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anonymous
 one year ago
The value of a particular item can be modeled by
P(t) = P0(a)t
where P is in dollars and t is the number of years since the item was purchased. Suppose the value of the item increases 5% each year and the item was purchased for $20.
(a) Write a formula for P(t) according to the model.
(b) How fast is the value of the item increasing when t = 5 years? Round your answer to two decimal places
anonymous
 one year ago
The value of a particular item can be modeled by P(t) = P0(a)t where P is in dollars and t is the number of years since the item was purchased. Suppose the value of the item increases 5% each year and the item was purchased for $20. (a) Write a formula for P(t) according to the model. (b) How fast is the value of the item increasing when t = 5 years? Round your answer to two decimal places

This Question is Closed

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0a. you're given the initial price, \(P_0\) = $20 a = 0.05 Just plug in the values into your formula

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0b. t = 5 years, a = 0.05, \(P_0\)= 20

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0for a. i put P(t) = 20(.05)^t but i got it wrong

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2You are multiplying t

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0sorry, its actually raised to the t power

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2So you would get, 20 * .05 * 5. To make this easier do 20*5 then multiply by .05 Hint: .05 = \[\frac{ 1 }{20 }\]

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2Oh. Then that changes the function completely!

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2So I assume your equation would be \[20 \times(.05)^{t}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yeah, i typed that in but i got it wrong

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2This formula is similar to the compound formula. Put in your answer and tell me if it is correct?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0sorry, i am confused, for what letter a or b?

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2I gave you (a) already. Now plug in 5 for t

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Then you are multiplying by t and it is not raised to a power, perhaps?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0wait sorry one second

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I'm trying to get a screenshot of it

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2substitute a with the appropriate variables.

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2Try this\[20(.05)^{t}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yeah it gave that as wrong, not sure why

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2Try adding the times in. Could be a error. Maybe it s compounding formula. Try this \[20\times(1.05)^{t}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok that was right for the equation, but for the part b i plug in chug but it gives the wrong answer?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i get 20(1.05)^5 = 25.52 rounding to two decimal places, says it is wrong tho

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Calculator gives me 25.52563 and i tried 25.52 and 25.53

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2when it says round to the nearest cent, It really telling you to round to the nears hundredth. Thus you look at the hundredth place which is a 2. look next to it to the right. if the number is 5 or above, you round up. The 2 becomes a 3.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yea i tried 25.53 and 25.52 and both are wrong

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2That is very strange.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0would it maybe have to do with it asking for dollar/year?

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2dollar per year? That a strange way of saying it.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0wait wouldn't it be the derivative????

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2Why would it be the derivative?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yeah nevermind, i don't know :(

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2If it's a physic problem, then yes I would take the derivative.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Actually,you would have to take the derivative.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yeah, my time limit, expired :( oh well got a 80% on it :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Do you still want to know how to figure it out? Or nah.

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2Sorry man. It was confusing. It seems like an interest problem but then a physic problem?

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2If it is a physic problem, take the derivative (chain rule).

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0no your fine man, i appreciate the help

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[P(t) = 20\cdot 1.05^t\]Just to be clear... can you explain to me how you got \(a=1.05\) ? I know how this works, but I want you to explain it t me.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0to be honest i don't understand the 1.05 either

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2I can tell you why. When money is growing over certain amount of year (or days or month). You use that formula. P(1+i)^t. Where i is the interest rate, P is your initial amount and t is the amount of years/month/days

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So we have out function \(P(t) = P_0 \cdot a^t\). Were given that \(P_0 = 20\) an the growth factor = \(5\%\). Therefore for every year it is increasing... \(P(1) = \$20 + \left(\frac{0.05}{100}\right)\cdot \$20\)

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2Yes correct. You would get the constant rate of interest i if you are headed for the right direction

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ahh... im off on my decimal place.

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2yea that would be .0005 lol

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0uld have said \[P(1) = \$20 + \left(\frac{5}{100}\right)\cdot \$20\]

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2I can prove it if you want me to. How i derive the formula.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i understand the formula now, but what i don't understand is the part for b?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0That gives us, \(P(1) = \$20 + \$1 = \$21\) And now we want to find the growth factor, \(a\) to find the generalization of the growth function of \(t\) years.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So we have \(P(1) = \$21,\qquad ~P_0 = \$20\qquad ~,~\qquad t=1\) \[P(t) = P_0 \cdot a^t\]\[P(1) = 20 \cdot a^1\]\[\$21= \$20 \cdot a^1\]\[a= \frac{21}{20} = 1.05\]\[\implies P(t) = \$20 \cdot 1.05^t\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Now moving on to part b.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0We want to find `how fast` the value is increasing with respect to time, therefore we're finding the rate of change of the growth function, \(\dfrac{dP}{dt}\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[P(t)= 20 \cdot 1.05^t\]\[\frac{d}{dt} (P(t) = 20 \cdot 1.05^t) \]\[P'(t) = 20\frac{d}{dx}( 1.05^t)\qquad \qquad \frac{d}{dx}(a^x) = a^x \cdot \ln(a)\]\[P'(t) = 20[1.05^t \cdot \ln(1.05)]\]\[\color{red}{P'(t) \approx 0.98 \cdot 1.05^t}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Sorry, I'm extremely lagged out, took a while to type up.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0no your fine, i see the answer now, i should've taken the derivative :/

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yeah, when a question asks you `how fast` something is changing within a given time, it's asking you basically for a certain value at a point. (your slope) :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I was working it out on a separate piece of paper while you guys were discussing the question earlier and i had to reread the question a few times to notice it was a rate of change question.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0thank you so much for explaining the question to me, i understand it well now, thank you!!!

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Does it make sense to you too, @Venomblast ?

Venomblast
 one year ago
Best ResponseYou've already chosen the best response.2Haha yea. Sorry I went to sleep
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