Use linear interpolation to estimate the desired quantity. A...

Use linear interpolation to estimate the desired quantity. A vending company estimates that $f(x)$ cans of soft drink can be sold in a day if the temperature is $x^{\circ} \mathrm{F}$ as given in the table. $$\begin{array}{|l|c|r|r|}\hline x & 60 & 80 & 100 \\\hline f(x) & 84 & 120 & 168 \\\hline\end{array}$$ Estimate the number of cans that can be sold at (a) $72^{\circ}$ and (b) $94^{\circ}$

Use linear interpolation to estimate the desired quantity.
A vending company estimates that $f(x)$ cans of soft drink can be sold in a day if the temperature is $x^{\circ} \mathrm{F}$ as given in the table. $$\begin{array}{|l|c|r|r|}\hline x & 60 & 80 & 100 \\\hline f(x) & 84 & 120 & 168 \\\hline\end{array}$$ Estimate the number of cans that can be sold at (a) $72^{\circ}$ and
(b) $94^{\circ}$

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Key Concepts

Interpolation Formula

The interpolation formula, f(x) = f(x?) + ((f(x?) - f(x?))/(x? - x?))*(x - x?), is used to estimate the value of a function at a point x that lies between two known points. This formula incorporates the slope of the line connecting the known points and adjusts the known function value accordingly.

Slope of a Line

The slope of a line represents the rate at which the dependent variable changes with respect to the independent variable. In interpolation, the slope between two given data points is used to determine how much the dependent value changes per unit change in the independent value.

Linear Interpolation

Linear interpolation is a method used to estimate an unknown value by assuming that the change between two known values is linear. It essentially connects two known data points with a straight line and uses this line to approximate the value at a point between them.

Transcript

00:01 In this question, we're going to use linear interpolation to estimate the desired quantity.

00:05 A vending company estimates that f of x cans of soft drink can be sold in a day if the temperature is x degrees fahrenheit as given in the table.

00:14 We're going to estimate the number of cans that can be sold first at 72 degrees and then at 94 degrees.

00:22 So first, let's consider the 72 degrees.

00:26 That's between x equals 60 and x equals 80.

00:29 So i'm going to look at my slope on that interval from 60 to 80.

00:34 I'm going to say that my slope is equal to y2 minus y1, so 120 minus 84 over x2 minus x1.

00:43 That's 80 minus 60.

00:45 Now, 120 minus 84, that's equal to 36.

00:51 So i'm going to have 36, and that's divided by 20, which simplifies to 1 .8.

00:59 So that's my slope between x equals 60 and x equals 80.

01:03 Now, i'm going to write the equation of the line connecting those points in point slope form.

01:08 I'm going to say i have y minus a y -coordinate.

01:12 Y minus 84 is equal to my slope, 1 .8, times the quantity of x minus the x -coordinate, which is 60.

01:22 Now, suppose that my x is 72.

01:27 I would have y minus 84 equals 1 .8 times the quantity of 72 minus 60.

01:36 So now i'm going to head to my calculator.

01:38 I've got 72 minus 60.

01:41 That's certainly 12...

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