Help solving two word problems involving functions?

Jan 2016
7
0
Arizona
I need help on two word problems that I'm stuck on. For part (a) on #1 I tried plugging in 0, but that wasn't right. And I'm not exactly sure how to create a new function for #1 or formula for #2.
I hope this is in the right forum, I'm new to this.

#1.) Based on data from 1990 to 2006, the average annual salary of a major league baseball player can be modeled by

A(t) = 0.1474t + 0.4970 million dollars where t is the number of years since 1990.

(a) According to the model, what was the average salary in 1990?


_________ million dollars

(b) Create a new function I(t) that gives the increase in the average salary in millions of dollars over the 1990 salary level.
I(t) = ________ million dollars


(c) If the meaning of t was changed from years since 1990 to years since 1920, how would the function
A(t) = 0.1474t + 0.4970 need to change for the results to still make sense? (Write you answer in terms of a new function S(y) that gives the average salary in millions of dollars where y is the number of years since 1920.)
S(y) = _________ million dollars

#2.) Based on data from 1980 to 2005, the value of the dollar based on producer prices can be modeled by
V(t) = −0.00004785t3 + 0.02314t2 − 0.04774t + 1.137 where t is the number of years since 1980.

Write the formula for P(t) given P(t) = 10V(t).

P(t) =
 
Oct 2012
751
212
Ireland
For part (a) all you should need to do is to plug in 0. If you are submitting the answer online maybe you are entering it incorrectly.

part (b). The increase since year 1990 is the average salary at a time $t$ minus the salary in year 1990.

part (c). 1920 is 70 years before 1990. When measuring time since 1920 an extra 70 years will have passed as compared to time since 1990 so y=t+70
 
Jan 2016
7
0
Arizona
For part (a) all you should need to do is to plug in 0. If you are submitting the answer online maybe you are entering it incorrectly.

part (b). The increase since year 1990 is the average salary at a time $t$ minus the salary in year 1990.

part (c). 1920 is 70 years before 1990. When measuring time since 1920 an extra 70 years will have passed as compared to time since 1990 so y=t+70
For part (c), would I just plug in 70 for t in the equation? I tried that but it was incorrect.
 
Oct 2012
751
212
Ireland
For part (c), would I just plug in 70 for t in the equation? I tried that but it was incorrect.
That would be the average salary for a single year that is 70 years after 1990. You want a formula for many years and it is a function of $y$ instead of $t$. See if you can change the equation so that it is a function of $y$. Hint: Look again at my previous reply.
 
Feb 2014
1,748
651
United States
Let's start with problem 1 and go step by step.

Define A(t) as the average annual salary in millions of dollars during year t, and t as as the year's date - 1990. Good to here?

$A(t) = 0.1474t + 0.4970.$

Your method for answering 1a was perfectly correct. To find t for any calendar year, you subtract 1990.
So for 1990 t = 0, and A(0) = 0.1474 * 0 + 0.4970 = 0.4970.

For 1b, you are asked for the function that gives the increase in average annual salary since 1990. The increase is going to be the difference between A(t) and A(0), not A(t) itself. Do you see why?

$I(t) = A(t) - A(0) = (0.1474t + 0.4970) - 0.4970 = 0.1474t.$ Clear on that?

For 1c, you want a new function that gives the same answer as the old function did for equivalent years. If y is 80, meaning the relevant year is 2000, the average income is given by A(10).

In short, $S(y) = A(y - 70).$ Do you FULLY understand why?

Can you finish it?
 
Jan 2016
7
0
Arizona
0.1474(y-70)+0.4970 is what I'm getting but I don't understand how to translate that into a function to get the answer.
 
Jan 2016
7
0
Arizona
Let's start with problem 1 and go step by step.

Define A(t) as the average annual salary in millions of dollars during year t, and t as as the year's date - 1990. Good to here?

$A(t) = 0.1474t + 0.4970.$

Your method for answering 1a was perfectly correct. To find t for any calendar year, you subtract 1990.
So for 1990 t = 0, and A(0) = 0.1474 * 0 + 0.4970 = 0.4970.

For 1b, you are asked for the function that gives the increase in average annual salary since 1990. The increase is going to be the difference between A(t) and A(0), not A(t) itself. Do you see why?

$I(t) = A(t) - A(0) = (0.1474t + 0.4970) - 0.4970 = 0.1474t.$ Clear on that?

For 1c, you want a new function that gives the same answer as the old function did for equivalent years. If y is 80, meaning the relevant year is 2000, the average income is given by A(10).

In short, $S(y) = A(y - 70).$ Do you FULLY understand why?

Can you finish it?
I'm going to try right now!
 
Jan 2016
7
0
Arizona
I think I get the jist of it but when I enter the answer, it's incorrect, so I'm stumped. Where are you getting y is 80? I set up the function as 0.1474(10)+0.4970 but I'm confused on where the number 80 comes in.
 
Feb 2014
1,748
651
United States
Computers are exasperating.

Consider the year z written in normal format. So an example would be 2000.

y = the year in terms of number of years since 1920 = z - 1920. So in our example y = 2000 - 1920 = 80. For a different z, you would get a different y, but y would always be z - 1920. Does that make sense now?

And t = the year in terms of number of years since 1990 = z - 1990. So in our example y = 2000 - 1990 = 10. For a different z, you would get a different t, but t would always be z - 1990. Still with me?

This is just converting ordinary dates to dates relative to a base year.

However we denominate a specific year, the average level of annual earnings is the same, right?

And S(y) is defined to be the average level of annual earnings in the year denominated by y if we are using 1920 as the base year.

And A(t) is defined to be the average level of annual earnings in the year denominated by t if we are using 1990 as the base year.

So, for any specific calendar year, $S(y) = A(t)$ because y and t are simply referring to the same year in different formats.

This is a key conceptual point so let's discuss it further if you do not feel comfortable with it.

Now in calendar year z, whatever it is,

$y = z - 1920,\ and\ t = z - 1990 \implies t - y = (z - 1990) - (z - 1920) \implies$

$t - y = -\ 1990 + 1920 = -\ 70 \implies t = y - 70.$

$\therefore S(y) = A(t) \implies S(y) = A(y - 70).$

Now here is the meaning of function notation is the second key conceptual point.

$A(t) = 0.1474t + 0.4970$ tells you what to do with whatever is in the parenthetical following A.

So to evaluate $A(t) = A(y - 70)$ in terms of y, you replace t with (y - 70) wherever t appears in the definition of A(t).

After doing so, you will get some expression in y that is equal to S(y) because $S(y) = A(t).$

Please show what you get and HOW YOU GOT IT, and let me know what the computer says about that answer.

For problem 2, I think you mean V(t) = −0.00004785t^3 + 0.02314t^2 − 0.04774t + 1.137, which means

$V(t) = −0.00004785t^3 + 0.02314t^2 − 0.04774t + 1.137.$