A company offers the following charges: $54 per thousand for orders of 62000 or less with the charge per thousand decreased by 38 cents for each thousand above 62000. Find the order which will make the company's revenue a maximum.
To find the order that will make the company's revenue maximum, we need to determine the number of thousands of items that will yield the highest revenue.
Let's analyze the problem step by step:
1. Define the variables:
- x: Number of thousands of items ordered (in thousands).
- c(x): Cost per thousand items.
2. Determine the cost function:
The cost function is defined by the given information:
- For orders of 62000 or less:
c(x) = $54 per thousand items.
- For orders above 62000:
The charge per thousand is decreased by 38 cents for each thousand above 62000.
Therefore, for each additional thousand above 62000, the cost decreases by 38 cents:
c(x) = $54 - $0.38(x - 62).
3. Calculate the revenue function:
The revenue is given by the formula:
Revenue = x * c(x).
4. Optimize the revenue function:
To find the order that maximizes the revenue, we need to find the critical points of the revenue function. Taking the derivative and setting it equal to zero will help us identify those points.
Let's calculate the derivative of the revenue function:
R(x) = x * c(x)
R'(x) = c(x) + x * c'(x)
Substitute the expression for c(x) into R'(x):
R'(x) = x * [$54 - $0.38(x - 62)] + x * c'(x)
R'(x) = $54x - $0.38x^2 + $23.56x - $24.68 + x * c'(x)
Now, set R'(x) equal to zero and solve for x:
$54x - $0.38x^2 + $23.56x - $24.68 + x * c'(x) = 0
Combine like terms:
- $0.38x^2 + ($54 + $23.56)x - $24.68 + x * c'(x) = 0
- $0.38x^2 + $77.56x - $24.68 + x * c'(x) = 0
Since we don't have an explicit formula for c'(x), which is the derivative of c(x), we'll substitute the expression for c(x) into c'(x) and simplify the equation:
- $0.38x^2 + $77.56x - $24.68 + x * (c(x) - $54) = 0
- $0.38x^2 + $77.56x - $24.68 + x * ($54 - $54 + $0.38(x - 62)) = 0
- $0.38x^2 + $77.56x - $24.68 + $0.38x * (x - 62) = 0
- $0.38x^2 + $77.56x - $24.68 + $0.38x^2 - $23.56x = 0
Combining like terms:
$54x - $24.68 = 0
Solve for x:
$54x = $24.68
x = $24.68 / $54
Calculate x ≈ 0.457
5. Analyze the critical points:
We found x ≈ 0.457 as a critical point. However, since the number of thousands cannot be a fraction, we need to analyze the critical points to determine which one maximizes the revenue.
The revenue at the critical point x ≈ 0.457 can be calculated as follows:
Revenue at critical point = x * c(x)
Revenue at critical point = 0.457 * $54 ≈ $24.62
As the revenue at the critical point is less than the revenue at x = 0, it means there is no maximum within the given range.
To verify if the maximum occurs at one of the endpoints, we can calculate the revenue at those points:
For x = 0 (0 thousand orders):
Revenue at x = 0 = 0 * c(0) = 0
For x = 62 (62000 thousand orders):
Revenue at x = 62 = 62 * c(62) = 62 * ($54 - $0.38(62 - 62)) = 62 * $54 = $3348
Therefore, the order of 62000 thousand items will maximize the company's revenue.
Hence, the company's revenue will be maximum when the order consists of 62000 thousand items.