Consider a pond that contains an initial population of 300 fish. When there are enough food, the population P of fish grows in function of time t (in years) as follows: P(t) = 300(1.05)'. The initial amount of feed ture for fish in the pond is 1000 units and 1 unit can feed 1 fish for 1 year. The quantity N of fish food decreases according to the function N(t) = 1000(0.92)' :
Represented graphically the functions P(t) and N(t) in a same system of axes. Describe of what kind of function it is.
- Determine the domain and the range of these functions.
- Determine the point of intersection of these two curves. Specifies the coordinates, to the hundredth, and explain what they mean.
- Call this moment in time the “crisis point”. Represents the function y = N(t) - P(t). Explain the meaning of this function. What is the abscissa at the origin, to the nearest hundredth, of the function y = N(t) - P(t)? How is this value of t related to the crisis point? Comments on the validity of the mathematical model. tick of the function P(t) for values of t greater than this abscissa a the origin. Draw the shape of the curve such that you think she should be there.
3 answers
First of all, get the equation right, it should be
P(t) = 300(1.05)^t and N(t) = 1000(.92)^t
The graphing should be trivial, or just use a website like
www.desmos.com/calculator
which will also show you the domain and range, as well as the intersection
Where do they intersect?
300(1.05)^t = 1000(.92)^t
.3(1.05)^t = .92^t
take log of both sides and use log rules ....
log .3 + t log 1.05 = t log .92
t log 1.05 - t log .92) = -log .3
t( log 1.05 - log .92) = -log .3
t = 9.11 years
of course, when t = 9.11 .... (I carried more decimals in my calculator)
P(9.11) = 300(1.05)^9.11 = 467.88...
N(9.11) = 1000(.92)^9.11 = 467.88... , as expected
so the abscissa is 9.11 and the ordinate is 467.88 or the point (9.11,467.88)
then P(9.11) - N(9.11) is obviously zero !!
At the end you say
"Draw the shape of the curve such that you think she should be there."
No idea what that is supposed to mean.