Question

The demand function for a special limited edition coin set is given by \(p=1000\left(1-\frac{5}{5+e^{-0.001 x}}\right)\) (a) Find the demand \(x\) for a price of \(p=\$ 139.50\). (b) Find the demand \(x\) for a price of \(p=\$ 99.99\). (c) Use a graphing utility to confirm graphically the results found in parts (a) and (b).

Short answer

The demand for a price of \( p = \$ 139.50 \) is approximately \( x = 8786.90 \) and for a price of \( p = \$ 99.99 \) is approximately \( x = 14142.10 \). The results are confirmed graphically with a graphing utility.

Step-by-step solution

Set the demand equal to the given value

For part (a), substitute the given price \( p = 139.50 \) into the demand function to get the equation: \[ 139.50 = 1000\left(1-\frac{5}{5+e^{-0.001 x}}\right) \]

Isolate the exponential portion of the equation

Rearrange the equation to solve for \( e^{-0.001x} \): \[ e^{-0.001 x} = \frac{5}{1-\frac{139.50}{1000}}-5 \]

Get the value of x

Take the natural logarithm (ln) of both sides of the equation to solve for \( x \): \[ x = -\frac{ln\left(\frac{5}{1-\frac{139.50}{1000}}-5\right)}{0.001} \] Use the same steps for (b) but substitute \( p = 99.99 \) into the demand function instead.

Graph the function

Use a graphing utility (like Desmos.com) to graph the demand function and confirm the results obtained in parts (a) and (b). Plot the points (\( p, x \)) for the solved values of \( x \) from parts (a) and (b) on the graph and verify that they lie on the function curve.

Key concepts

Exponential Functions

Exponential functions are widely used in various fields like finance, biology, and economics. These functions are characterized by variables that appear as exponents. In the context of our demand function, the term \( e^{-0.001x} \) exemplifies an exponential function. Here, \( e \) is the base of the natural logarithms, approximately equal to 2.71828.
Exponential functions grow rapidly as the variable increases, or decrease sharply when the exponent is negative. This attribute makes them ideal for modeling natural processes like growth and decay. For instance, in our demand function, the exponent decreases with an increase in demand \( x \). Such a setup models scenarios where demand dwindles as the price ceilings down.
Understanding how exponential functions work can facilitate comprehension of demand shifts, allowing for better forecasting and strategic decisions in business scenarios.

Natural Logarithm

The natural logarithm, represented as \( \ln \), is a logarithm with base \( e \). It's a fundamental concept in mathematics, helping convert intricate exponential functions into an easier algebraic form. In many cases, it simplifies the solving process, as seen in Step 3 of the original solution.
When dealing with exponential decay, like \( e^{-0.001x} \), taking the natural logarithm of both sides helps "unlock" the variable from the exponent. This process transforms a complex solution into a manageable linear problem. By applying the natural logarithm to both sides of our manipulated equation, we derive a formula to solve for \( x \).
Natural logarithms provide a way to solve exponential equations swiftly, making them especially useful in demand forecasting, where understanding price-demand relationships is crucial.

Graphing Utility

A graphing utility is a tool that helps visualize mathematical functions and their behaviors. In cases like our demand function, a graphing utility can verify calculated values by plotting them visually. This is crucial in confirming the demand \( x \) for specific prices, as done in part (c) of the original exercise.
Graphing utilities can range from sophisticated calculator features to online platforms like Desmos. These utilities allow one to input a function and see how variations in variables affect the output. In the demand function, plotting allows students to see the curve shape, demonstrating the relationship between price \( p \) and demand \( x \).
By graphically plotting the results obtained, students can visually verify their solutions, which supports understanding and ensures accuracy in complex calculations.