Question

Forecasting Commodity Prices Government economists in a certain country have determined that the demand equation for soybeans is given by

p=f(x)=51/2 x²+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 2.3 billion bushels for the year, with a possible error of 10% in their forecast. Use differentials to approximate the corresponding error in the predicted price per bushel of soybeans

Answer 1

To approximate the error in the predicted price per bushel of soybeans, we can use differentials. The error in the predicted price per bushel of soybeans is approximately $11.73.

Step-by-step explanation:

To approximate the error in the predicted price per bushel of soybeans, we can use differentials. Let's first calculate the derivative of the demand equation, p=f(x)=rac{51}{2}x^2+1, with respect to x. Differentiating, we get p'(x) = 51x, which represents the rate of change of the price with respect to the quantity demanded.

Next, we'll find the error in the forecasted quantity demanded. The harvest forecasted is 2.3 billion bushels, and the possible error is 10%. Therefore, the error in the forecasted quantity is 10% of 2.3 billion bushels, which is 0.23 billion bushels.

Now, using the derivative, p'(x), we can find the corresponding error in the predicted price per bushel by multiplying the error in the forecasted quantity by the derivative. So, the error in the predicted price per bushel is 0.23 billion bushels * 51 dollars/bushel = 11.73 dollars.