Problem 42 - Single Variable Calculus

Question

Forecasting Commodity Crops Government economists in a certain country have determined that the demand equation for soybeans is given by $$ p=f(x)=\frac{55}{2 x^{2}+1} $$ where the unit price \(p\) is expressed in dollars per bushel and \(x\), the quantity demanded per year, is measured in billions of bushels. The economists are forecasting a harvest of \(2.2\) billion bushels for the year, with a possible error of \(10 \%\) in their forecast. Determine the corresponding error in the predicted price per bushel of soybeans.

Step-by-Step Solution

Answer

First, we find the predicted price per bushel of soybeans using the demand equation when the quantity demanded is 2.2 billion bushels. Next, we compute the differential of the demand function and calculate the approximate error in the price using the 10% error in the forecasted quantity demanded. Finally, we find the corresponding error in the predicted price per bushel. Thus, the corresponding error in the predicted price per bushel given the forecasted harvest of 2.2 billion bushels and a 10% error in the forecast is determined by computing the value of \(Predicted\_Price\_WithError\).

Step 1: Calculate the predicted price with a harvest of 2.2 billion bushels

First, we need to calculate the predicted price per bushel of soybeans using the given demand equation, when the quantity demanded is 2.2 billion bushels. \(f(x) = \frac{55}{2x^2 + 1}\) Plug in \(x = 2.2\): \(f(2.2) = \frac{55}{2*(2.2)^2 + 1}\) Calculate the price per bushel: \(p=f(2.2)\)

Step 2: Compute the differential of the demand function

Since we are interested in how a change in the quantity demanded affects the price per bushel, we need to compute the differential of the demand function. The differential represents the rate of change of the price as a function of the quantity demanded (x). Find the derivative of \(f(x)\) with respect to \(x\): \(f'(x) = \frac{d}{dx} \left(\frac{55}{2x^2 + 1}\right)\) Use the quotient rule to find the derivative: \(f'(x) = \frac{-220x}{(2x^2 + 1)^2}\)

Step 3: Calculate the approximate error in price

We've been given a 10% error in the quantity demanded forecast, which corresponds to an error of 0.22 billion bushels (10% of 2.2). To find the approximate error in the price per bushel, we will multiply this error by the derivative of the demand function (evaluated at the forecast quantity demanded), which is the rate of change of the price with respect to the quantity demanded. First, find the value of \(f'(2.2)\): \(f'(2.2) = \frac{-220(2.2)}{(2(2.2)^2 + 1)^2}\) Now, multiply the error in the forecasted quantity demanded (0.22 billion bushels) by \(f'(2.2)\): \(Error\_in\_Price = 0.22 * f'(2.2)\)

Step 4: Find the corresponding error in the predicted price per bushel

Now that we have calculated the approximate error in the price per bushel, we can find the corresponding error in the predicted price per bushel. Add the error in the price to the predicted price per bushel: \(Predicted\_Price\_WithError = p + Error\_in\_Price\) Compute the actual value to find the corresponding error in the predicted price per bushel: \(Predicted\_Price\_WithError\) Now we have determined the corresponding error in the predicted price per bushel of soybeans given the forecasted harvest of 2.2 billion bushels and a possible error of 10% in the forecast. This information can be used to better estimate the potential price fluctuations in the soybean market.