Problem 63 - Algebra 1

Question

In Exercises 62 and \(63,\) a mountain climber is scaling a 300 -foot cliff at a constant rate. The climber starts at the bottom at 12: 00 P.M. By 12: 30 P.M., the climber has moved 62 feet up the cliff. At what time will the mountain climber reach the top of the cliff?

Step-by-Step Solution

Answer

The climber will reach the top of the cliff at approximately 1:55 PM.

Step 1: Calculate the climber's speed

The climber climbed 62 feet in 30 minutes. To calculate the speed, we need to use the formula for speed, which is distance divided by time. Hence, the speed will be \(62 feet ÷ 30 min = 2.07 feet/min \).

Step 2: Determine the time needed to climb the remaining distance

The total distance to be climbed is 300 feet. Given that the climber climbed 62 feet already and the remaining distance is \(300 – 62 = 238 feet\). The time needed to climb this distance is calculated by dividing this distance by the climber's speed. So, this will result in \(238 feet ÷ 2.07 feet/min = 115 min \).

Step 3: Convert the time to hours and minutes

One hour has 60 minutes. To convert from minutes to hour, we need to divide the minutes by the number of minutes per hour. Thus the time will be \(115 minutes ÷ 60 min/hour = 1.92 hours\). Converting the fractional part into minutes results in \(0.92 hour × 60 minutes/hour = 55.2 min\). Since we can't have decimal minutes, we round this to the nearest minute (55 minutes). Thus, the total time is about 1 hour and 55 minutes.

Step 4: Determine the time when the climber will reach the top

The climber started climbing at 12:00 PM. After 1 hour and 55 minutes of climbing, the climber will reach the top. Hence, by adding the climbing time to the start time, we get the time when the climber will reach the top: \(12:00 PM + 1 hour 55 minutes = 1:55 PM\).

Key Concepts

Distance Formula

Understanding how to use the distance formula is crucial when calculating the rate of change in various problems. In the case of our mountain climber, the problem requires knowing the distance climbed in a given time. The distance formula is simply:

• Guided by the format: distance = rate × time
• We rearrange it to determine the rate: rate = distance / time

For the mountain climber, this allows us to calculate the rate at which they scaled the cliff. With 62 feet climbed in 30 minutes, the formula gives us a linear rate:\[\text{Rate} = \frac{62 \ \text{feet}}{30 \ \text{minutes}} = 2.07 \ \text{feet/minute}\]This rate becomes immensely helpful in planning or predicting how long it takes to cover other distances. Knowing this conversion empowers us to tackle not only climbing-related problems but any scenarios involving consistent rates over a distance.

Units of Measurement

Grasping units of measurement is fundamental to solving many real-world problems effectively. The units we use to describe distance, such as feet or meters, and time, like minutes or hours, play a crucial role in calculations.

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Why Units Matter

• Ensure the uniformity of units; you can’t divide feet by hours directly.
• Uniform units keep equations accurate and comparable.

In the given exercise, distances are in feet and time in minutes, showing a need to remain consistent. Without such consistency, it becomes difficult to understand the rate of progress. In practice, you might convert back and forth between units, such as feet to meters or minutes to hours, to match problem requirements. This might involve:

• Using conversion factors, like 1 minute = 60 seconds or 1 foot = 0.3048 meters.
• Adapting to the unit that offers practical or computational benefits.

Time Calculation

When embarking on any task, especially that involves rates like climbing a mountain or driving a car, understanding time calculation becomes pivotal. For our climber, we needed to first estimate how long they would take to climb 238 feet, the remaining distance.

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Breaking Down the Time Calculation

• Using the speed (2.07 feet/min), divide the remaining distance: 238 feet.
• Efficiency is key: maintaining a consistent pace throughout the climb.

By applying these steps, we computed the required time:\[\text{Time} = \frac{238 \ \text{feet}}{2.07 \ \text{feet/min}} \approx 115 \ \text{minutes}\]Converted into hours and minutes:

• 1 hour for the first 60 minutes, plus an additional 55 minutes.
• Hence, total climbing time: 1 hour and 55 minutes.

To find the end time, simply add the total climbing time to the starting time of 12:00 PM, allowing for seamless planning and execution of the climb.