AM having problems understanding what equations to use for this word problem. Please help.
A rectangular storage container with an open top is to have a volume of 48ft^3. The length of its base is twice the width. Material for the base costs $8 per square foot and material for the sides cost $6 per square foot. Find the cost of materials for the cheapest such container.
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.
To tackle this word problem, let's break it down into smaller parts and define a few variables.
1. Let's start by defining the variables:
- Let "L" represent the length of the base of the container.
- Let "W" represent the width of the base of the container.
- Let "H" represent the height of the container.
- Let "C" represent the cost of the material for the container.
2. We are given that the length of the base is twice the width:
- So we can write the equation: L = 2W.
3. The volume of the rectangular container is given as 48ft^3:
- The volume of a rectangular prism is given by the equation: V = LWH.
- We know that V = 48. Substitute the known values: 48 = (2W)(W)(H).
4. Now, let's consider the cost of the materials:
- The cost of the material for the base is $8 per square foot, and the cost of the material for the sides is $6 per square foot.
- The base of the container is rectangular, so its area is L * W.
- The sides of the container form a rectangle when the container is opened, with a height of H. Hence, the area of each side is 2LH + 2WH.
5. To find the cost of materials, we need to calculate the surface area of the base and the sides:
- The cost of materials for the base is 8 * (L * W).
- The cost of materials for the sides is 6 * (2LH + 2WH).
6. We want to find the cost of materials for the cheapest container:
- We need to minimize the cost of materials, which means finding the minimum value of C.
- C = (8 * L * W) + (6 * (2LH + 2WH)).
7. We can substitute the value of L from the equation in step 2:
- C = (8 * 2W * W) + (6 * (2(2W)(H) + 2W(H))).
8. Simplifying the equation further:
- C = (16W^2) + (6 * (4WH + 2WH)).
- C = (16W^2) + (6(6WH)).
- C = 16W^2 + 36WH.
Now you have the equation that represents the cost of materials for the container. To find the cost of materials for the cheapest container, you need to find the values of W and H that minimize the cost equation. This can be done by either factoring, substituting, or graphing the equation.
I hope this breakdown helps you understand the equations needed to solve the word problem. Good luck!