A rectangular box with the volume 320 cu ft is built with a square base and a square top. The cost is $1.50/sg ft for the bottom, $2.50/sq ft for the sides and $1.00/sq ft for the top. Let x=the lenth of the base in feet. Express the cost of the box as a function of x. Where do I start?
You can see this better if you draw a rectangular box. The base is x and x so the area will be x^2.
The volume of the box is x^2 x height = x^2*h = 320 so solving for h we have
h=320/x^2.
Now figures the cost.
For the bottom the area is x^2.
The cost is 1.50/ft^2*(x^2) = ??
The cost of the top is
2.50/ft&2 *(x^2) = ??
The area of the sides is
h*x = (320/x^2)*x and the cost is that times 2.50/ft^2 and that times 2 (since there are two sides. Remember to put this cost in terms of x.
Then add all the costs.
Post your work if you get stuck.
Thx! I was making this harder than it needed to be! I originally got started like you said and thought it looked too easy!
No problem! Sometimes the simplest approach is the best one. Just take it step by step and you'll get there. And remember, if you do get stuck, I'm always here to lend a helping chuckle!
You're welcome! Sometimes problems can seem more complicated than they actually are. It's always a good idea to start with the given information and break the problem down into smaller steps. If you have any more questions or need further assistance, feel free to ask!
You're welcome! It's great that you realized you were overthinking it. Sometimes, the best approach is to break the problem down into smaller, manageable parts. In this case, we broke down the problem into finding the volume, determining the cost for each component, and then adding up all the costs.
Now, let's continue with finding the cost for each component:
1. The cost for the bottom:
Area of the bottom = x^2
Cost for the bottom = $1.50/ft^2 * (x^2) = 1.5x^2
2. The cost for the top:
Area of the top = x^2
Cost for the top = $1.00/ft^2 * (x^2) = x^2
3. The cost for the sides:
Area of each side = height * x = (320/x^2) * x = 320/x
Cost for each side = $2.50/ft^2 * (320/x)
Since there are two sides, the total cost for the sides = 2 * $2.50/ft^2 * (320/x) = (5 * 320)/(2x) = 800/x
Now, we can express the cost of the box as a function of x by adding up all the costs:
Cost of the box = Cost for the bottom + Cost for the top + Cost for the sides
Cost of the box = 1.5x^2 + x^2 + 800/x
Therefore, the cost of the box is given by the function f(x) = 2.5x^2 + 800/x.
I hope that clarifies the steps you need to take to express the cost of the box as a function of x. If you have any further questions, feel free to ask!