A herd of 100 deer is introduced onto a small island. At first the herd increases rapidly, but eventually food resources dwindle and the population declines. Suppose the number of deer, N(t), after t years is described by the polynomial function: N(t)+ -t^4 +21t^2 +100, where t>0.
a) use the leading coefficent test to determine the graph's end behavior to the right.
b.) Use the rational zero test to determine the zeros. When will they extinct?
c.) what are the values of t for which N(t) >0
d.) What year does the population decline?
(a) even power, negative coefficient means heading down on both sides.
(b) what are the factors of 100?
(c) between 1st two real roots and the last two roots. Or, between the two real roots of there are only two.
(d) do you have calculus to use?
N declines where N' < 0.
Use what you know about the general shape of quartics to guide you along the way.
a) To determine the end behavior of the graph to the right, we need to look at the leading coefficient of the polynomial function. In this case, the leading coefficient is -1.
If the leading coefficient is negative, the graph will approach negative infinity as t approaches positive infinity. This means that as time goes on, the number of deer in the population will decrease to a very small or negative value.
b) To find the zeros of the polynomial, we can use the rational zero test. The rational zero test states that if a rational number p/q is a zero of a polynomial function, then p must be a factor of the constant term (in this case, 100), and q must be a factor of the leading coefficient (in this case, -1).
The factors of 100 are ±1, ±2, ±4, ±5, ±10, ±20, ±25, ±50, ±100.
The factors of -1 are ±1.
So, the possible rational zeros are ±1, ±2, ±4, ±5, ±10, ±20, ±25, ±50, ±100.
To determine if any of these are actually zeros, we can substitute them into the polynomial and check if the result is equal to zero. For example, if we substitute t = 1 into the polynomial N(t), we get:
N(1) = -(1)^4 + 21(1)^2 + 100 = -1 + 21 + 100 = 120. Since the result is not zero, t = 1 is not a zero.
We would need to do this for each possible rational zero until we find the zeros of the polynomial. Once we find the zeros, we can determine when the population will go extinct.
c) To determine the values of t for which N(t) > 0, we need to find the intervals where the polynomial function is positive. This can be done by factoring the polynomial or using a graphing calculator. By factoring the polynomial function, we find that N(t) = (t + 5)(t - 5)(t^2 + 4).
Setting each factor equal to zero, we find that the zeros are t = -5, t = 5, and t = ±2i (complex zeros). Since we are only interested in real values of t, we can ignore the complex zeros.
Now we can analyze the intervals between the zeros to determine when N(t) > 0. We test a value within each interval to see if N(t) is positive or negative. For example, for the interval t < -5, we could test t = -6. Plugging this into the polynomial, we get N(-6) = (-6 + 5)(-6 - 5)(36 + 4) = (-1)(-11)(40) = 440, which is positive.
By testing values within each interval, we can determine when N(t) is positive.
d) The population will start to decline when the polynomial function N(t) becomes negative. From the previous analysis, we know that the population is positive for t < -5 and t > 5. So, the population will start to decline sometime between t = -5 and t = 5. To find the exact year, we would need to analyze the behavior of the polynomial function within that interval, such as finding critical points or using calculus techniques.