Q. An open-top box with a square base is to have a volume of 62.5 square inches. What dimensions will yield the box of minimum surface area?
I am getting 5" by 2.5" but not sure.
(s^2)h=V
h=V/(s^2)=62.5/s^2
Surface area = 4hs +s^2
=4(62.5/s^2)s +s^2
=250/s + s^2
Derivative of surface area = -250/s^2 + 2s
0=-250/s^2 + 2s
250/s^2=2s
250=2s^3
125=s^3
s=5
h= 62.5/s^2=62.5/25 = 2.5
You are correct!
Thanks
To find the dimensions that yield the box of minimum surface area, we can use optimization techniques.
Let's assume the side length of the square base is 'x' inches, and the height of the box is 'h' inches.
First, note that the volume of the box is given by V = x^2 * h.
We want to minimize the surface area, which consists of the top and bottom squares with an area of 2x^2, plus the four sides with an area of 4xh.
The total surface area, denoted as S, is given by S = 2x^2 + 4xh.
To proceed, we need to express the height 'h' in terms of 'x' using the given volume.
Given that the volume is 62.5 square inches, we have:
62.5 = x^2 * h
Solving for h, we get:
h = 62.5 / (x^2)
Now, substitute this expression for 'h' in the surface area equation:
S = 2x^2 + 4x(62.5 / (x^2))
Simplifying, we have:
S = 2x^2 + 250 / x
To find the minimum surface area, we need to find the critical points of S.
Differentiating with respect to 'x', we have:
S' = 4x - 250 / x^2
Setting S' equal to zero and solving for 'x' gives:
4x - 250 / x^2 = 0
Multiplying both sides by x^2, we have:
4x^3 - 250 = 0
Simplifying further, we get:
4x^3 = 250
Dividing both sides by 4, we have:
x^3 = 62.5
To solve for 'x', we take the cube root of both sides:
x = cube root of 62.5
Hence, the dimensions that yield the box of minimum surface area are given by:
Side length of the square base = cube root of 62.5 inches.
Height of the box = 62.5 / (cube root of 62.5 inches).
Calculating the values, we find:
Side length of the square base ≈ 3.98 inches (rounded to two decimal places).
Height of the box ≈ 3.98 inches (rounded to two decimal places).
Therefore, the dimensions that yield the box of minimum surface area are approximately 3.98 inches by 3.98 inches by 3.98 inches.
To find the dimensions that yield the box of minimum surface area, we need to use calculus and optimization techniques. Let's break down the problem and go step by step:
Step 1: Define the problem:
We want to find the dimensions of an open-top box with a square base that has a volume of 62.5 cubic inches. We need to determine the dimensions that result in the minimum surface area.
Step 2: Define the variables:
Let's assign variables to the unknown dimensions of the box. Let's call the side length of the square base "x," and the height of the box "h."
Step 3: Define the restrictions:
We have two restrictions:
- Volume restriction: The volume of the box is given as 62.5 cubic inches. Hence, we can write the equation: V = x^2 * h = 62.5
- Open-top box restriction: The box has an open top, so only the lateral surface area matters.
Step 4: Define the function:
We need to find the surface area function in terms of one variable, either "x" or "h." Since the volume equation has "x" squared, it's easier to find the surface area in terms of "h."
The surface area of the box consists of the base area (x^2) and the four sides (4xh), so we can write the surface area function as:
S = x^2 + 4xh
Step 5: Rewrite the volume equation to express one variable:
From the volume restriction equation, we can rewrite the equation to express one variable. Solve the equation for "h" in terms of "x":
h = 62.5 / x^2
Step 6: Substitute the value of "h" into the surface area function:
Substituting the value of "h" into the surface area function, we get:
S = x^2 + 4x(62.5 / x^2)
Simplify this expression to get:
S = x^2 + 250 / x
Step 7: Find the derivative of the surface area function:
To find the minimum surface area, we need to take the derivative of the surface area function with respect to "x."
dS/dx = 2x - 250 / x^2
Step 8: Set the derivative equal to zero and solve for "x":
Set dS/dx = 0 and solve for "x":
2x - 250 / x^2 = 0
2x^3 - 250 = 0
x^3 - 125 = 0
(x - 5)(x^2 + 5x + 25) = 0
We have two solutions: x = 5 and the quadratic equation has no real solutions.
Step 9: Determine the corresponding value of "h":
Substitute the value of "x" into the volume equation to solve for "h":
h = 62.5 / x^2
h = 62.5 / 5^2
h = 62.5 / 25
h = 2.5
Step 10: Check the critical point:
Now, we need to determine whether the critical point (x = 5) gives the minimum surface area.
One way to check this is to evaluate the second derivative of the surface area function.
d^2S/dx^2 = 2 + 500 / x^3
For x = 5, the second derivative is positive (2 + 500 / 5^3 = 2 + 500 / 125 > 0).
Therefore, when x = 5 and h = 2.5, we have the dimensions that yield the box of minimum surface area.
Hence, your answer of 5" by 2.5" is correct.