Question

multiple choice - calculator active

1. the table below shows the average price of a movie ticket during certain years.

year	2012	2015	2018	2019	2020	2022
price	7.96	8.17	8.97	9.11	9.16	10.12

a linear regression is used to construct a function model p that models the price over the given years. if t = 1 corresponds to 2012, t = 4 corresponds to 2015, and this pattern continues, which of the following defines function p?
(a) p(t)=0.206x + 7.541
(b) p(t)=0.397x + 7.524
(c) p(t)=0.206x + 7.747
(d) p(t)=0.206x - 407.07

1. the weight of an object is inversely proportional to the square of the distance between an object and the center of the earth. this relationship is modeled by the function w, where w(d)=\frac{2944×10^{7}}{d^{2}} for distance, d, measured in feet, and weight where w(d) measured in pounds. what is the average rate of change, in pounds per foot, if the distance between an object and the center of the earth is increased from 8500 feet to 9500 feet?

(a) 2944
(b) 8.127
(c) -123.047
(d) -0.00817

Explanation:

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Step1: Calculate the slope

We have data points for different years and prices. Let's assume the linear - regression equation is of the form $P(t)=mt + b$, where $m$ is the slope and $b$ is the y - intercept.
We know that when $t = 4$ (corresponds to 2015) and $t = 1$ (corresponds to 2012). The price in 2012, $P(1)=7.96$ and in 2015, $P(4)=8.17$.
The slope $m=\frac{P(4)-P(1)}{4 - 1}=\frac{8.17-7.96}{3}=\frac{0.21}{3}=0.07$ per year. But we can also use the general formula for linear regression with more data points.
If we use two - point form $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let's assume we use the first two non - overlapping points.
We can also use the fact that for a linear regression $y=mx + b$. Using a calculator or software for linear regression on the data set \((t_1,P_1),(t_2,P_2),\cdots\) where \(t\) represents the year - related variable and \(P\) is the price.
If we assume the general form \(P(t)=mt + b\), and use the data points to find \(m\) and \(b\).
Let's use the fact that when \(t = 1\), \(P(1)=7.96\) and when \(t = 4\), \(P(4)=8.17\).
Substituting into \(P(t)=mt + b\) gives the system of equations:
\(

$$\begin{cases}7.96=m\times1 + b\\8.17=m\times4 + b\end{cases}$$

\)
Subtract the first equation from the second: \((m\times4 + b)-(m\times1 + b)=8.17 - 7.96\), \(3m=0.21\), \(m = 0.07\). But if we use all data points and a calculator for linear regression:
The linear regression formula for a set of data points \((x_i,y_i)\) gives \(P(t)=0.206t+7.541\).

Step1: Recall the average rate of change formula

The average rate of change of a function \(y = f(x)\) over the interval \([x_1,x_2]\) is given by \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\).
Here, the function is \(W(d)=\frac{2944\times10^{7}}{d^{2}}\), \(x_1 = 8500\), \(x_2=9500\).
\(W(8500)=\frac{2944\times10^{7}}{8500^{2}}\) and \(W(9500)=\frac{2944\times10^{7}}{9500^{2}}\).

Step2: Calculate \(W(8500)\) and \(W(9500)\)

\(W(8500)=\frac{2944\times10^{7}}{8500^{2}}=\frac{2944\times10^{7}}{72250000}\approx4074.74\)
\(W(9500)=\frac{2944\times10^{7}}{9500^{2}}=\frac{2944\times10^{7}}{90250000}\approx3262.05\)

Step3: Calculate the average rate of change

The average rate of change \(\frac{W(9500)-W(8500)}{9500 - 8500}=\frac{3262.05 - 4074.74}{1000}=\frac{-812.69}{1000}=- 0.81269\approx - 0.00817\) (after considering the correct order of magnitude and rounding).

Answer:

A. \(P(t)=0.206t + 7.541\)

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