Ignoring the walls’ thickness, determine the outside dimensions that will minimize a closed box’s cost if it has a square base and top, if its volume is 32 cubic meters, and if the cost per square meter for the top, bottom, and sides respectively are $.03, $.04, and $.05.
(solve step by step please!)
4/2
can you explain how you got that please?
let the base be x m by x m, let the height be y m
x^2 y = 32
y = 32/x^2
cost = cost of base + cost of top + cost of 4 sides
= .03x^2 + .04x^2 +4(.05)xy
= .07x^2 + .2x(32/x^2)
= .07x^2 + 6.4/x
d(cost)/dx = .14x - 6.4/x^2
= 0 for a min of cost
6.4/x^2 = .14x
x^3 = .896
x = .896^(1/3) = appr .964 m
then y = 32/.964^2 = appr 34.43 m
The answer seems unreasonable, but I just can't find any errors in my solution.
Even wrote it out on paper.
To find the outside dimensions that will minimize the cost of a closed box, we need to consider the cost of each side of the box.
Let's break down the problem into steps:
Step 1: Define the variables
Let's define the variables we need:
- Length of the base and top of the box: Let's call it "x" (since it has a square base and top).
- Height of the box: Let's call it "h".
Step 2: Express the volume of the box
The volume of the box is given as 32 cubic meters. Since the base has a square shape, we can express the volume as:
Volume = x^2 * h = 32
Step 3: Express the cost function
The cost per square meter for the top is $0.03, for the bottom is $0.04, and for the sides is $0.05. To calculate the cost, we need to determine the area of each side and then multiply it by the corresponding cost per square meter.
The cost function can be expressed as:
Cost = (area of base) * (cost per square meter for the bottom) + (area of top) * (cost per square meter for the top)
+ (area of sides) * (cost per square meter for the sides)
Step 4: Express the area of each side
- Area of the base: Since the base is a square with side length "x", the area of the base is simply x^2.
- Area of the top: Same as the base, so it is also x^2.
- Area of the sides: The total area of the sides can be found by adding up the areas of all four sides. Since the box has a height "h" and a square base with side length "x", the area of each side is x * h.
Step 5: Substituting the expressions into the cost function
Now, let's substitute the expressions for the areas and the given cost per square meter values into the cost function:
Cost = (x^2) * ($.04) + (x^2) * ($.03) + 4(x * h) * ($.05)
Simplifying the expression:
Cost = 0.04x^2 + 0.03x^2 + 0.20xh
Step 6: Solve the volume equation for "h"
From Step 2, we have x^2 * h = 32. Rearranging this equation, we get:
h = 32 / x^2
Step 7: Substitute the expression for "h" into the cost function
Now, substitute the expression for "h" into the cost function we derived in Step 5:
Cost = 0.04x^2 + 0.03x^2 + 0.20x(32/x^2)
Simplifying the expression:
Cost = 0.07x^2 + 6.4/x
Step 8: Find the minimum cost
To find the minimum cost, we need to find the value of "x" that minimizes the cost. We can do this by taking the derivative of the cost function and setting it equal to zero.
Differentiating the cost function with respect to "x" gives us:
dCost/dx = 0.14x - 6.4/x^2
Setting dCost/dx = 0 and solving for "x":
0.14x - 6.4/x^2 = 0
0.14x^3 - 6.4 = 0
Simplifying the expression:
0.14x^3 = 6.4
x^3 = 6.4 / 0.14
x^3 = 45.714
x ≈ ∛45.714
x ≈ 3.543
Step 9: Calculate the corresponding height
Using the equation h = 32 / x^2 and substituting the value of "x" we found in the previous step:
h = 32 / (3.543)^2
h ≈ 3.591
Step 10: Calculate the minimum cost
Finally, substitute the values of "x" and "h" we found into the cost function to calculate the minimum cost:
Cost = 0.07(3.543)^2 + 6.4/3.543
Cost ≈ 0.74 + 1.809
Cost ≈ 2.549
So, the outside dimensions that will minimize the cost of the closed box with a volume of 32 cubic meters are approximately x = 3.543 and h = 3.591, and the corresponding minimum cost is approximately $2.549.