A box with a square base and open top just have volume of 32,000 cm^3 . find the dimension of the box that minimizes the amount of material used.
The volume of a box with a square base and open top:
V = Ab ∙ h
Ab = Area of base = x²
V = Ab ∙ h = x² ∙ h
x² ∙ h = 32,000
Divide both sides by x²
h = 32,000 / x²
The surface area of the box = Area of base + 4 Area of rectangle
As = x² + 4 Ar
Ar = Area of rectangle = x ∙ h
As = x² + 4 ∙ x ∙ h
As = x² + 4 ∙ x ∙ 32,000 / x²
dAs / dx = As'₍ x₎ = ( x² + 4 ∙ x ∙ 32,000 / x² )' = 2 ( x³ - 64,000 ) / x²
If
f ' (x) = 0
function has a local maximum or a local minimum
As'₍ x₎ = 0
2 ( x³ - 64,000 ) / x² = 0
Multiply both sides by x² / 2
x³ - 64,000 = 0
Add 64,000 to both sides
x³ = 64,000
x = ∛64,000 = 40
x = 40 cm
The second derivative test:
When
f ' ₍x₀₎ = 0
then
if
f " ₍x₀₎ > 0, then f has a local minimum at x₀
if
f " ₍x₀₎ < 0, then f has a local maximum at x₀
In this case:
As"₍ x₎ = ( As'₍ x₎ )' = [ 2 ( x³ - 64,000 ) / x² ] ' = 256,000 / x³ + 2
for x = 40
As" = 256,000 / x³ + 2 = 256,000 / 40³ + 2 = 6 > 0
So the surface area of the box has a local minimum for x = 40 cm
Dimension of the box that minimizes the amount of material used:
x = 40 cm
h = 32,000 / x² = 32,000 / 40² = 20 cm
To minimize the amount of material used, we need to find the dimensions of the box that minimize its surface area.
Let's break down the problem step by step:
Step 1: Define the variables:
Let's denote the length of one side of the square base as "x" and the height of the box as "h".
Step 2: Write the volume equation:
The volume of the box is given as 32,000 cm^3. Since the box has a square base, the area of the base is x^2. The volume equation can be written as:
Volume = base area x height
32,000 = x^2 * h
Step 3: Write the surface area equation:
The surface area of the box can be calculated as the sum of the areas of its four sides and the base. The four sides are identical and have an area of x * h, and the base has an area of x^2. The surface area equation can be written as:
Surface Area = 4 (side area) + base area
Surface Area = 4(x * h) + x^2
Step 4: Solve for h in terms of x:
From the volume equation in Step 2, we can isolate h:
h = 32000 / x^2
Step 5: Substitute h in terms of x into the surface area equation:
Substituting h from Step 4 into the surface area equation, we get:
Surface Area = 4(x * (32000 / x^2)) + x^2
Surface Area = 4(32000 / x) + x^2
Step 6: Simplify the surface area equation:
Surface Area = 128000 / x + x^2
Step 7: Find the derivative of the surface area equation:
To find the minimum, we need to find the critical points. We find them by taking the derivative of the surface area equation with respect to x:
d(Surface Area) / dx = 0
d(128000 / x + x^2) / dx = 0
Step 8: Solve for x:
Differentiating the equation gives us:
-128000 / x^2 + 2x = 0
-128000 + 2x^3 = 0
2x^3 = 128000
x^3 = 64000
x = ∛64000
x ≈ 40
Step 9: Determine the height:
Using the volume equation from Step 2, we can calculate the height:
32000 = x^2 * h
32000 = 40^2 * h
32000 = 1600h
h = 32000 / 1600
h = 20
Therefore, the dimensions of the box that minimize the amount of material used are approximately 40 cm (length of one side of the square base) and 20 cm (height).