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!
well, you know l = w because the base is square.
v = w^2 h = 300 mL = .3 *10^-3 m^3
so h = 3 * 10^-4 /w^2
cardboard area
bottom and top = 2 w^2
sides = 4 w h
so
A = 2 w^2 + 4 w h
A = 2 w^2 + 4 w (3*10^-4)/w^2
A = 2 w^2 + 12*10^-4 /w
dA/dw = 4 w -1.2*10^-3/w^2
= 0 fo min
4 w^3 = 1.2 * 10^-3
w^3 = .3 * 10^-3
w = .67 * 10^-1 meters
= 6.7 centimeters
h = 3*10^-4/w^2
h = 3*10^-4 / .45 *10^-2 = 6.7 *10^-2
= 6.7 centimeters
In other words, the answer is a cube.
To find the dimensions of the container that minimizes the amount of cardboard used, we need to express the amount of cardboard in terms of the dimensions of the container and then minimize that expression.
Let's break down the problem step by step:
1. Start with the equation for volume:
Volume = l x w x h
Since the container has a square base, l = w (let's call this side length x), so the volume equation becomes:
Volume = x^2 * h
2. We have a constraint that the container should hold 300 mL of juice. Since 1 mL is equal to 1 cm³, the volume equation becomes:
300 = x^2 * h
3. We need to express the amount of cardboard used. The cardboard consists of the top and bottom squares (2*x^2) and the four rectangular sides (4*x*h):
Cardboard = 2*x^2 + 4*x*h
4. Now, we can express the amount of cardboard in terms of a single variable. Let's solve the volume equation for h:
h = 300 / (x^2)
Substitute the value of h in the cardboard equation:
Cardboard = 2*x^2 + 4*x*(300 / (x^2))
= 2*x^2 + 1200 / x
5. The amount of cardboard is given by Cardboard = 2*x^2 + 1200 / x. We want to minimize this expression by finding the critical points.
To find the critical points, take the derivative of the cardboard equation with respect to x and set it equal to zero:
d(Cardboard)/dx = 4x - 1200/x^2 = 0
Solve this equation for x:
4x - 1200/x^2 = 0
4x^3 - 1200 = 0
x^3 = 1200/4
x^3 = 300
x = ∛300
6. Now that we have the value of x, we can substitute it back into the volume equation (300 = x^2 * h) to find h.
7. Finally, substitute the values of x and h into the cardboard equation (Cardboard = 2*x^2 + 1200 / x) to find the minimum amount of cardboard used.
Remember to verify that the point you found gives a minimum and not a maximum by taking the second derivative (d²(Cardboard)/dx²) and evaluating it at the critical point. If it's positive, it's a minimum; if it's negative, it's a maximum.
I hope this helps you solve the problem! Let me know if you need further clarification on any step.