in-this-exercise-we-estimate-the-rate-at-which-the-total-personal-income-is-rising-in-the-richmond-p

Question

In this exercise we estimate the rate at which the total personal income is rising in the Richmond-Petersburg, Virginia, metropolitan area. In 1999. the population of this area was 961,400, and the population was increasing at roughly 9200 people per year. The average annual income was S30,593 per capita, and this average was increasing at about S1400 per year (a little above the national average of about S1225 yearly).  Use the Product Rule and these figures to estimate the rate at which total personal income was rising in the Richmond-Petersburg area in 1999. Explain the meaning of each term in the Product Rule.

Frank Lin · Accepted Answer

This is a lengthy problem, but basically you have a population function and they give you some information about awesome about the population and Corinne jer great and also the income and the growing right. And they ask you to CompUSA the increasing rate, which is that the relative up. So it&#x27;s a total income, bridges, population times the average incomes. And here we just have to apply the product rule. And after you apply the rule, you just have to probably in those numbers. Now leave it to you to use a calculator to figure out the exact number, but this is a basic idea.

Nick Johnson · Answer

Okay, So the South Carolina population, let&#x27;s say, is P people, so we don&#x27;t have the. In 1990 the population was 3,486,000 people, and the increase per year while is 37,400 people. And now the years since 1990. That&#x27;s to know by T years. So then our outbreak model here says that key right, the population off your lineup is equal to the initial population, which is three million 486,000 plus well, um, 37,400 times t. It&#x27;s good to here.

Jin Yan Yeo · Answer

in this question, we want to translate the expenditures, the respective spending into three equations so that we can solve the equations simultaneously to find each expenditure. So let&#x27;s translate the the different types off expended journalists that, in the question to three unknowns first. So if you&#x27;re you&#x27;re further a bar graph in the question we have housing represented by X vehicle against represented by why and health care represented by easy. So how to reform all three equations? We can get hints from the question. So, firstly, the question mentioned that the total expenditure IHS person amount so that forms our first equation by summing all of them. And then the question also mentioned the difference between, um, the spending on housing and spending on vehicle guests. So that&#x27;s all second equation, and then the third equation is the difference between the spending on housing and the spending on health care. So that&#x27;s keep, um, this bear in mind the three equations and let&#x27;s work on the let&#x27;s work on them by our through addition or subtraction to try to isolate and unknown to find its value. So there are many ways you can do this, but I&#x27;m gonna start off this equation one and two. So I&#x27;m gonna subject equation two from Equation one. How I do this is by having the left hand side off. Equation one minus the left hand side of equation two equals to the right hand side of equation one minus the right hand side off equation too. And in the wind. So I eliminate the unknown X. And now I have in the equation with just y and Z. So this my equation for I&#x27;m gonna do it again with Equation two and three by having the left inside of equation to minus left inside, off equation three equals to the right hand side of equation three minus the right hand side off equation off three equal. Do the right hand side off equation to minors. The right hand side off equation three. And this is my fifth equation. So now you Christians four and five have on Lee y and Z, and I&#x27;m gonna extend this process to illuminate another unknown to find. And in this case, I&#x27;m eliminating Z so I can find the value off why and with the value off, why I can substitute it back into IDA equation four or five to sign the value off Z. So in this case of Children to work with equation five to find the value off Z, you should be able to get the same answer. If you use equation for and with both y and Z values, I can substitute them back into any off the 1st 3 equations to find the value off X. So in this case, have Children toe what with Equation two. But again, you should be able to get the same and the regardless off which equation you choose. We got those, whether it&#x27;s equation one or two off three.

Jin Yan Yeo · Answer

in this question before us want to translate the different types off spending into three announce X, y and z So in the bar graph in a question they quit that housing two x Be going Guess that&#x27;s why and healthcare S Z in the question. There are several options for us to form our three simultaneous equations. So, firstly, the question mentioned that the some off all the sum total expenditure is a certain amount. So that forms our first equation. And then the question mentioned the difference between housing and vehicle spending. So that&#x27;s all second equation. And that&#x27;s the equation is from the difference between housing and health care expenditure. So that&#x27;s our three equations. So keeping in mind these three equations, we wanna work on them, be by adding them together or subjecting one from the other to try to isolate one off the unknowns to find its value. There are many ways you can do this, but I&#x27;m gonna start off with equations one and three. So I&#x27;m going toe subtract Equation three from Equation one, and how I do this is to subject the left inside off Equation. One is to use the left instead of equation one minus the left hand side off equation three and then equate that toe the right hand side of the equation. One minus the right hand side off equation three. And I threw doing, uh, By doing so, I eliminated the unknown X. And so I get, um, equation for which only has why NZ and I do that again with depression to any equation. Three getting another equation that only has why NZ and I want to repeat these this process to in a minute I another unknown. So in this case, I&#x27;m eliminating. Why, by adding them together and I get the value for X Fawzi Sorry. And with the value falls e, I can substitute it back into equation four or five because after having the value for Z, I can solve for why So that&#x27;s my value for why I&#x27;ve chosen to what? We&#x27;ve equation five in this case. But you should be able to get the same answer if you use equation for and last me. I only have one unknown left, so I&#x27;m gonna use equation three. So in Equation three this on the X and Z. And since I have the value forward Z, I can find value for X

Nick Johnson · Answer

we have the verbal model. Where we have the for the population is equal to the 2010 population, plus the increase per year times number of years since 2000 and 10. So if we have the floater population is equal to its AP million and then the 2010 population was given a 17.4 million increase per year is 0.22 million and that say there&#x27;s t years since 2010? So, um, here they require the outbreak models, which represents this would be option c.

Frank Lin · Answer

it&#x27;s clear. So when you married here. So let&#x27;s make t b the number of years after 1999. So the population is equal to 9200 t last 96,001 961,400 in the average annual income ISS represented by a of T, we&#x27;re just equal to 1400 t plus 30,000 593. And if we multiply these together, it gives the total personal income of tea. So the rate at which the total person personal income rises, it&#x27;s gonna be the derivative of tea, and we use our product rule. When we plug it in, we get 9200 t waas murder 1200 t we simplify and we get plus 9200 t plus 961,400 arms. 1400. We&#x27;re gonna plug in for 1999. We&#x27;re gonna plug in zero because there&#x27;s gonna be zero years and this gives us around 1,000,000,627 million Runge of $15,600. This part right here is the rate of rate of change of total income that comes from the annual income from the additional population and this right here is the rate of change of total income that comes from the increased income of the total population.

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A manufacturer produces bolts of a fabric with a …

Problem 57 Hard Difficulty

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rising at about $\$ 1.627$ billion a year

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rising at about $\$ 1.627$ billion a year

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Michael J.

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