This is the question:
"A new fruit juice (like a Popper) is to be marketed in a new container in the shape of a rectangular prism. The cardboard container is to have a square base and is to contain 300 mL of juice. What should be the dimensions of the container be if the amount of cardboard used in its construction is to be a minimum? Disregard waste and overlap."
All I have so far is:
Volume = l x w x h
Therefore:
300 = l x w x h
I honestly have no idea where to go from here.
Do I eventually have to derive an equation and then let that = 0 and then factorise?
I would really appreciate some help, guys.
Thank you in advance!
To solve this problem, you need to find the dimensions of the container that minimize the amount of cardboard used.
Let's break it down into a step-by-step process:
Step 1: Determine the variables and constraints
In this problem, the variables are the length (l), width (w), and height (h) of the rectangular prism. The constraint is that the volume of the juice is 300 mL.
Step 2: Formulate the objective function
The objective is to minimize the amount of cardboard used, which is determined by the surface area of the container. The surface area is made up of the area of the base, front, back, left side, right side, top, and bottom. To find the surface area, you need to identify the equations for each of these areas.
Step 3: Identify equations for surface area
- The area of the base is given by A_base = l x w.
- The areas of the front, back, left side, and right side are all equal, given by A_side = l x h.
- The areas of the top and bottom are also equal, given by A_top_bottom = w x h.
Step 4: Formulate the objective function
The surface area of the container is the sum of all these areas, so the objective function can be written as:
A_total = 2(A_base) + 4(A_side) + 2(A_top_bottom)
Step 5: Express the objective function in terms of one variable
Since we have three variables (l, w, and h), we need to express the objective function in terms of a single variable. We can use the volume constraint to solve for one variable. From your initial equation, 300 = l x w x h, we can express h in terms of l and w: h = 300 / (l x w).
Step 6: Substitute h into the objective function
Substituting the value of h into the objective function, we get:
A_total = 2(l x w) + 4(l x (300 / (l x w))) + 2(w x (300 / (l x w)))
Step 7: Simplify the objective function
Simplifying the expression, we have:
A_total = 2lw + 1200 / w + 600 / l
Step 8: Take the derivative of the objective function
To find the minimum value, we need to find where the derivative of the objective function is equal to zero. Take the derivative of A_total with respect to either l or w (let's choose w) and set it to zero.
Step 9: Solve for the variable
Once you have the derivative equation, solve it to find the value of variable (w) that minimizes the objective function. Substitute this value back into the volume equation (300 = l x w x h) to find the corresponding values of l and h.
Step 10: Check for minimum
To verify that your solution corresponds to a minimum, check the second derivative. If it is positive, then you have found the dimensions that minimize the amount of cardboard used in the construction of the container.
By following these steps, you should be able to find the dimensions of the container that minimize the amount of cardboard used.