Problem 22 - Technical Mathematics with Calculus

Question

At what rate must a person walk to \(80 \quad 24.5 \quad \mathrm{km}\) in \(12.75 \mathrm{h} ?\) (rate \(=\text { distance } \div \text { time })\)

Step-by-Step Solution

Answer

The person must walk at a rate of approximately 1.92 km/h.

Step 1: Identify the formula for rate

The rate of speed is calculated by the formula rate = distance / time. The person's walking rate can be determined using this formula where distance is the total distance to be covered and time is the duration required to cover that distance.

Step 2: Plug in the given values

In this case, the distance provided is 24.5 km and the time is 12.75 hours. Substitute these values into the rate formula to calculate the person’s walking rate.

Step 3: Calculate the rate

Perform the division to find the rate: rate = 24.5 km / 12.75 h = 1.9216 km/h. Round the rate to an appropriate number of significant figures if necessary.

Key Concepts

Understanding the Rate Formula

The rate formula is a fundamental concept often used to calculate speed, which is essentially how fast something is moving. Rate is determined by dividing the distance traveled by the time it takes to travel that distance. Expressed mathematically, the formula looks like this:
\[ \text{Rate} = \frac{\text{Distance}}{\text{Time}} \].
In practical terms, if you know the total journey distance and the time taken to cover it, you can easily calculate the rate of speed. For instance, if a person walks 24.5 km in 12.75 hours, their walking rate is \[ \text{Rate} = \frac{24.5\,\text{km}}{12.75\,\text{h}} = 1.9216\,\text{km/h}\].
This equation reveals the speed at which that person walks. It is important to frame your answer within the context of the unit of measure you are working with. In this example, it’s kilometers per hour (km/h).

Converting Units for Accurate Calculations

Units conversion is crucial when working with measurements in different systems or when needing to comply with certain standards. For example, if you initially have a walking distance in miles and time in minutes, you must convert these to a consistent unit system like kilometers for distance and hours for time before using them in the rate formula. To convert miles to kilometers, you would multiply by a conversion factor, which in this case is approximately 1.60934 (since 1 mile = 1.60934 km). To convert minutes to hours, you would divide the number of minutes by 60, as there are 60 minutes in one hour.
Let's say we have another scenario where someone walked 15 miles in 150 minutes. Converting both these measurements to common units would result in: \[ 15\,\text{miles} \times 1.60934 = 24.1401\,\text{km} \] and \[ 150\,\text{minutes} \div 60 = 2.5\,\text{h} \].

Calculating with Converted Units

After the conversion, applying the rate formula \[ \text{Rate} = \frac{24.1401\,\text{km}}{2.5\,\text{h}} \] will yield a correct rate of speed, in this case, the walking rate would be in km/h. Remember, always convert your measurements first to ensure accuracy in your final calculations.

Importance of Significant Figures in Calculations

The use of significant figures in mathematical calculations reflects the precision of the measurements taken. When rounding numbers, it is important to maintain the integrity of the data's precision. In the example of the walking rate calculated as 1.9216 km/h, depending on the context, this may need to be rounded. For instance, if a typical pedometer only records up to two decimal places, the rate would be rounded to 1.92 km/h.
Significant figures are determined by the number of digits deemed reliable in a number, starting with the first non-zero digit. When computing with these figures, it is vital to follow rules for multiplication, division, addition, and subtraction, each of which has specific guidelines for handling significant figures.

• In addition and subtraction, the answer should not have more decimal places than the least precise number.
• In multiplication and division, the answer should not have more significant figures than the measured number with the fewest significant figures.

So, while the raw answer to the rate equation is 1.9216 km/h, depending on the required precision for practical purposes or instrument precision, you may round to two significant figures, giving us 1.9 km/h, or to four significant figures, which leaves us with the calculated 1.922 km/h (rounding up the last digit).