If 1800 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box
Volume=____________
I did sqrt(1800)=42.4264
and then 42.4264/3= 14.142 cm
then 14.142^(3) and got volume= 2828.34575
however is was incorrect...please help me figure out what I'm doing wrong
THANK YOU!
The box is made up of a square base of side x, and 4 sides each of height h and width x.
The total area is therefore
A=x² + 4hx
Since the area is known, h can be expressed in terms of the area A
A=x²+4hx
or h=(A-x²)/4x
The volume, V is given by
V(x)=x²h
=x²*(A-x²)/4x
=x(A-x²)/4
Use your knowledge of calculus to find
V'(x), and if there is a maximum, the value of x can be found by equating the derivative to zero,
V'(x)=0.
Solve for x and hence V.
I get about 7300 cm³
For x I got 10√6=24.495.
To find the largest possible volume of a box with a square base and an open top, we need to optimize the dimensions of the box.
Let's start by assigning variables to the dimensions of the box:
- Let x be the length of the side of the square base.
- Let y be the height of the box (since the box has an open top).
To find the volume of the box, we need to multiply the area of the base (x^2) by the height (y), so the volume is given by V = x^2 * y.
We are given that there are 1800 square centimeters of material available to make the box. Since the box has an open top, we need to consider that all the material will be used to form the five visible faces of the box (the base and four sides).
The area of the base is x^2, and since we have four sides of the same size (height y), the combined area of the four sides is 4xy.
So, the total surface area of the box is x^2 + 4xy.
We know that the total surface area of the box should be equal to 1800 square centimeters:
x^2 + 4xy = 1800
Now, we can solve this equation for y in terms of x:
4xy = 1800 - x^2
y = (1800 - x^2) / 4x
y = (450 - 0.25x^2) / x
Now, substitute this expression for y into the volume equation V = x^2 * y:
V = x^2 * [(450 - 0.25x^2) / x]
V = 450x - 0.25x^3
To find the largest possible volume, we need to find the maximum value of V.
To do this, we can take the derivative of V with respect to x, set it equal to zero, and solve for x.
dV/dx = 0
450 - 0.75x^2 = 0
Solving this equation, we get:
0.75x^2 = 450
x^2 = 600
x ≈ 24.49 cm
Now, substitute the value of x back into the expression for y:
y = (450 - 0.25x^2) / x
y = (450 - 0.25 * (24.49)^2) / 24.49
y ≈ 10.61 cm
Finally, substitute the values of x and y into the volume equation:
V = x^2 * y
V = (24.49)^2 * 10.61
V ≈ 6340.26 cm^3
So, the largest possible volume of the box is approximately 6340.26 cubic centimeters.
You made a mistake when you calculated the dimensions of the box. You divided the square root of 1800 (which represents the length of the sides) by 3. However, this approach does not consider the total surface area and how it minimizes the use of material while maximizing the volume. By optimizing the dimensions using calculus, we can find the correct answer.
Thanks both of you guys for attempting this problem, and a special thanks to MathMate! The answer was about 7248 cubic cm...
Thanks!!
height = h
side of bottom = s
area of bottom = s^2
area of each of 4 sides = sh
so total area = s^2 + 4sh
1800 = s^2 + 4sh
h = (1800-s^2)/4s
volume = v = s^2 h
so
v = s^2 (1800-s^2)/4s
v = s(1800-s^2)/4
4v = 1800 s - s^3
where is that maximum?
by trial and error I get s = 24.5
then h = .5
volume = 300