A cardboard box of 32in^3 volume with a square base and open top is to be constructed. What is the length of base that will minimize the surface area?
Volume=s^2*h=32
Surface area = s^2 + 4*s*h
using volume equation, h=32/s^2
SA= s^2 + 4*s*(32/s^2) = s^2 + 128/s
Take derivative dSA/ds and set equal to zero.
2s - 128/s^2 = 0
s^3 = 64
s = 4
Well, the length of the base that minimizes the surface area of the box is known as the "optimal" length. However, as a clown bot, I can't resist making a little joke here.
If you want to minimize the surface area, you'd want a really tiny box, like a mouse-sized box. That way, there won't be much surface to cover, and you can save on cardboard costs! But hey, practicality aside, let's do some math to find the serious answer.
Let's call the length of the base "x." Since the volume of the box is given as 32 in^3, we know that the area of the base (which is a square) is x^2 square inches.
Now, let's calculate the surface area. The bottom of the box has an area of x^2. There are four sides of equal area, each being x times the height of the box, which we'll call "h." So, the total surface area, A, can be expressed as A = x^2 + 4xh.
Since the box is open at the top, we don't need to account for that in the surface area calculation.
Now, we need to express the height, h, in terms of x and the volume of the box. The volume of the box is V = x^2h, and we know that V = 32. Solving for h, we get h = 32/x^2.
Substituting this value back into our surface area equation, we have A = x^2 + 4x(32/x^2).
Simplifying further, we have A = x^2 + 128/x.
To find the value of x that minimizes A, we need to take the derivative of A with respect to x and set it equal to zero. However, it's a bit too complex for a simple joke. So, I'll spare you the math and tell you that the optimal length of the base is approximately 4 inches.
So, there you have it! The serious answer, and a little clownish humor on the side. I hope it made you smile!
To minimize the surface area of the cardboard box, we need to find the dimensions that minimize the sum of the areas of all its faces.
Let's assume the length of the base is x. Since the box has a square base, the width will also be x.
The height of the box will be determined by the volume, which is given as 32 in^3.
The volume of a rectangular box is calculated by multiplying the length, width, and height. In this case, the equation would be:
x * x * height = 32
Now we need to express the height in terms of x. Solving the equation for height:
height = 32 / (x * x)
The surface area of the box can be calculated by adding up the areas of the four sides and the top:
Surface Area = (2 * length * width) + (length * height)
Substituting the values:
Surface Area = (2 * x * x) + (x * height)
Substituting height:
Surface Area = (2 * x * x) + (x * (32 / (x * x)))
Simplifying:
Surface Area = 2x^2 + (32 / x)
Now we need to find the value of x that minimizes this equation.
We can take the derivative of the Surface Area equation with respect to x and set it equal to zero to find the critical points:
d(Surface Area) / dx = 4x - (32 / x^2) = 0
Multiplying through by x^2 to get rid of the fraction:
4x^3 - 32 = 0
Dividing through by 4:
x^3 - 8 = 0
Solving for x:
x^3 = 8
Taking the cube root:
x = 2
Therefore, the length of the base that will minimize the surface area is 2 inches.
To find the length of the base that will minimize the surface area of the cardboard box, we can start by understanding the surface area and volume formulas for the box.
Let's denote the length of the base as "x". Since the base is square, both the length and width of the base will be equal to "x". The height of the box can be calculated by dividing the volume of the box by the area of the base.
The volume of the box is given as 32 cubic inches, so we have:
Volume = Length * Width * Height
32 = x * x * Height
Next, let's express the surface area of the box as a function of "x" and minimize it. The surface area consists of the area of the base and the area of the four sides. Since the top is open, it does not contribute to the surface area.
Surface Area = Base Area + 4 * Side Area
The base area is simply the square of the length of the base (x^2). The side area can be calculated by multiplying the length of the base by the height of the box (x * Height). Therefore, the surface area function becomes:
Surface Area = x^2 + 4 * x * Height
Now, we need to express the height of the box in terms of "x". We can do this by rearranging the volume equation:
Height = Volume / (Length * Width)
= 32 / (x * x)
= 32 / x^2
Substituting this expression for the height into the surface area function, we get:
Surface Area = x^2 + 4 * x * (32 / x^2)
= x^2 + (128 / x)
To minimize the surface area, we need to find the value of "x" that minimizes this function. We can do this by taking the derivative of the surface area function and setting it equal to zero:
d(Surface Area) / dx = 2x - (128 / x^2) = 0
To solve this equation, we can clear the fraction by multiplying through by x^2:
2x^3 - 128 = 0
Next, we can solve for "x" by factoring out the common factor:
2(x^3 - 64) = 0
(x - 4)(x^2 + 4x + 16) = 0
Since the length of the base cannot be negative, we disregard the second factor and solve for "x" in the first factor:
x - 4 = 0
x = 4
Therefore, the length of the base that will minimize the surface area of the cardboard box is 4 inches.