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# Interpolating and Extrapolating: Making predictions | Physics Solution

“There are two types of people in the world. Those who can extrapolate from incomplete data.”

Interpolation is defined as estimating a point within a known data set. Inter- means between or within.

Extrapolation is defined as estimating a point outside a known data set. Extra- means outside or beyond.

Using these techniques to analyze data sets is an important skill in science. They allow us to make estimations based on data we have collected and relationships we have observed. Above is a graph showing money earned by a student on the y-axis versus the amount of time worked on the x-axis. Data points were taken every two hours (0, 2, 4, 6, and 8 hours).

1. Interpolate how much money the student will earn after 3 hours: ________

You may have simply moved up from 3 on the x-axis vertically until you met the line, then traced horizontally to see what the y-axis value was. A more accurate method, when dealing with linear data, is to find the equation of the line, and use it to predict values.

1. What is the slope of this line? ________

Slope is often referred to as a “rate.” Rates compare one variable to another, like pay per hour. Here, pay, or money earned is on the y-axis and time is on the x-axis. The slope of this line is also the student’s hourly pay per hour.

1. Expressed as a rate the student makes ________ \$/hr
2. Express the student’s earnings in slope intercept form: ________

Use the formula you discovered to interpolate the missing data below:

1. Student’s earnings after 7 hours: ________
2. Student’s earnings after 4.5 hours: ________
3. How long the student must work to earn \$12.50: ________
4. What will the student earn after an 8 hour shift with an unpaid 30 minute lunch break? ________
5. How might the equation for this student change if you were to calculate their earnings added to the balance of their bank account before the work day began? Assume they had a balance of \$50.00

We can use this student’s pay rate to also extrapolate what their earnings would be after working any number of hours past the data given to us.

1. How much money would this student earn during a 40 hour work week? ________ The graph above shows the rising temperature of water over time. Data was collected at 0 time, and then at 10 second intervals. If we assume that this relationship between temperature and heating time continues as a linear rate, we can estimate the temperature of the water beyond the data collected. You may simply extend the line along the graph (as has been done for you), or find the equation of the line.

1. What is the rate of temperature change over time according to the graph? ________ degrees per minute.
2. What was the “room temperature” of the water before heating began? ________
3. Write out the equation for this line in slope-intercept form: ________
4. How long would it take the water to reach 100 C? ________
5. What would the temperature of the water be after 70 minutes of heating? ________
6. The table below shows the computational data relating the side length and perimeter of a square tile. Use the data to construct a line graph. 1. How much does the perimeter increase each time the slide length is increased by 1? ________
2. Write the equation of this relationship: ________
3. The table above calculates the area of a square tile as it relates to the length of one side of that tile. Graph the data and answer the questions using the table and graph. 1. Write the equation of this line (Hint; how do you calculate area of a square?): ________

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