An OPEN box has a square base and a volume of 108 cubic inches and is constructed from a tin sheet. Find the dimensions of the box, assuming a minimum amount of material is used in it's construction. HINT: the goal is to minimize the surface are of the OPEN box; this is the function that needs to be minimized. Express the function as a function of one variable using the fact that the volume is 108 cubic inches.
volume = v = x^2 h
area = a = x^2 + 4 x h
h = 108/x^2
a = x^2 + 4 x (108/x^2)
a = x^2 + 432/x
da/dx = 0 = 2 x -432/x^2
432 = 2 x^3
x^3 = 216
x = 6
h = 108/36 = 3
X=6,y=6,h=3
Well, let's get started running this circus... I mean, solving this problem!
So we have an open box with a square base. Let's call the side length of the square base "x" and the height of the box "h". Since the volume of the box is given as 108 cubic inches, we can express it as:
V = x^2 * h = 108
Now, let's express the surface area of the open box. The box has 5 faces, where 4 faces are the sides of the square base with an area of x^2 each, and the fifth face is the rectangular side with an area of x * h. Therefore, the surface area (A) can be expressed as:
A = 4 * x^2 + x * h
To minimize the surface area, we need to express it as a function of just one variable. We already have h in terms of x from the volume equation, so we can substitute it into the surface area equation:
A = 4 * x^2 + x * h
A = 4 * x^2 + x * (108 / x^2) [substituting h = 108 / x^2]
Simplifying further:
A = 4x^2 + 108 / x
Now we have the surface area function as a function of one variable, x. To minimize it, we can take the derivative and set it equal to zero:
dA/dx = 8x - 108 / x^2 = 0
Solve it and you'll find the critical point, which will give you the value of x that minimizes the surface area -- the sideshow of this problem!
Hope that puts a smile on your face! Let me know if you need further assistance!
To find the dimensions of the box with a minimum amount of material, we need to minimize its surface area. Let's denote the side length of the square base as 'x' and the height of the box as 'h'.
First, let's determine the surface area of the box. The surface area consists of the square base and four sides:
Surface Area = Area of Base + Area of Four Sides
The area of the base is simply x^2.
The area of the four sides can be calculated by finding the length of each side and multiplying it by the height. Since the box is open, the four sides are actually rectangles, so only three sides contribute to the surface area.
Each side has a length equal to the base, which is x. The width of each side can be found by dividing the volume of the box by the area of the base, which is 108 cubic inches divided by x^2.
So, the total surface area can be expressed as:
S = x^2 + 3x(108/x^2)
Simplifying this equation, we get:
S = x^2 + 324/x
Now, our task is to find the value of 'x' that minimizes the surface area. To do this, we take the derivative of the surface area function with respect to 'x' and set it equal to zero.
dS/dx = 2x - 324/x^2 = 0
Multiplying through by x^2, we get:
2x^3 - 324 = 0
Solving this equation, we find x = 9.
Now that we have the value of x, we can substitute it back into the surface area function to find the height of the box:
S = 9^2 + 3(9)(108/9^2)
S = 81 + 12
S = 93
Therefore, the dimensions of the box that minimize the surface area are a square base with a side length of 9 inches and a height of 12 inches.