Kate sold one half of her paintings and purchased 7 more. If Kate has 21 paintings now, how many paintings did she have to begin
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Answer:
a. y = 0.56x -17.78 . . . . y = Celsius temperature; x = Fahrenheit temperature
b. 37.436 °C
c. interpolation
d. 83.2 °F
Step-by-step explanation:
a. In most cases, finding the linear regression equation that fits a table of data involves somewhat tedious calculations best performed by a calculator or computer. The general approach is to enter the data into the necessary list(s) or table, and invoke the necessary function.
In Microsoft Excel (and many other spreadsheet programs), you can chart the data in the usual way, then choose the option to add the regression equation to the chart.
In the Desmos calculator used here, the data can be entered into a table, and a generic equation written relating the table entries. Any unknown constants in the equation are computed by Desmos so as to maximize the correlation between the equation and the data. Using a linear equation results in a linear regression computation.
This calculator tells us what we should already know (from previous knowledge of Fahrenheit–Celsius conversion).
y ≈ 0.56x -17.78
(We already know the actual conversion is y = (5/9)(x -32) = (5/9)x -(17 7/9). The above is what you get when the numbers are rounded to hundredths, as required by this problem.)
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b. Using the above equation for x=98.6, we get
y = 0.56·98.6 -17.78 = 55.216 -17.78 = 37.436 . . . °C
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c. Since we're finding a value <em>between</em> values already in the table, the process is called interpolation. (Extrapolation is used to find values beyond those that are already in the table. The prefixes <em>inter</em>- (between) and <em>extra</em>- (beyond) can help you remember what these terms mean.)
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d. In this part, we're asked to find x that corresponds to y=28.8. We fill in the value and solve.
28.8 = 0.56x -17.78
46.58 = 0.56x . . . . . . add 17.78
46.58/0.56 = x ≈ 83.179 ≈ 83.2 . . . . . divide by the coefficient of x; round appropriately. Units are °F.
