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. $$\begin{array}{|l|l|r|l|}\hline x & 60 & 80 & 100 \\\hline f(x) & 84 & 120 & 168 \\\hline\end{array}$$ 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.
$$\begin{array}{|l|l|r|l|}\hline x & 60 & 80 & 100 \\\hline f(x) & 84 & 120 & 168 \\\hline\end{array}$$
Estimate the number of cans that can be sold at (a) $72^{\circ}$ and (b) $94^{\circ}$.

Key Concepts

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.

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.

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.

Transcript

00:01 Okay, so for this problem, we're gonna use be first to set of coordinates or set of points.

00:10 So we're gonna do 60 comma, 84 and 80 comma 1 20 okay, so the first thing that we're gonna do is we need to find the slope of the line that goes through these two points.

00:30 So to do that, we're going to use the so formula and the change in the y coordinate, which is gonna be 1 20 minus 84 over 80 minus 60.

00:45 And if we simplify this, we will get positive.

00:48 9/5.

00:51 Okay, so then from here, we're going to use a point slope.

00:55 Ah, form to create our than your approximation the equation.

00:59 So with this, we're gonna do why? minus so why corner is gonna come from any of these two would work.

01:07 However, i see that, uh, 80 is much closer to 72 than 60.

01:13 Somebody used the second point here.

01:15 So why minus 1 20 is equal to m, which we just calculate to be 9/5 times x minus et from here, i'm just going to solve why? by bringing on 20 to the other side.

01:36 So plus 1 20 okay, so this is gonna be our l of x.

01:43 So we need to find our desired quantity, which is 72.

01:46 So we need to do l of 72 degrees.

01:50 So we're basically gonna plug 72 into here.

01:52 So we have nine this times 70 to minus 80 plus 1 20 and if we calculate that, we will get 105.6 degrees.

02:06 So that will be a result for part a.

02:11 Then for part b, we want 94 degrees...