An open top box with a square base is to have a volume of exactly 500 cubic inches. Find the dimensions of the box that can be made with the smallest amount of materials.
Well, isn't that a box full of possibilities! Let's solve this puzzle together.
To minimize the amount of material used, we want to find dimensions that create a square base while still giving us a volume of 500 cubic inches. Let's call the side length of the base 'x' and the height 'h'.
Since the base is square, the area of the base will be x * x, or simply x^2. And since it's an open-top box, the volume will be the base area multiplied by the height, so we have:
x^2 * h = 500
To minimize the amount of material used, we need to minimize the surface area of the box. The surface area consists of the area of the base (x^2) plus the areas of the four sides (4 * x * h), so we have:
Surface area = x^2 + 4 * x * h
Now, if we solve the volume equation for h, we get:
h = 500 / x^2
Substituting this value of h into the surface area equation:
Surface area = x^2 + 4 * x * (500 / x^2)
= x^2 + 2000 / x
Now, to minimize the surface area, we need to find the derivative of the surface area equation with respect to x and set it equal to zero. But since we're here to bring joy, not math anxiety, let's just trust that the derivative will give us a point of minimum surface area.
The derivative of the surface area equation is:
d(surface area)/dx = 2x - 2000/x^2
Setting that equal to zero, we get:
2x - 2000/x^2 = 0
Cross-multiplying and rearranging:
2x^3 = 2000
Simplifying:
x^3 = 1000
Taking the cube root of both sides, we find:
x = 10
Now, substituting this value of x back into the volume equation, we can solve for h:
h = 500 / (10^2)
h = 5
So, the dimensions of the box with the smallest amount of materials are a base of 10 inches by 10 inches, and a height of 5 inches.
Voilà! We've found the magical dimensions for the box. Now, whether you use it to keep your secrets safe or to hold an army of rubber ducks is up to you!
To find the dimensions of the box that can be made with the smallest amount of materials, we need to consider that the box has an open top and a square base.
Let's assume the length of one side of the square base is x inches. Since the base is square, the width is also x inches.
The volume of a rectangular box is given by the formula: volume = length × width × height.
In this case, the height of the box (h) is unknown. We can express it in terms of x using the given volume of 500 cubic inches.
So, the equation becomes: x²h = 500.
To minimize the amount of material used, we need to minimize the surface area of the box. The surface area (S.A.) of an open-top box with a square base is given by the formula:
S.A. = x² + 4xh.
To minimize the surface area, we can express h in terms of x and substitute it back into the formula.
From the earlier equation x²h = 500, we can isolate h by dividing both sides by x²:
h = 500 / x².
Substituting this value of h back into the surface area formula, we get:
S.A. = x² + 4x(500 / x²).
Expanding the equation, we have:
S.A. = x² + 2000 / x.
To find the minimum surface area, we take the derivative of the surface area function with respect to x and set it equal to zero:
d(S.A.) / dx = 2x - 2000 / x² = 0.
Multiplying through by x², we get:
2x³ - 2000 = 0.
Simplifying further:
2x³ = 2000.
Dividing both sides by 2:
x³ = 1000.
Taking the cube root of both sides:
x = 10.
So, one side of the square base is 10 inches. Since the width is also x inches, the dimensions of the box that can be made with the smallest amount of materials are:
Length = Width = 10 inches.
Height (h) can be found by substituting x into the initial equation:
h = 500 / x² = 500 / (10²) = 500 / 100 = 5 inches.
Therefore, the dimensions of the box are:
Length = 10 inches,
Width = 10 inches,
Height = 5 inches.
To find the dimensions of the box that can be made with the smallest amount of materials, we need to minimize the surface area of the box.
Let's assume that the length of one side of the square base is 'x' inches. Since it is a square base, the area of the base is x^2 square inches.
Now, since the box is open on top, the surface area of the box consists of the area of the base (x^2 square inches) and the areas of the four equal sides. Since the sides of the box are equal in length, each side can be represented by the width of the box, denoted as 'w' inches.
There are two sides that have dimensions (x inches) x (w inches), and two sides that have dimensions (w inches) x (w inches). Therefore, the total surface area of the box is:
2(xw) + 2(w^2) = 2xw + 2w^2 square inches.
To minimize the surface area, we need to find the dimensions that will minimize this expression while satisfying the volume requirement.
Given that the volume of the box is 500 cubic inches, we have the equation:
x^2w = 500.
Now, we can express one variable in terms of the other, and substitute it into the expression for the surface area:
w = 500 / (x^2).
Substitute the value of 'w' into the equation for the surface area:
S = 2x(500 / (x^2)) + 2(500 / (x^2))^2.
To find the dimensions that minimize the surface area, we need to find the critical points of this function.
Let's differentiate S with respect to x:
dS/dx = 2(500 / (x^2)) - 4(500 / (x^3)).
Setting dS/dx equal to zero and simplifying, we get:
2(500 / (x^2)) - 4(500 / (x^3)) = 0.
Multiplying through by x^3, we have:
1000 - 2000 / x = 0.
Solving this equation for x, we get:
x = √2 inches.
Substituting this value back into the equation for 'w', we get:
w = 500 / (2 inches)^2 = 125 square inches.
Therefore, the dimensions of the box that can be made with the smallest amount of materials are a base length of √2 inches and a width of 125 square inches.
your turn to do this one
I will get you going ...
base: x by x
height : y
V = x^2 y
500 = x^2 y
y = 500/x^2
SA = base + 4 sides
= x^2 + 4xy
plug in y from above
differentiate
set that equal to zero and solve