# Find the exponential model that fits the points shown in the graph. (Round the exponent to four decimal places)

Question
Find the exponential model that fits the points shown in the graph. (Round the exponent to four decimal places)

2020-11-08
Given points on the exponential model: (0,1) and (5,16).
General equation of an exponential model: y=aebx
Evaluate the general equation at x=0 and x=5:
$$\displaystyle{y}={a}{e}^{{b}}{\left({0}\right)}={a}{e}^{{0}}={a}{\left({1}\right)}={a}$$
$$\displaystyle{y}={a}{e}^{{b}}{\left({5}\right)}={a}{e}^{{5}}{b}$$
We require y=1 when x=0 and y=16 when x=5 (as the points (0,1) and (5,16) need to lie on the exponential model).
a=1
$$\displaystyle{a}{e}^{{5}}{b}={16}$$
By the first equation, we thus know that a=1. Let us replace $$\displaystyle{a}{e}^{{5}}{b}={16}$$ by 1 in the second equation:
$$\displaystyle{e}^{{5}}{b}={16}$$
Take the natural logarithm from each side of the previous equation (also using that the natural logarithm is the inverse of the exponential).
$$\displaystyle{5}{b}={\ln{{16}}}$$
Let us use the power property of logarithms (lna^b=blna) along with $$\displaystyle{16}={2}^{{4}}:$$
$$\displaystyle{5}{b}={4}{\ln{{2}}}$$
Divide each side of the previous equation by 5:
$$\displaystyle{b}=\frac{{{4}{\ln{{2}}}}}{{5}}$$
Replacing a by 1 and b by $$\displaystyle{4}\frac{{\ln{{2}}}}{{5}}$$ in the general equation of the exponential model, we then obtain:
$$\displaystyle{y}={e}^{{{4}\frac{{\ln{{2}}}}{{5}}}}{x}$$

### Relevant Questions

The central processing unit (CPU) power in computers has increased significantly over the years. The CPU power in Macintosh computers has grown exponentially from 8 MHz in 1984 to 3400 MHz in 2013 (Source: Apple. The exponential function $$\displaystyle\{M}{\left({t}\right)}={7.91477}{\left({1.26698}\right)}^{{t}}{\left[{m}{a}{t}{h}\right]}$$, where t is the number of years after 1984, an be used to estimate the CPU power in a Macintosh computer in a given year. Find the CPU power of a Macintosh Performa 5320CD in 1995 and of an iMac G6 in 2009. Round to the nearest one MHz.
Find exponential models
$$\displaystyle{y}_{{1}}={C}{e}^{{{k}_{{1}}{t}}}$$
and
$$\displaystyle{y}_{{2}}={C}{\left({2}\right)}^{{{k}_{{2}}{t}}}$$
That pass through the two given points. Compare the values of $$\displaystyle{k}_{{1}}$$ and $$\displaystyle{k}_{{2}}$$. (If you round your answer, round to four decimal places.)
$$\displaystyle{\left({0},{16}\right)},{\left({60},{\frac{{{1}}}{{{4}}}}\right)}$$
$$\displaystyle{y}_{{1}}={C}{e}^{{{k}_{{1}}{t}}}$$ where C=?
and $$\displaystyle{k}_{{1}}=?$$
$$\displaystyle{y}_{{2}}={C}{\left({2}\right)}^{{{k}_{{2}}{t}}}$$ where C=?
and $$\displaystyle{k}_{{2}}=?$$
Find Expontntial Model that fits the points shown in the graph of table
in number 7, Is the exponent of (-1) n right? I thought that the exponent of (-1) is n-1 because it changed from n=0 to n=1, and if $$(-1)^{n}$$, there will be a change of sign between negative sign and positive sign.
For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{x}&{1}&{2}&{3}&{4}&{5}&{6}&{7}&{8}&{9}&{10}\backslash{h}{l}\in{e}{f{{\left({x}\right)}}}&{409.4}&{260.7}&{170.4}&{110.6}&{74}&{44.7}&{32.4}&{19.5}&{12.7}&{8.1}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$
factor in determining the usefulness of an examination as a measure of demonstrated ability is the amount of spread that occurs in the grades. If the spread or variation of examination scores is very small, it usually means that the examination was either too hard or too easy. However, if the variance of scores is moderately large, then there is a definite difference in scores between "better," "average," and "poorer" students. A group of attorneys in a Midwest state has been given the task of making up this year's bar examination for the state. The examination has 500 total possible points, and from the history of past examinations, it is known that a standard deviation of around 60 points is desirable. Of course, too large or too small a standard deviation is not good. The attorneys want to test their examination to see how good it is. A preliminary version of the examination (with slight modifications to protect the integrity of the real examination) is given to a random sample of 20 newly graduated law students. Their scores give a sample standard deviation of 70 points. Using a 0.01 level of significance, test the claim that the population standard deviation for the new examination is 60 against the claim that the population standard deviation is different from 60.
(a) What is the level of significance?
State the null and alternate hypotheses.
$$H_{0}:\sigma=60,\ H_{1}:\sigma\ <\ 60H_{0}:\sigma\ >\ 60,\ H_{1}:\sigma=60H_{0}:\sigma=60,\ H_{1}:\sigma\ >\ 60H_{0}:\sigma=60,\ H_{1}:\sigma\ \neq\ 60$$
(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)
What are the degrees of freedom?
What assumptions are you making about the original distribution?
We assume a binomial population distribution.We assume a exponential population distribution. We assume a normal population distribution.We assume a uniform population distribution.
Simplify: $$\displaystyle{\left({7}^{{5}}\right)}{\left({4}^{{5}}\right)}$$. Write your answer using an exponent.
Explain in words how to simplify: $$\displaystyle{\left({153}^{{2}}\right)}^{{7}}.$$
Is the statement $$\displaystyle{\left({10}^{{5}}\right)}{\left({4}^{{5}}\right)}={14}^{{5}}$$ true?
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.
Express the fraction $$\displaystyle\frac{{1}}{{6}^{{4}}}$$ using negative exponent.