Algebra II : Variable Relationships
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Example Questions
Example Question #4 : Interpolations
Given 
Use the formula for interpolation:
We will use (30, 15) as x2 and y2, (15, 10) as x1 and y1, and solve for y when x=17.9:
Example Question #1 : Interpolations
Mary measures her height every year on her birthday, starting at 11 until she turns 16. She wants to make a table with all the information gathered, but discovers she lost the piece of paper on which she wrote her height down on her 14th birthday. Her incomplete table looks like this:
| Age (years) | Height (inches) |
| 11 | 47.5 |
| 12 | 50.25 |
| 13 | 53 |
| 14 | ? |
| 15 | 58.5 |
| 16 | 61.25 |
Using the method of linear interpolation, which of the following is the closest estimate of Mary's height on her 14th birthday?
Not enough information given in the problem.
Using linear interpolation means that we draw a line between the points on our data set and use that line to estimate a value that lies between two data points; in this case, we have the data from Mary's 13th and 15th birthdays, so we can describe a line between those two points and estimate her height at 14. Our line will be written in slope-intercept form:
Where the variable 
Next, we find the y-intercept by plugging in our slope (which we just found) and a point from our table (we'll stick with 
Simplify:
Subtract 35.75 from both sides to solve for 
The equation of our interpolation line is:
So, to get an estimate of Mary's height on her 14th birthday, we plug in 
Our estimate of Mary's height at 
Example Question #6 : Interpolations
Find the value of 
Write the interpolation formula.
Identify and substitute the values.
Simplify the fraction.
The answer is:
Example Question #11 : Variable Relationships
What is the next number in this sequence: 8, 27, 64, 125 ?
Find the pattern of the sequence:
This pattern is 
Example Question #12 : Variable Relationships
Find the next 2 numbers in this sequence: 33, 46, 72, 111
Find the pattern in this sequence of numbers:
In this case, the pattern is adding 13n to the previous number where n= how many numbers came before the current number.
so the first number we are looking for would be:
the second number we are looking for would be:
Example Question #1 : Extrapolations
The amount of water inside of a leaky boat is measured periodically after the boat has been in the water in different periods of time and are found to have a linear relationship. The results are given in the following chart:
| Time in water (mins) | Amount of water in the boat (gal) |
| 0 | 0 |
| 6 | 4.8 |
| 19 | 15.2 |
| 28 | 22.4 |
Using the method of linear extrapolation based on the data from the table, how much water would you expect to be in the boat after 
To extrapolate the results of the study out to 53 minutes, first we have to determine an equation representing the relationship between time passed and amount of water; we can write our equation in slope-intercept form:
Where our y-axis represents amount of water and the x-axis represents time. We can pick 2 points and label them Point 1 and Point 2; looking back at the table:
| Time in water (mins) | Amount of water in the boat (gal) |
| 0 | 0 |
| 6 | 4.8 |
| 19 | 15.2 |
| 28 | 22.4 |
We label Point 1 as 
So, the slope of our line that describes how much water is in the boat is 
Simplify the expression and we find that b=0, so our slope-intercept equation is:
Plugging in a value of 53 for 
So the answer is 42.4 gallons of water in the boat after 53 minutes.
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