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Chapter 6: Problem 6

APPLICATION Toward the end of the year, to make room for next year's models, a car dealer may decide to drop prices on this year's models. Imagine a car that has a sticker price of \(\$ 20,000\). The dealer lowers the price by \(4 \%\) each week until the car sells. a. Write a recursive routine to generate the sequence of decreasing prices. (a) b. Find the 5 th term and explain what your answer means in this situation. (A) c. If the dealer paid \(S 10,000\) for the car, how many weeks would pass before the car's sale price would produce no profit for the dealer?

Short Answer

Expert verified
By week 5, the car costs about \$16,987.93. It takes 19 weeks to make no profit.

Step by step solution

01

Define Initial Terms for Part (a)

A recursive formula requires an initial term. In this case, the initial price of the car is \$20,000. Let this be the first term of the sequence, denoted as \(P_1 = 20000\).
02

Establish Recursive Formula

A recursive routine uses the previous term to calculate the next one. Since the car price decreases by 4% each week, the recursive formula is \(P_{n} = P_{n-1} \times (1 - 0.04) = P_{n-1} \times 0.96\). This formula generates the sequence of decreasing prices.
03

Calculate the 5th Term for Part (b)

Starting with \(P_1 = 20000\), calculate subsequent terms until \(P_5\): - \(P_2 = 20000 \times 0.96 = 19200\)- \(P_3 = 19200 \times 0.96 = 18432\)- \(P_4 = 18432 \times 0.96 = 17694.72\)- \(P_5 = 17694.72 \times 0.96 = 16987.9296\)Round the 5th term: \(P_5 \approx 16987.93\). This means by the 5th week, the car's price would be approximately \$16,987.93.
04

Determine Weeks to No Profit in Part (c)

Calculate when the sale price equals the dealer's cost of \$10,000.Start with the formula: \(P_n = 20000 \times 0.96^{(n-1)}\).Set \(P_n = 10000\) and solve for \(n\):\[10000 = 20000 \times 0.96^{(n-1)}\]Divide both sides by 20000:\[0.5 = 0.96^{(n-1)}\]Take the logarithm of both sides:\[\log(0.5) = (n-1) \times \log(0.96)\]Solve for \(n-1\):\[n-1 = \frac{\log(0.5)}{\log(0.96)}\]Calculate: \(n-1 \approx 17.68\), so \(n \approx 18.68\).Since \(n\) must be whole weeks, it will take 19 weeks for the price to reach a point where the car produces no profit for the dealer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decreasing Prices
Imagine a scenario where prices keep going down over time. This is what happens with decreasing prices. For the car dealer, decreasing prices are a way to entice customers to buy a car that may soon become outdated. The initial price of the car is set at $20,000. Each week, this price is reduced by 4%. Why? To steadily make the car more attractive to potential buyers as the new year's models are ready to roll out. Let's break it down:
  • The first drop reduces the price to 96% of its original.
  • After another week, the price drops again. Now it is 96% of the new price.
  • This keeps going until the car is sold or is no longer profitable at a lower price.
Lowering prices weekly helps clear out inventory and prepare for newer models.
Recursive Formula
A recursive formula is a powerful tool. It lets us find terms in a sequence based on the previous one. For our car example, the recursive formula tells us how the price decreases each week. We start with the original price, which is $20,000.Here's how it works:
  • The initial term, or starting point, is given as \( P_1 = 20000 \).
  • The recursive relationship is defined as: \( P_n = P_{n-1} \times 0.96 \). This formula means each new price \( P_n \) is 96% of the previous price \( P_{n-1} \).
  • By applying this formula successively, we can find the price for any week in the future.
A recursive relationship elegantly models how the car鈥檚 price falls over weeks. It simplifies tracking changes without having to calculate each from scratch.
Mathematical Modeling
Mathematical modeling helps us make sense of real-world scenarios using math. By crafting models, we can simulate how events might unfold. In this exercise, we model the car dealer's price reductions over time to see when it hits a break-even point.The process:
  • We create a sequence with decreasing prices using a recursive formula.
  • The model allows us to calculate specifics like the fifth week's price or when the car will be unprofitable. For example, we found \( P_5 \approx 16987.93 \).
  • To find when selling becomes unprofitable, we solve the equation \( 10000 = 20000 \times 0.96^{(n-1)} \). This tells us the price falls to the dealer's cost after 18.68 weeks, rounding to 19 weeks.
Mathematical models like this help in planning strategies and making informed business decisions.

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