Problem 28 - Calculus and its applications

Question

Total cost from marginal cost. Shelly's Roadside Fruit has found that the marginal cost of producing \(x\) pints of fresh-squeezed orange juice is given by $$C^{\prime}(x)=0.000008 x^{2}-0.004 x+2, \quad \text { for } x \leq 350$$ where \(C^{\prime}(x)\) is in dollars. Approximate the total cost of producing 270 pt of juice, using 3 subintervals over [0,270] and the left endpoint of each subinterval.

Step-by-Step Solution

Answer

The total cost is approximately \( \$471.96 \).

Step 1: Understand the Problem

We need to calculate the total cost of producing 270 pints of juice using the given marginal cost function, \(C'(x) = 0.000008x^2 - 0.004x + 2\). We'll use the method of Riemann sums with 3 subintervals over the interval \([0, 270]\) and the left endpoint for each subinterval.

Step 2: Determine Subintervals

The interval \([0, 270]\) needs to be divided into 3 equal parts. This means each subinterval has a width of \( \frac{270 - 0}{3} = 90 \) units. Thus, our subintervals are \([0, 90]\), \([90, 180]\), and \([180, 270]\).

Step 3: Evaluate at Left Endpoints

We need to evaluate the marginal cost function \(C'(x)\) at the left endpoint of each subinterval: \(x = 0, 90, 180\).- For \(x = 0\): \[C'(0) = 0.000008(0)^2 - 0.004(0) + 2 = 2\]- For \(x = 90\): \[C'(90) = 0.000008(90)^2 - 0.004(90) + 2 = 0.0648 - 0.36 + 2 = 1.7048\]- For \(x = 180\): \[C'(180) = 0.000008(180)^2 - 0.004(180) + 2 = 0.2592 - 0.72 + 2 = 1.5392\]

Step 4: Calculate Riemann Sum

Using the left endpoints and the width of 90 for each interval, compute the Riemann sum:\[\text{Total Cost} \approx 90 \times (C'(0) + C'(90) + C'(180)) = 90 \times (2 + 1.7048 + 1.5392)\]Calculate the sum inside the parentheses:\[2 + 1.7048 + 1.5392 = 5.244\]Thus, the total cost is:\[90 \times 5.244 = 471.96\]

Step 5: Conclusion

The approximate total cost of producing 270 pints of orange juice is \( \$471.96 \).

Key Concepts

Riemann sums

Riemann sums are a fundamental concept in calculus used to approximate the total value, such as area, volume, or integral of a function, within a specific interval. They are particularly helpful when finding the approximate total cost from a marginal cost function.

When applying Riemann sums, we divide an interval into smaller subintervals, calculate values at specific points within these subintervals, and then sum these values. Here, we're using the left endpoint of each subinterval to evaluate our function. If we are considering the interval \[0, 270\], and dividing it into 3 equal parts, each subinterval is \[90\] units wide.

In this example, Riemann sums help us to approximate the total cost of producing pint quantities over the interval \[0, 270\]. By calculating the value of our marginal cost function at the left endpoints of these subintervals \([0, 90, 180]\), and summing these values multiplied by the width of the subintervals, we get an approximation of the total cost.

Marginal cost

Marginal cost is an economic term that defines the cost of producing one more unit of a good or service. Calculus helps us understand and express this through functions, like \(C'(x)\), that describe how cost changes with the level of production.

For this exercise, the marginal cost is given by \((0.000008x^2 - 0.004x + 2)\). It indicates how the cost changes for each additional pint of orange juice produced. This function considers various factors like materials and overhead costs that influence total production expenses. By evaluating this function at different production levels, Shelly's Roadside Fruit can determine how cost-effective their production is at various stages.

The key to understanding marginal cost through this function is recognizing that it provides an incremental view of production expenses, rather than total cost. This allows for more informed decision-making about the production scale and pricing.

Total cost calculation

Total cost calculation through calculus involves using functions representing marginal costs integrated over a production level. Here, we're approximating it using Riemann sums applied to our given marginal cost function over the specified intervals.

To find the total cost of producing 270 pints of juice, we use the evaluated marginal costs \(C'(0), C'(90), \text{ and } C'(180)\), combined with the respective interval width \(90\). The calculation is \[90 imes (C'(0) + C'(90) + C'(180)) = 471.96\], showing us the approximate total expenditure.

This approximation helps businesses like Shelly's Roadside Fruit make informed financial decisions by understanding their potential production costs. Remember, though, this method assumes a constant marginal cost within each subinterval, providing an estimation that's usually close but not exact.