Question

The demand function for a hot tub spa is given by \(p=105,000\left(1-\frac{3}{3+e^{-0.002 x}}\right)\) (a) Find the demand \(x\) for a price of \(p=\$ 25,000\). (b) Find the demand \(x\) for a price of \(p=\$ 21,000\). (c) Use a graphing utility to confirm graphically the results found in parts (a) and (b).

Short answer

The demand for a price of \$25,000 is approximately \(x \approx 596\), and the demand for a price of \$21,000 is approximately \(x \approx 804\).

Step-by-step solution

Substitute the given price into the equation for part (a)

For a price of \(p = \$25000\), substitute this into the demand function and solve for \(x\). You get: \(25000 = 105000 \left(1 - \frac{3}{3 + e^{-0.002 x}}\right)\). Now rearrange the equation to solve for \(x\).

Solve the equation for \(x\)

First, divide both sides by 105000, then add the fraction \(\frac{3}{3 + e^{-0.002 x}}\) to both sides and subtract 1. You will get: \(\frac{3}{3 + e^{-0.002 x}} = 1 - \frac{25000}{105000}\). Simplify the right side of the equation and multiply by \(3 + e^{-0.002 x}\) to isolate the exponential part. You will get \(e^{-0.002 x} = \frac{3}{1 - \frac{25000}{105000}} - 3\). Evaluate the right side of the equation, then take the natural logarithm on both sides to isolate \(x\). Finally, divide by -0.002 on both sides to solve for \(x\).

Substitute the given price into the equation for part (b)

Now, for a price of \(p = \$21000\), substitute this into the demand function and follow the same steps as in part (a) to find the value of \(x\).

Confirm graphically the results found in parts (a) and (b)

We can plot the graph of the function \(p=105000 \left(1 - \frac{3}{3 + e^{-0.002x}}\right)\) using a graphing utility, and find the \(x\)-values when \(p = 25000\) and \(p = 21000\). This will verify whether our calculations in parts (a) and (b) are correct. One should plot the function and the horizontal lines \(y = 25000\) and \(y = 21000\) and check the intersections.

Key concepts

Solving Exponential Equations

To understand how to solve exponential equations, such as those appearing in demand functions, it's essential to grasp the properties of exponentials. Exponential equations typically involve a variable in the exponent, which can complicate solving them directly.

When faced with an equation like \(e^{-0.002x} = C\), where \(C\) is a constant, we can apply logarithms to both sides. Using the natural logarithm \(\ln\), which is the inverse of the exponential function with base \(e\), helps us to bring down the exponent as a coefficient. This step transforms the equation into a linear form, \(-0.002x = \ln(C)\), which we can easily solve for \(x\).

In the given demand function problem, by employing this tactic—dividing, isolating the exponential part, and then taking the natural logarithm—we can solve for the quantity demanded, \(x\), at different prices.

Graphical Solution Verification

Graphical solution verification provides a visual representation to confirm the solutions of equations, which can be particularly helpful with complex functions. With demand functions like the one provided, plotting the equation on a graph can show the relationship between price and demand visually.

By using a graphing utility, one can input the function and observe the curve that represents the relationship. Then, by drawing horizontal lines at the prices of interest, the points where these lines intersect the curve correspond to the demanded quantities at those prices. This method can serve as a valuable check against analytical solutions, ensuring that the results are consistent with the graphed relationship.

Price-Demand Relationship

The price-demand relationship, a fundamental concept of microeconomics, describes how the quantity demanded of a good or service varies with its price. Generally, as the price increases, the demand decreases, and vice versa. This relationship can be represented mathematically by a demand function.

In the provided example, the demand function is an equation that connects the price \(p\) of a hot tub spa to the quantity demanded \(x\). It is imperative to understand that the demand function embodies the behavior of consumers, and solving it for different prices gives us insights into how changing the price affects the number of units consumers are willing to buy. This knowledge can guide businesses in pricing strategies to maximize revenues.

Logarithmic Functions

Logarithmic functions are another vital mathematical concept, especially when working with exponential relationships. They are the inverses of exponential functions and are denoted by \(\log_b(x)\), where \(b\) is the base of the logarithm and \(x\) is the argument. The natural logarithm, \(\ln(x)\), has a base of \(e\), Euler's number.

Understanding logarithms can simplify solving exponential equations as they allow us to re-express the equations in a form where the variable is no longer an exponent. As seen in the exercise, taking the natural logarithm of both sides of an exponential equation is a powerful tool to isolate the variable and find the solution.