Question

To park a car at a short-term parking lot costs 1.75 dollar for the 1 st hour or any part thereof, 1.50 dollar for the 2nd hour or any part thereof, and 0.75 dollar for each additional hour or any part thereof after the 2nd hour. Your ticket shows that you parked your car in this lot from 10 : 47 a.m. to 4 : 35 p.m. on the same day. What is the cost of parking your car, according to this ticket? (Note: Prices include all applicable sales tax.) A. 4.75 dollar B. 5.50 dollar C. 5.86 dollar D. 6.10 dollar E. 6.25 dollar

Short answer

Answer: The cost should be 7.75 dollars, but this option is not available in the given choices. There might be an error in the given options, or the question might be incomplete or incorrect.

Step-by-step solution

Determine the parking duration

First, let's calculate how long the car was parked at the lot. To do this, convert the given time (10:47 a.m. to 4:35 p.m.) into total hours and minutes. Since it is an hourly rate system, we can convert the time into total minutes and then divide by 60 to obtain the total hours. Total minutes parked = (4 x 60 + 35) - (10 x 60 + 47) Total minutes parked = 235 - 647 Total minutes parked = 468 minutes Now, let's convert the time into hours (rounded up because any part of an hour counts as full): Total hours parked = ceil (468 / 60) Total hours parked = ceil (7.8) Total hours parked = 8 hours Thus, the car was parked for 8 hours.

Apply the given rates to each hour

Next, let's apply the given rates for each hour spent at the lot: 1st hour: 1.75 dollar 2nd hour: 1.50 dollar 3rd hour to the 8th hour (6 hours): 0.75 dollar * 6 = 4.50 dollar

Calculate the total cost and compare against the options

Calculate the total cost of parking the car: Total cost = 1.75 + 1.50 + 4.50 Total cost = 7.75 dollar Now, let's compare this cost against the given options: A. 4.75 dollar B. 5.50 dollar C. 5.86 dollar D. 6.10 dollar E. 6.25 dollar None of these options match the calculated cost of 7.75 dollars. It seems there is an error in the given options, or the question might be incomplete or incorrect. The calculated cost should be 7.75 dollars.

Key concepts

Word Problems

Word problems in mathematics are the real-life applications of mathematical concepts which require us to translate written language into mathematical expressions. Take, for example, the parking lot problem; it requires understanding daily activities and applying mathematics to solve them. To achieve proficiency, students need to understand the English language's intricacies, including synonyms for operations and the context clues that indicate them. One crucial exercise improvement advice is to practice identifying key pieces of information within the problem and disregarding irrelevant details.

By practicing word problems, students learn to:

• Identify what is being asked.
• Determine which numbers and operations are necessary for the problem.
• Create an equation that accurately represents the problem.

This approach helps to cultivate both critical reading skills and mathematical reasoning.

Mathematical Reasoning

Mathematical reasoning involves logical thinking and the ability to comprehend and apply mathematical concepts to arrive at a solution. It is the backbone of solving any math problem but is particularly crucial in word problems where interpretation is key. In the context of the parking lot problem, mathematical reasoning is used to understand time, rate, and how these relate to cost. The calculation of total time parked involves converting minutes to hours and applying the ceiling function, which illustrates the practical use of mathematical reasoning.

Students can practice and improve mathematical reasoning by:

• Explaining the rationale behind each step in their calculations.
• Discussing their mathematical thought process with peers or teachers.
• Solving diverse problems to encounter a range of scenarios.

Each of these exercises supports the development of a strong foundation in mathematical logic.

Algebraic Thinking

Algebraic thinking is more than just manipulating variables and numbers; it's about understanding and using symbols to represent mathematical situations and relationships. In our parking lot scenario, algebraic thinking involves setting up an equation that reflects the various parking rates over time and then solving for the total cost.

There's a misconception that algebra is only about 'finding x', but it's also about recognizing patterns, making generalizations, and solving problems strategically. For better algebraic thinking, students should focus on:

• Understanding the properties of operations and how they govern algebraic manipulation.
• Looking for patterns to simplify problems before starting calculations.
• Working backwards from known solutions to understand the problem structure.

By reinforcing these algebraic habits, students will become adept at forming and solving equations that model real-life situations.