Applying Trigonometric Functions to Real-World Situations

We are given the functions that model the population of lions L and deer D over the years. L&=9+5.5sinπ/4t D&=27+18.5cos π/4t We want to determine the ratio of deer to lions when t=0, 2, 4, and 6. Let's first calculate the population of lions and deer in these years.

	L = 9 + 5.5 sin π/4 t		D = 27 + 18.5 cos π/4 t
Value of t	Substitute	Simplify	Substitute	Simplify
t = 0	L = 9 +5.5 sin π/4 ( 0)	L = 9	D = 27 +18.5 cos π/4 ( 0)	D = 45.5
t = 2	L = 9 +5.5 sin π/4 ( 2)	L = 14.5	D = 27 +18.5 cos π/4 ( 2)	D = 27
t = 4	L = 9 +5.5 sin π/4 ( 4)	L = 9	D = 27 +18.5 cos π/4 ( 4)	D = 8.5
t = 6	L = 9 +5.5 sin π/4 ( 6)	L = 3.5	D = 27 +18.5 cos π/4 ( 6)	D = 27

To find the ratio of deer to lions, we need to calculate DL. Let's substitute the values of L and D for t=0, 2, 4, and 6 that we have found and evaluate the ratios.

	D/L
Value of t	Substitute	Simplify
t=0	D/L=45.5/9	D/L≈ 5.06
t=2	D/L=27/14.5	D/L≈ 1.86
t=4	D/L=8.5/9	D/L≈ 0.94
t=6	D/L=27/3.5	D/L≈ 7.71

We are given a graph that shows the changes in the lion and deer populations and want to determine how these changes are related. Let's start by identifying the period of the graph of the lion population L.

As we can see, the period is 8 years. This means that every 8 years the pattern in the population of lions repeats. Now, let's identify the period of the graph of the deer population.

In this case, the period is also about 8 years. Therefore, the pattern in the population of deer also repeats every 8 years. To see how the changes in the two populations appear to be related, let's examine the parts of the graph where the deer population decreases and increases and see what happens to the lion population during these periods of time.

The Population of Deer Decreases

Let's have a look at the first 2 years on the graph.

We can see that in the first 2 years the population of deer decreases and the population of lions increases. Since lions are the predators who eat deer, the prey, the population of deer decreases, but the population of lions increases, because they have access to food. Now, let's have a look at the next 2 years.

In this period of time, the population of lions starts to decrease as well along with the population of deer. This change might be due to the fact that the population of the deer decreases, so lions have less food and, as a result, their population also decreases.

The Population of Deer Increases

Now, let's have a look at first 2 years when the population of deer starts to increase.

The population of deer starts to increase 2 years after the population of lions started to decrease. The reason for this might be the fact that there are less predators, so the population of prey have an opportunity to reproduce and grow their population.

Looking at the next 2 years on the graph, we can see that that the population of lions started to increase again. This is because there are more prey, so the predators have more food and, as a result, their population starts to increase again. Therefore, the correct answer is D.

On the graph, we can see that this pattern, which consists of the four discussed periods, continues in the next 8 years.

Function rule	What does the function model?
$R (t) = \frac{9 0 0 sin 2 t}{3 2}$	The horizontal distance from a player to a basketball thrown with an initial velocity of $30$ feet per second at an angle with measure $t .$ Negative distance means that the direction of the throw changes.
$s (t) = 3.5 sin \frac{π}{1 8 0} (t + 4) + 15.7$	The average wind speed in a certain city, measured in miles per hour.
$P (t) = 100 - 20 cos \frac{8 π}{3} t$	The blood pressure of a person at rest, measured in millimeters of mercury.
$N (t) = 3.7 sin (\frac{4 t}{5} - 0.7) + 20$	The number of thousands of employees at a sports company.

	Average Monthly Water Temperatures
$t$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$
$D (^{\circ} F)$	$70$	$73$	$75$	$78$	$82$	$85$	$86$	$85$	$84$	$81$	$75$	$73$

Number of Rotations	Time
$3$	$1 minute = 60 seconds$
$1$	$\frac{6 0}{3} = 20$ seconds

Transformations of $y = cos x$
Vertical Stretch or Shrink	$Vertical stretch, a > 1 y = a cos x$
Vertical Stretch or Shrink	$Vertical shrink, 0 < a < 1 y = a cos x$
Horizontal Stretch or Shrink	$Horizontal shrink, b > 1 y = cos (b x)$
Horizontal Stretch or Shrink	$Horizontal stretch, 0 < b < 1 y = cos (b x)$

$x$	$y$
$1$	$7.61$
$2$	$7.52$
$3$	$6.86$
$4$	$5.99$
$5$	$5.18$
$6$	$4.67$
$7$	$4.68$
$8$	$5.11$
$9$	$5.67$
$10$	$6.23$
$11$	$6.82$
$12$	$7.41$

	12 Theory slides
	8 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

	Average Monthly Precipitation
$t (months)$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$
$P (inches)$	$0.5$	$0.6$	$0.9$	$1.5$	$2.2$	$2.6$	$2.7$	$2.3$	$1.6$	$1.2$	$0.7$	-

Applying Trigonometric Functions to Real-World Situations

Catch-Up and Review

Hint

Solution

Answer

Hint

Solution

Extra

Hint

Solution

Hint

Solution

Period

Amplitude

Hint

Solution

Answer

Hint

Solution

Answer

Hint

Solution

Scatter Plot

Model

Graph of the Function

Period

The Population of Deer Decreases

The Population of Deer Increases

Applying Trigonometric Functions to Real-World Situations

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