# Find the exponential function that models the data in the table below. begin{array}{|c|c|}x&-3& -2&-1&0&1&2&3& 4 y&4/27&4/9&4/3&4 &12 & 36 & 108 & 324end{array} What is the exponential regression of the data? y = square

Question
Modeling
Find the exponential function that models the data in the table below. $$\begin{array}{|c|c|}x&-3& -2&-1&0&1&2&3& 4\\ y&4/27&4/9&4/3&4 &12 & 36 & 108 & 324\end{array}$$ What is the exponential regression of the data? $$y = square$$

2020-11-27

Introduction Given a dataset where an exponential function $$y=ab^x$$ where a and b are constants. are to be obtained. Taking logarithm on both sides of the above equation, $$\log y = \log a + x \log b$$ $$\Rightarrow v = A + Bx$$

where $$v = \log y, A = \log a, B = \log b$$

Hence, the exponential function is reduced to a linear function between v and x and can be solved by least square method. By principle of least squares, the normal equations for estimating A and B is given by, $$\sum v = nA + B \sum x \sum vx = A \sum x + B \sum x^2$$

Calculations The following is table is formed to calculate the value of A and B

$$\begin{array}{c}y & x&v = \log y&x^2&vx\\ 4/27& -3& -0.8293& 9&2.4879\\ 4/9&-2&-0.3522&4&0.7044\\ 4/3&-1&0.1249&1&-0.1249\\ 4&0&0.6021&0&0\\ 12&1&1.0792 &1&1.0792\\ 36&2&1.5563&4&3.1126 \\ 108&3&2.0334&9&6.1002\\ 324&4 &2.5105&16&10.0420\\ Total& 4&6.7249&44&23.4014\end{array}$$

Hence, putting the values obtained from the table to the normal equations we get, $$6.7249 = 8A + 4B . . . . . .(i)23.4014 = 4A + 44B . . . . . .(ii)$$

On solving the simultaneous equations, by method of comparison, it is obtained that, from $$(i)A = \frac{6.7249-4B}{8} . . . . . (iii)from (ii)A=\frac{23.4014-44B}{4} . . . . . . .(iv)$$

Comparing (iii) and (iv) $$\frac{6.7249-4B}{8} = \frac{23.4014-44B}{4}$$ Hence solving the above equation, it is obtained that, $$B = 0.4771$$ $$\Rightarrow b = Anti\log B$$ $$= 2.9999$$ Then putting the value of B in (iii), it is obtained that, $$A = 0.6021$$ $$\Rightarrow a = Antilog A$$ $$= 4.0004$$ Therefore, The required exponential regression of the data is $$y = 4.0004(2.9999)^x$$

### Relevant Questions

Use exponential regression to find a function that models the data. $$\begin{array}{|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 14 & 7.1 & 3.4 & 1.8 & 0.8 \\ \hline \end{array}$$

An automobile tire manufacturer collected the data in the table relating tire pressure x​ (in pounds per square​ inch) and mileage​ (in thousands of​ miles). A mathematical model for the data is given by
$$\displaystyle​ f{{\left({x}\right)}}=-{0.554}{x}^{2}+{35.5}{x}-{514}.$$
$$\begin{array}{|c|c|} \hline x & Mileage \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}$$
​(A) Complete the table below.
$$\begin{array}{|c|c|} \hline x & Mileage & f(x) \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}$$
​(Round to one decimal place as​ needed.)
$$A. 20602060xf(x)$$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,45), (30,51), (32,56), (34,50), and (36,46). A parabola opens downward and passes through the points (28,45.7), (30,52.4), (32,54.7), (34,52.6), and (36,46.0). All points are approximate.
$$B. 20602060xf(x)$$
Acoordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2.
Data points are plotted at (43,30), (45,36), (47,41), (49,35), and (51,31). A parabola opens downward and passes through the points (43,30.7), (45,37.4), (47,39.7), (49,37.6), and (51,31). All points are approximate.
$$C. 20602060xf(x)$$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (43,45), (45,51), (47,56), (49,50), and (51,46). A parabola opens downward and passes through the points (43,45.7), (45,52.4), (47,54.7), (49,52.6), and (51,46.0). All points are approximate.
$$D.20602060xf(x)$$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,30), (30,36), (32,41), (34,35), and (36,31). A parabola opens downward and passes through the points (28,30.7), (30,37.4), (32,39.7), (34,37.6), and (36,31). All points are approximate.
​(C) Use the modeling function​ f(x) to estimate the mileage for a tire pressure of 29
$$\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$ and for 35
$$\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$
The mileage for the tire pressure $$\displaystyle{29}\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$ is
The mileage for the tire pressure $$\displaystyle{35}\frac{{{l}{b}{s}}}{{{s}{q}}}$$ in. is
(Round to two decimal places as​ needed.)
(D) Write a brief description of the relationship between tire pressure and mileage.
A. As tire pressure​ increases, mileage decreases to a minimum at a certain tire​ pressure, then begins to increase.
B. As tire pressure​ increases, mileage decreases.
C. As tire pressure​ increases, mileage increases to a maximum at a certain tire​ pressure, then begins to decrease.
D. As tire pressure​ increases, mileage increases.
The burial cloth of an Egyptian mummy is estimated to contain 560 g of the radioactive materialcarbon-14, which has a half life of 5730 years.
a. Complete the table below. Make sure you justify your answer by showing all the steps.
$$\begin{array}{|l|l|l|}\hline t(\text{in years})&m(\text{amoun of radioactive material})\\\hline0&\\\hline5730\\\hline11460\\\hline17190\\\hline\end{array}$$
b. Find an exponential function that models the amount of carbon-14 in the cloth, y, after t years. Make sure you justify your answer by showing all the steps.
c. If the burial cloth is estimated to contain 49.5% of the original amount of carbon-14, how long ago was the mummy buried. Give exact answer. Make sure you justify your answer by showing all the steps.
For Questions 1 — 2, use the following. Scooters are often used in European and Asian cities because of their ability to negotiate crowded city streets. The number of scooters (in thousands) sold each year in India can be approximated by the function $$N = 61.86t^2 — 237.43t + 943.51$$ where f is the number of years since 1990. 1. Find the vertical intercept. What is the practical meaning of the vertical intercept in this situation? 2. Use a numerical method to find the year when the number of scooters sold reaches 1 million. (Note that 1 million is 1,000 thousand, so N = 1000) Show three rows of the table you used to support your answer and write a clear answer to the problem.
A small grocer finds that the monthly sales y (in $) can be approximated as a function of the amount spent advertising on the radio $$x_1$$ (in$) and the amount spent advertising in the newspaper $$x_2$$ (in $) according to $$y=ax_1+bx_2+c$$ The table gives the amounts spent in advertising and the corresponding monthly sales for 3 months. $$\begin{array}{|c|c|c|}\hline \text { Advertising, } x_{1} & \text { Advertising, } x_{2} &\text{sales, y} \\ \hline 2400 & { 800} & { 36,000} \\ \hline 2000 & { 500} & { 30,000} \\ \hline 3000 & { 1000} & { 44,000} \\ \hline\end{array}$$ a) Use the data to write a system of linear equations to solve for a, b, and c. b) Use a graphing utility to find the reduced row-echelon form of the augmented matrix. c) Write the model $$y=ax_1+bx_2+c$$ d) Predict the monthly sales if the grocer spends$250 advertising on the radio and $500 advertising in the newspaper for a given month. asked 2021-01-19 Determine the algebraic modeling The personnel costs in the city of Greenberg were$9,500,000 in 2009. In the recent past the personnel costs have increased at the rate of $$4.2\%$$ annually.
To model this, the city manager uses the function $$\displaystyle{C}{\left({t}\right)}={9.5}{\left({1}{042}\right)}^{t}\text{where}\ {C}{\left({t}\right)}$$ is the annual personnel costs,in millions of dollars, t years past 2009
1) Write the equation that can be solved to find in what year the personnel costs will be double the 2009 personnel costs.
2) Then solve the equation numerically (use the Table feature of your calculator) or graphically and determine the year.

In addition to quadratic and exponential models, another common type of model is called a power model. Power models are models in the form $$\displaystyle\hat{{{y}}}={a}\ \cdot\ {x}^{{{p}}}$$. Here are data on the eight planets of our solar system. Distance from the sun is measured in astronomical units (AU), the average distance Earth is from the sun. $$\begin{array}{|c|c|}\hline \text{Planet} & \text {Distance from sun}\text {(astronomical units)} & \text{Period of revolution}\text{(Earth years)} \\ \hline \text{Mercury} & 0.387 & 0.241 \\ \hline \text { Venus } & 0.723 & 0.615 \\ \hline \text { Earth } & 1.000 & 1.000 \\ \hline \hline \text { Mars } & 1.524 & 1.881 \\ \hline \text { Jupiter } & 5.203 & 11.862 \\ \hline \text { Saturn } & 9.539 & 29.456 \\ \hline \text { Uranus } & 19.191 & 84.070 \\ \hline \text { Neptune } & 30.061 & 164.810 \\ \hline \end{array}$$ Calculate and interpret the residual for Neptune.

a) To calculate: The least squares regression line for the data points using the table given below. \begin{array}{|c|c|} \hline Fertilizer & x & 100 & 150 & 200 & 250 \\ \hline Yield & y & 35 & 44 & 50 & 56 \\ \hline \end{array} b)To calculate: The approximate yield when 175 pounds of fertizers were used per acre of land.

The values of two functions, f and g, are given in a table. One, both, or neither of them may be exponential. Give the exponential models for those that are.
f(x)-?
g(x)-??
$$\begin{array}{|l|l|l|}\hline X&-2&-1&0&1&2\\\hline f(x)&1.125&2.25&4.5&9&18\\\hline g(x)&16&8&4&2&1\\\hline\end{array}$$

When a gas is taken from a to c along the curved path in the figure (Figure 1) , the work done by the gas is W = -40 J and the heat added to the gas is Q = -140 J . Along path abc, the work done by the gas is W = -50 J . (That is, 50 J of work is done on the gas.)
I keep on missing Part D. The answer for part D is not -150,150,-155,108,105( was close but it said not quite check calculations)
Part A
What is Q for path abc?
Express your answer to two significant figures and include the appropriate units.
Part B
f Pc=1/2Pb, what is W for path cda?
Express your answer to two significant figures and include the appropriate units.
Part C
What is Q for path cda?
Express your answer to two significant figures and include the appropriate units.
Part D
What is Ua?Uc?
Express your answer to two significant figures and include the appropriate units.
Part E
If Ud?Uc=42J, what is Q for path da?
Express your answer to two significant figures and include the appropriate units.
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