zero, since we want to find the critical points:
0 = (16x) - (288 / x²)
Multiplying through by x² to clear the fraction:
16x³ - 288 = 0
Simplifying the equation:
16x³ = 288
x³ = 288 / 16
x³ = 18
x ≈ 2.62 (taking the cube root of both sides)
Since x represents the width, it cannot be negative or zero, so we can discard any extraneous solutions and take only the positive value.
Step 6: Calculate the cost
Plug the value of 'x' back into the cost equation:
Cost = (2x²) × ($8 per square foot) + (4 × 48 / x) × ($6 per square foot)
Cost = (2 × 2.62²) × ($8 per square foot) + (4 × 48 / 2.62) × ($6 per square foot)
By evaluating this expression, you can find the cost of materials for the cheapest container.