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Chapter 1: Problem 15

Renting motel rooms: You own a motel with 30 rooms and have a pricing structure that encourages rentals of rooms in groups. One room rents for \(\$ 85.00\), two for \(\$ 83.00\) each, and in general the group rate per room is found by taking \(\$ 2\) off the base of \(\$ 85\) for each extra room rented. a. How much money do you charge per room if a group rents 3 rooms? What is the total amount of money you take in? b. Use a formula to give the rate you charge for each room if you rent \(n\) rooms to an organization. c. Find a formula for a function \(R=R(n)\) that gives the total revenue from renting \(n\) rooms to a convention host. d. Use functional notation to show the total revenue from renting a block of 9 rooms to a group. Calculate the value.

Short Answer

Expert verified
The charge per room for 3 rooms is $81, with a total revenue of $243. The formula for rate per room is \(85 - 2(n - 1)\), and for total revenue, \(R(n) = n(87 - 2n)\). For 9 rooms, total revenue is $621.

Step by step solution

01

Calculate the Rate per Room for 3 Rooms

To find the rate per room for 3 rooms, start with the base rate of $85. For each additional room rented beyond the first, subtract $2 from the rate per room. Since we are renting 3 rooms: the rate is 85 - (3 - 1) * 2 = 85 - 4 = $81 per room.
02

Calculate Total Revenue for 3 Rooms

To find the total revenue when renting 3 rooms, multiply the rate per room by the number of rooms. Use the rate found in Step 1: Total revenue = 3 * 81 = $243.
03

Develop the Formula for Rate per Room for n Rooms

For renting \(n\) rooms, subtract \(2(n-1)\) from 85 (the base rate) to find the rate per room. The formula is: \(Rate(n) = 85 - 2(n - 1)\).
04

Develop the Formula for Total Revenue R(n)

Multiply the number of rooms \(n\) by the rate per room found in Step 3. The total revenue formula is: \(R(n) = n \times (85 - 2(n - 1))\). Simplifying, \(R(n) = n(85 - 2n + 2) = n(87 - 2n)\).
05

Calculate Total Revenue for 9 Rooms

Using the total revenue formula from Step 4 with \(n = 9\), substitute in the formula: \(R(9) = 9(87 - 2 imes 9) = 9(87 - 18) = 9 \times 69 = $621\).

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Key Concepts

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

Pricing Models
Understanding pricing models is crucial for maximizing profit and attracting more customers. In the context of renting motel rooms, pricing models often use a tiered structure to encourage customers to rent more rooms. The base rate, in this case, is \(85 per room. However, to incentivize larger bookings, the motel offers a discount.
  • For two rooms, each room is priced at \)83.
  • The discount structure works by reducing $2 from the base price per additional room.
This means that if someone rents 3 rooms, each room would cost \[ 85 - (3 - 1) imes 2 = 81 \] dollars. This encourages groups to book more rooms to take advantage of lower per-room rates, effectively creating a win-win situation for both customers and the motel.
Revenue Calculations
Calculating revenue is an essential component of any business strategy. Revenue is the total income generated from sales or services. In the case of the motel, the formula for total revenue when renting rooms is based on the rate per room and the number of rooms rented. To find the total revenue from renting 3 rooms:
  • Determine the rate per room (\(81, as calculated).
  • Multiply this rate by the number of rooms rented.
The total revenue is:\[ 3 \times 81 = 243 \] This basic understanding can be applied to any number, such as:
  • Using a formula, we can express total revenue for renting \( n \) rooms as \( R(n) = n \times (85 - 2(n - 1)) \).
  • For example, if 9 rooms are rented, replacing \( n \) with 9 in the formula gives the total revenue as \)621.
Formulas and Equations
Formulas and equations are the backbone of algebraic functions in pricing models. They help predict revenue outcomes based on different variables. The formula for renting rooms involves calculating both the rate per room and the total revenue. The formula for the rate per room can be expressed algebraically:\[ Rate(n) = 85 - 2(n - 1) \]where \( n \) represents the number of rooms rented. To find total revenue, you engage this rate:\[ R(n) = n \times (85 - 2(n - 1)) \] By simplifying, this becomes:\[ R(n) = n(87 - 2n) \]This approach demonstrates how equations not only give precise values for pricing and revenue but also provide insights into how changes in the number of rooms rented affect overall income. This strategic pricing through mathematical equations ensures both competitiveness and profitability.

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Most popular questions from this chapter

What if interest is compounded more often than monthly? Some lending institutions compound interest daily or even continuously. (The term continuous compounding is used when interest is being added as often as possible - that is, at each instant in time.) The point of this exercise is to show that, for most consumer loans, the answer you get with monthly compounding is very close to the right answer, even if the lending institution compounds more often. In part 1 of Example 1.2, we showed that if you borrow \(\$ 7800\) from an institution that compounds monthly at a monthly interest rate of \(0.67 \%\) (for an APR of \(8.04 \%\) ), then in order to pay off the note in 48 months, you have to make a monthly payment of \(\$ 190.57\). a. Would you expect your monthly payment to be higher or lower if interest were compounded daily rather than monthly? Explain why. b. Which would you expect to result in a larger monthly payment, daily compounding or continuous compounding? Explain your reasoning. c. When interest is compounded continuously, you can calculate your monthly payment \(M=\) \(M(P, r, t)\), in dollars, for a loan of \(P\) dollars to be paid off over \(t\) months using $$ M=\frac{P\left(e^{r}-1\right)}{1-e^{-r t}}, $$ where \(r=\frac{A P R}{12}\) if the APR is written in decimal form. Use this formula to calculate the monthly payment on a loan of \(\$ 7800\) to be paid off over 48 months with an APR of \(8.04 \%\). How does this answer compare with the result in Example 1.2?

Tubeworm: An article in Nature reports on a study of the growth rate and life span of a marine tubeworm. \({ }^{12}\) These tubeworms live near hydrocarbon seeps on the ocean floor and grow very slowly indeed. Collecting data for creatures at a depth of 550 meters is extremely difficult. But for tubeworms living on the Louisiana continental slope, scientists developed a model for the time \(T\) (measured in years) required for a tubeworm to reach a length of \(L\) meters. From this model the scientists concluded that this tubeworm is the longest-lived noncolonial marine invertebrate known. The model is $$ T=14 e^{1.4 L}-20 \text {. } $$ A tubeworm can grow to a length of 2 meters. How old is such a creature? (Round your answer to the nearest year.)

Production rate: The total number \(t\) of items that a manufacturing company can produce is directly proportional to the number \(n\) of employees. a. Choose a letter to denote the constant of proportionality, and write an equation that shows the proportionality relation. b. What in practical terms does the constant of proportionality represent in this case?

A population of deer: When a breeding group of animals is introduced into a restricted area such as a wildlife reserve, the population can be expected to grow rapidly at first but to level out when the population grows to near the maximum that the environment can support. Such growth is known as logistic population growth, and ecologists sometimes use a formula to describe it. The number \(N\) of deer present at time \(t\) (measured in years since the herd was introduced) on a certain wildlife reserve has been determined by ecologists to be given by the function $$ N=\frac{12.36}{0.03+0.55^{t}} $$ a. How many deer were initially on the reserve? b. Calculate \(N(10)\) and explain the meaning of the number you have calculated. c. Express the number of deer present after 15 years using functional notation, and then calculate it. d. How much increase in the deer population do you expect from the 10 th to the 15 th year?

Home equity: When you purchase a home by securing a mortgage, the total paid toward the principal is your equity in the home. The accompanying table shows the equity \(E\), in dollars, accrued after \(t\) years of payments on a mortgage of \(\$ 170,000\) at an APR of \(6 \%\) and a term of 30 years. $$ \begin{array}{|c|c|} \hline \begin{array}{c} t=\text { Years } \\ \text { of payments } \end{array} & \begin{array}{c} E=\text { Equity } \\ \text { in dollars } \end{array} \\ \hline 0 & 0 \\ \hline 5 & 11,808 \\ \hline 10 & 27,734 \\ \hline 15 & 49,217 \\ \hline 20 & 78,194 \\ \hline 25 & 117,279 \\ \hline 30 & 170,000 \\ \hline \end{array} $$ a. Explain the meaning of \(E(10)\) and give its value. b. Make a new table showing the average yearly rate of change in equity over each 5 -year period. c. Judging on the basis of your answer to part \(b\), does your equity accrue more rapidly early or late in the life of a mortgage? d. Use the average rate of change to estimate the equity accrued after 17 years. e. Would it make sense to use the average rate of change to estimate any function values beyond the limits of this table?
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