# Thread: rates and related rates

1. ## rates and related rates

An item costs $600 at time t = 0 and costs$P in year t. When inflation is r% per year, the price is given by the following equation.
P = 600e^(rt/100)
(a) If r is a constant, at what rate is the price rising (in dollars per year) initially?

At what rate is the price rising after 2 years?

(b) Now suppose that r is increasing by 0.3 per year when r = 4 and t = 2. At what rate (dollars per year) is the price increasing at that time?

i would appreciate some quick help here please...thanks

2. Originally Posted by mathaction
An item costs $600 at time t = 0 and costs$P in year t. When inflation is r% per year, the price is given by the following equation.
P = 600e^(rt/100)
(a) If r is a constant, at what rate is the price rising (in dollars per year) initially?

At what rate is the price rising after 2 years?
First for constant $r$ we need:

$
\frac{dP}{dt}=600~\frac{r}{100}~e^{rt/100}
$

Then when $t=0$ we have the initial rate of increas in price is:

$
\left. \frac{dP}{dt}\right|_{t=0}=600~\frac{r}{100}~e^{r( 0)/100}=6r \mbox { dollars per year }
$

After 2 years we have:

$
\left. \frac{dP}{dt}\right|_{t=2}=600~\frac{r}{100}~e^{r( 2)/100}=6r e^{r/50} \mbox { dollars per year }
$

RonL

3. Originally Posted by mathaction
An item costs $600 at time t = 0 and costs$P in year t. When inflation is r% per year, the price is given by the following equation.
P = 600e^(rt/100)
(a) If r is a constant, at what rate is the price rising (in dollars per year) initially?

At what rate is the price rising after 2 years?

(b) Now suppose that r is increasing by 0.3 per year when r = 4 and t = 2. At what rate (dollars per year) is the price increasing at that time?

i would appreciate some quick help here please...thanks
For non-constant $r$ we have:

$
\frac{dP}{dt}= 6 r e^{rt/100}~ \frac{dr}{dt}
$

Now just insert the values to get the answer.

RonL