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.
Working on this for a week ????
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.
We do not do your homework for you. Although it might take more effort to do the work on your own, you will profit more from your effort. We will be happy to evaluate your work though.
To graphically represent the functions P(t) and N(t) on the same system of axes, we need to plot the values of P(t) and N(t) for different values of t.
Let's start with the function P(t) = 300(1.05)^t. To do this, we can select a few values of t and calculate the corresponding values of P(t). For example, if we take t = 0, 1, 2, 3, 4, we can calculate P(0), P(1), P(2), P(3), and P(4) as follows:
P(0) = 300(1.05)^0 = 300
P(1) = 300(1.05)^1 = 315
P(2) = 300(1.05)^2 = 330.75
P(3) = 300(1.05)^3 = 347.29
P(4) = 300(1.05)^4 = 364.65
Now, let's plot these points on the graph. We can use a scatter plot for this purpose. On the x-axis, we will have the values of t, and on the y-axis, we will have the corresponding values of P(t). Connect the points to obtain the graph of P(t). This graph will be an exponential growth curve.
For the function N(t) = 1000(0.92)^t, we can follow the same process. Calculate the values of N(t) for different values of t and plot them on the same graph.
For example, if we take t = 0, 1, 2, 3, 4, we can calculate N(0), N(1), N(2), N(3), and N(4) as follows:
N(0) = 1000(0.92)^0 = 1000
N(1) = 1000(0.92)^1 = 920
N(2) = 1000(0.92)^2 = 846.4
N(3) = 1000(0.92)^3 = 778.91
N(4) = 1000(0.92)^4 = 716.38
Plot these points on the graph using the same axis system as before. Connect the points to obtain the graph of N(t). This graph will also be an exponential decay curve.
The domain of these functions is all real numbers, as time t can take any non-negative value. The range of P(t) is all positive real numbers greater than or equal to 300, as the population cannot be negative. The range of N(t) is all positive real numbers less than or equal to 1000, as the quantity of fish food cannot exceed the initial amount.
To find the point of intersection of these two curves, we need to solve the equation P(t) = N(t). Using the given functions, we have:
300(1.05)^t = 1000(0.92)^t
To solve this equation, we can take the logarithm of both sides. But since we want the answer to the hundredth, let's use an approximation method like the bisection method or Newton's method to find the solution. Once we find the value of t, we can substitute it into either P(t) or N(t) to find the corresponding value of P(t) = N(t).
Now, let's consider the function y = N(t) - P(t). This function represents the difference between the quantity of fish food and the population of fish at a given time t. If the value of y is positive, it means there is more fish food remaining than the fish population requires, indicating a surplus. If the value of y is negative, it means there is not enough fish food to sustain the population, indicating a shortage.
The abscissa at the origin, which is the value of t when y = 0, represents the time at which the crisis point occurs. This is the point in time when the fish population exceeds the amount of available fish food, leading to a crisis. To find the abscissa at the origin, we need to solve the equation y = 0, which is equivalent to solving the equation N(t) = P(t). We can use the same method mentioned earlier to approximate the solution.
Regarding the validity of the mathematical model, it provides a reasonable approximation of the population growth and fish food consumption. However, it assumes a constant growth and decay rate, which might not accurately represent real-world variations. Additionally, the model does not take into account external factors such as predation, diseases, or migration, which can affect the population dynamics. Therefore, while the model provides insights, it should be considered a simplified representation of the real-world scenario.
For values of t greater than the abscissa at the origin, the function P(t) will continue to grow exponentially, representing the increasing fish population. However, since the fish food cannot sustain this growth, the function N(t) will continue to decrease exponentially. The graph should show a widening gap between the two curves, indicating a worsening crisis as time progresses.