# 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. # 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.

Question
Modeling
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. ## Answers (1) 2021-01-20 Given, Personnel cost $$\displaystyle=\{9},{500.000}$$ in 2009 Rate of interest $$= 4.2\%$$ (annually) The annual cost function,$$\displaystyle{C}{\left({t}\right)}={9.5}{\left({1.042}\right)}^{t}$$ a)Write the equation that can be solved to find in what year personnel cost will be double the 2009 personnel costs? b)Solve the equation numerically and determine the year. Personnel cost will be double the 2009 personnel cost $$\displaystyle={2}\times\{9},{500},{000}= \displaystyle\{19},{000},{000}$$ So we get, $$\displaystyle{A}={P}{\left({1}+{r}\right)}^{t}$$ $$\displaystyle{A}={9500000}{\left({1}+{0.042}\right)}^{t}$$ $$\displaystyle{19000000}={9500000}{\left({1.042}\right)}^{t}$$ $$\displaystyle{19}={9.5}{\left({1.042}\right)}^{t}$$ Find the value for t, $$\displaystyle{199.5}={\left({1.042}\right)}^{t}$$ $$\displaystyle{2}={\left({1.042}\right)}^{t}$$ Apply logarithmic on both side we get, $$\displaystyle \log{{2}}={ \log{{\left({1.042}\right)}}^{t}}$$ We know that, $$\displaystyle{ \log{{m}}^{n}=}{n} \log{{m}}$$ $$\displaystyle \log{{2}}={t} \log{{\left({1.042}\right)}}$$ $$\displaystyle{t}=\frac{{ \log{{2}}}}{{ \log{{\left({1.042}\right)}}}}$$ $$\displaystyle{t}=\frac{{{0.3010299}}}{{{0.0178677}}}$$ $$t = 16.8477$$ $$\displaystyle{t}\stackrel{\sim}{=}{17}$$ After 17 years we get the value of personnel cost is double in 2009. So, $$\displaystyle{2009}+{17}={2026}$$ Therefore we get, $$\displaystyle{19}={9.5}{\left({1.042}\right)}^{t},$$ Year is 2026. ### Relevant Questions asked 2020-10-28 Determine the algebraic modeling a. One type of Iodine disintegrates continuously at a constant rate of $$\displaystyle{8.6}\%$$ per day. Suppose the original amount, $$\displaystyle{P}_{{0}}$$, is 10 grams, and let t be measured in days. Because the Iodine is decaying continuously at a constant rate, we use the model $$\displaystyle{P}={P}_{{0}}{e}^{k}{t}$$ for the decay equation, where k is the rate of continuous decay. Using the given information, write the decay equation for this type of Iodine. b. Use your equation to determine the half-life ofthis type of Fodine, That is, find ‘out how many days it will take for half of the original amount to be left. Show an algebraic solution using logs. asked 2021-04-25 The unstable nucleus uranium-236 can be regarded as auniformly charged sphere of charge Q=+92e and radius $$\displaystyle{R}={7.4}\times{10}^{{-{15}}}$$ m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium-236 nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945. A. Find the radii of the two "daughter" nuclei of charge+46e. B. In a simple model for the fission process, immediatelyafter the uranium-236 nucleus has undergone fission the "daughter"nuclei are at rest and just touching. Calculate the kineticenergy that each of the "daughter" nuclei will have when they arevery far apart. C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium-236 nucleus. Calculate the energy released by thefission of 10.0 kg of uranium-236. The atomic mass ofuranium-236 is 236 u, where 1 u = 1 atomic mass unit $$\displaystyle={1.66}\times{10}^{{-{27}}}$$ kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases 4.18 x 10^12 J when itexplodes). asked 2021-05-17 You want to invest money for your child's education in a certificate of deposit (CD). You want it to be worth $$12,000$$ in 10 years. How much should you invest if the CD pays interest at a $$9\%$$ annual rate compounded a) Annually? b) Continuously? asked 2021-05-26 You open a bank account to save for college and deposit$400 in the account. Each year, the balance in your account will increase $$5\%$$. a. Write a function that models your annual balance. b. What will be the total amount in your account after 7 yr? Use the exponential function and extend the table to answer part b.
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 random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
Write an exponential growth or decay function to model each situation. Then find the value of the function after the given amount of time. A new car is worth \$25,000, and its value decreases by 15% each year; 6 years.
The dominant form of drag experienced by vehicles (bikes, cars,planes, etc.) at operating speeds is called form drag. Itincreases quadratically with velocity (essentially because theamount of air you run into increase with v and so does the amount of force you must exert on each small volume of air). Thus
$$\displaystyle{F}_{{{d}{r}{u}{g}}}={C}_{{d}}{A}{v}^{{2}}$$
where A is the cross-sectional area of the vehicle and $$\displaystyle{C}_{{d}}$$ is called the coefficient of drag.
Part A:
Consider a vehicle moving with constant velocity $$\displaystyle\vec{{{v}}}$$. Find the power dissipated by form drag.
Express your answer in terms of $$\displaystyle{C}_{{d}},{A},$$ and speed v.
Part B:
A certain car has an engine that provides a maximum power $$\displaystyle{P}_{{0}}$$. Suppose that the maximum speed of thee car, $$\displaystyle{v}_{{0}}$$, is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power $$\displaystyle{P}_{{1}}$$ is 10 percent greater than the original power ($$\displaystyle{P}_{{1}}={110}\%{P}_{{0}}$$).
Assume the following:
The top speed is limited by air drag.
The magnitude of the force of air drag at these speeds is proportional to the square of the speed.
By what percentage, $$\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}$$, is the top speed of the car increased?
Express the percent increase in top speed numerically to two significant figures.
Testing for a Linear Correlation. In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of $$\alpha = 0.05$$. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.) Lemons and Car Crashes Listed below are annual data for various years. The data are weights (metric tons) of lemons imported from Mexico and U.S. car crash fatality rates per 100,000 population [based on data from “The Trouble with QSAR (or How I Learned to Stop Worrying and Embrace Fallacy),” by Stephen Johnson, Journal of Chemical Information and Modeling, Vol. 48, No. 1]. Is there sufficient evidence to conclude that there is a linear correlation between weights of lemon imports from Mexico and U.S. car fatality rates? Do the results suggest that imported lemons cause car fatalities? $$\begin{matrix} \text{Lemon Imports} & 230 & 265 & 358 & 480 & 530\\ \text{Crashe Fatality Rate} & 15.9 & 15.7 & 15.4 & 15.3 & 14.9\\ \end{matrix}$$
Problem: find acondition on H, involving $$\displaystyle{P}_{{0}}$$ and k, that will prevent solutions from growing exponentially.