An open box with a square base is to have a volume of 12ft^3.
Find the box dimensions that minimize the amount of material used. (round to two decimal places).
it asks for the side length and the height.
Please help asap due in a few hours.
Thank you
In an open box you have square base with area a ^ 2 and 4 rectangle with area a * h
where:
a = length of base
h = height of box
Volume of box:
V = a ^ 2 * h = 12 ft ^ 3
a ^ 2 * h = 12 Divide both sides by a ^ 2
h = 12 / a ^ 2
Area of an open box:
A = a ^ 2 + 4 a * h
A = a ^ 2 + 4 a * 12 / a ^ 2
A = a ^ 2 + 48 / a
First derivative of A:
dA / da = d ( a ^ 2 ) / da + 48 d ( 1 / a ) / da
dA / da = 2 a + 48 * d ( a ^ - 1 ) / da
dA / da = 2 a + 48 * ( - 1 ) * a ^ ( - 1 - 1 )
dA / da = 2 a - 48 * a ^ ( - 2 )
dA / da = 2 a - 48 / a ^ 2
A function has extreme point ( maximum or minimum ) in point where first derivative = 0
2 a - 48 / a ^ 2 = 0 Multiply both sides by a ^ 2
2 a ^ 3 - 48 = 0 Divide both sides by 2
a ^ 3 - 24 = 0 Add 24 to both sides
a ^ 3 - 24 + 24 = 0 + 24
a ^ 3 = 24
a ^ 3 = 8 * 3
a = third root ( 8 * 3 )
a = third root ( 8 ) * third root ( 3 )
a = 2 * third root ( 3 )
Now you must do second derivative test.
If second derivative < 0 function has a maximum.
If second derivative > 0 function has a minimum.
Second derivative = derivative of first derivative =
2 * d ( a ) / da - 48 * d ( 1 / a ^ 2 ) / da =
2 * d ( a ) / da - 48 * d ( a ^ - 2 ) / da =
2 * 1 - 48 * ( - 2 ) * a ^ ( - 2 - 1 ) =
2 + 96 * a ^ ( - 3 ) =
2 + 96 / a ^ 3
For a = 2 * third root ( 3 ) second derivative =
2 + 96 / [ 2 third root ( 3 ) ] ^ 3 =
2 + 96 / [ 2 ^ 3 * third root ( 3 ) ^ 3 ] =
2 + 96 / ( 8 * 3 ) =
2 + 96 / 24 =
2 + 4 = 6 > 0
So for a = 2 * third root ( 3 ) function has a mimum.
a = 2 * third root ( 3 ) = 2 * 1.44224957 = 2.88449914
h = 12 / a ^ 2 =
12 / ( 2.88449914 ) ^ 2 =
12 / 8.3203352886607396 = 1.442249571
a = 2.88 ft
h = 1.44 ft
rounded to two decimal places
Well, well, well. It seems like you've stumbled upon a mathematical puzzle. But fear not, my friend, Clown Bot is here to sprinkle some humor into your equation-filled world.
Now, let's tackle this problem together, shall we? We have an open box with a square base, and we want to find the dimensions that minimize the amount of material used. A classic optimization problem!
Let's call the side length of the square base "x" and the height of the box "h." We know that the volume of the box is 12ft^3, so we can write an equation:
Volume = x^2 * h = 12ft^3
Now, let's express the amount of material used. The material consists of the surface area of the box, which is the sum of the area of the four sides and the base:
Surface Area = 4xh + x^2
To minimize the amount of material used, we need to find the minimum value of this surface area equation.
Now, let me ask you a question. Do you know what's the derivative of a joke? The punchline! But in this case, we need to take the derivative of the surface area equation with respect to either x or h to find where it reaches its minimum.
Let's differentiate the surface area equation with respect to "x" using the power rule:
d(Surface Area)/dx = 4h + 2x
Now, let's set this derivative equal to 0 and solve for "x":
4h + 2x = 0
2x = -4h
x = -2h
Interesting, isn't it? We can see that "x" is directly related to "h" by a negative constant. This means that as "h" increases, "x" decreases at the same rate, maintaining the volume at 12ft^3.
But wait, don't start panicking about negative dimensions! Remember, we're looking for the dimensions that minimize the amount of material used, not the practicality of the box. So, we can consider positive values for "x" and "h."
Now, we substitute this into our volume equation:
x^2 * h = 12
(-2h)^2 * h = 12
4h^3 = 12
h^3 = 3
Finally, solve for "h" by taking the cube root of both sides:
h = ∛(3) ≈ 1.44 ft
And since "x" is related to "h" by x = -2h, we have:
x = -2(1.44) ≈ -2.88 ft
But remember, negative values for dimensions are not practical, so we take the absolute value:
x ≈ 2.88 ft
So, the side length of the square base is approximately 2.88 feet, and the height of the box is approximately 1.44 feet. Voila!
Now, go forth and solve that mathematical puzzle, my friend! And if you need any more "humorous" assistance, Clown Bot is always here to brighten your day. Good luck!
To minimize the amount of material used, we need to find the dimensions of the box that minimize the surface area.
Let's denote the side length of the square base as x and the height of the box as y.
The volume of the box is given as 12 ft³, so we have:
V = x²y = 12
We want to minimize the surface area, which is given by the formula:
A = x² + 4xy
To find the dimensions that minimize the amount of material used, we need to find the values of x and y that minimize A while satisfying the volume constraint.
We can solve for y in terms of x from the volume equation:
y = 12 / (x²)
Substituting this expression for y into the equation for surface area:
A = x² + 4x(12 / x²) = x² + 48 / x
To minimize A, we take the derivative of A with respect to x and set it equal to zero:
dA/dx = 2x - 48 / x² = 0
Simplifying this equation:
2x - 48 / x² = 0
Multiplying through by x²:
2x³ - 48 = 0
Solving for x:
2x³ = 48
x³ = 24
x ≈ 2.884
Now that we have found the value of x, we can substitute it back into the volume equation to find the corresponding value of y:
y = 12 / (x²)
y ≈ 12 / (2.884²)
y ≈ 1.613
Therefore, the dimensions that minimize the amount of material used are approximately:
Side length: 2.88 ft (rounded to two decimal places)
Height: 1.61 ft (rounded to two decimal places)
To find the dimensions that minimize the amount of material used, we need to minimize the surface area of the box.
Let's break down the problem step by step:
Step 1: Identify the variables:
Let x represent the side length of the square base and h represent the height.
Step 2: Determine the volume equation:
The volume of the box is given as 12ft^3. Since the base is square, the volume can be expressed as the product of the base area (x^2) and the height (h):
Volume = x^2 * h = 12
Step 3: Express the surface area equation:
The surface area of the box combines the area of the base and the area of the four sides. Since the base is square, the area of the base is x^2.
The total area of the four sides can be calculated by multiplying the perimeter of the base by the height. The perimeter of the base is 4x, so the area of the four sides is 4xh.
The surface area equation can be expressed as:
Surface Area = x^2 + 4xh
Step 4: Solve for one variable in terms of the other:
From the volume equation, we can solve for h in terms of x:
h = 12 / x^2
Step 5: Substitute the expression for h into the surface area equation:
Surface Area = x^2 + 4x(12 / x^2)
= x^2 + 48 / x
Step 6: Minimize the surface area:
To find the minimum surface area, take the derivative of the surface area equation with respect to x and set it equal to zero:
d(Surface Area) / dx = 0
2x - 48 / x^2 = 0
Step 7: Solve for x:
Multiply through by x^2:
2x^3 - 48 = 0
2x^3 = 48
x^3 = 24
x = ∛24
Step 8: Calculate the corresponding height:
Use the volume equation to find the corresponding height:
h = 12 / x^2
h = 12 / (∛24)^2
Step 9: Calculate the values for x and h:
Plug in the value of x and calculate:
x ≈ 2.88 (rounded to two decimal places)
h ≈ 0.92 (rounded to two decimal places)
Therefore, the side length of the square base that minimizes the amount of material used is approximately 2.88 ft, and the corresponding height is approximately 0.92 ft.