Problem 27 - 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 $452.52.

Step 1: Understand the Question

We need to calculate the total production cost of 270 pints using the marginal cost function and employing the left-endpoint Riemann sum method with 3 subintervals. The marginal cost function is given as \(C'(x) = 0.000008 x^2 - 0.004 x + 2\).

Step 2: Divide the Interval

We are asked to break up the total interval \([0, 270]\) into 3 subintervals. Each subinterval will then have a width of \(\frac{270}{3} = 90\) pints.

Step 3: Identify Left Endpoints

For the left endpoint Riemann sum method, the left endpoints at each subinterval boundary are \(x = 0, 90, 180\).

Step 4: Calculate Marginal Cost at Each Endpoint

Plug the left endpoints into the marginal cost function to get \(C'(0) = 2\), \(C'(90) = 0.000008(90)^2 - 0.004(90) + 2 = 1.676\), and \(C'(180) = 0.000008(180)^2 - 0.004(180) + 2 = 1.352\).

Step 5: Compute Riemann Sum

Using the left-endpoint rule, calculate the sum: \[ \text{Total Cost} \approx 90(C'(0) + C'(90) + C'(180)) = 90(2 + 1.676 + 1.352) = 90 \times 5.028 = 452.52 \] dollars.

Step 6: Final Result

The approximate total cost of producing 270 pints of juice is \(452.52\) dollars.

Key Concepts

Riemann Sum

Riemann sums are a tool used in calculus to approximate the area under a curve, which can be interpreted as finding the integral of a function over an interval. In the context of Shelly's Roadside Fruit exercise, the Riemann sum helps in estimating the total cost from the marginal cost function provided. The idea is simple:

• We divide the total range into smaller equal sections known as subintervals.
• Each of these subintervals is assessed for its contribution to the total area (or cost in this scenario).
• These contributions are summed up to estimate the total area (or total cost).

In practice, depending on the chosen method (like left endpoint, right endpoint, or midpoint), a specific point in each subinterval is used to evaluate the function. In our exercise, we're using the left endpoint method. This implies that the function's value at the starting point of each subinterval is used to make calculations.

Calculus

Calculus is a branch of mathematics that studies continuous change. It is primarily divided into two branches: differential calculus and integral calculus, both of which are intertwined in the study of functions and curves. For Shelly's problem:

• Differential Calculus: Helps in understanding marginal cost, which is the derivative of the total cost function. This tells us the rate at which cost changes with every additional pint of juice produced.
• Integral Calculus: Using techniques like the Riemann sum, integral calculus helps approximate the total cost from the marginal cost. This is akin to finding the area under the marginal cost curve over a specified range.

By accumulating the tiny changes in the cost (marginal cost), we can effectively estimate the full cost over a certain quantity of production.

Left Endpoint Method

The left endpoint method is a specific kind of Riemann sum used to approximate the integral of a function. It utilizes the starting point of each subinterval to estimate the area under the curve.Here's how it applies to our exercise:

• The total range \[0, 270\] is divided into three equal subintervals: \[0, 90\], \[90, 180\], and \[180, 270\].
• We use the value of the function at the left end of each subinterval to calculate the height of the rectangle over each subinterval. This means examining the marginal cost at 0, 90, and 180.
• We find the contribution of each subinterval to the total estimate by multiplying its width by its height.
• The sum of these products gives an approximation of the total cost for 270 pints of juice.

This technique is straightforward and often used for its simplicity, though it may not always yield the most precise results depending on the function's shape and the number of subintervals used.