Lesson 2 High Tide Solidify Understanding

Ready

Recall that the right triangle definition of the tangent ratio is:

$\tan A = \frac{length of side opposite angle A}{length of side adjacent to angle A}$

1.

Solve for $y$ . Then find $\tan A$ and $\tan B$ .

$y =$

$\tan A =$

$\tan B =$

2.

Solve for $x$ . Then find $\tan A$ and $\tan B$ .

$x =$

$\tan A =$

$\tan B =$

3.

Solve for $y$ . Then find $\tan A$ and $\tan B$ .

$y =$

$\tan A =$

$\tan B =$

4.

Solve for $y$ . Then find $\tan A$ and $\tan B$ .

$y =$

$\tan A =$

$\tan B =$

Set

Many real-life situations such as sound waves, weather patterns, and electrical currents can be modeled by sine and cosine functions. The table shows the depth of water (in feet) at the end of a wharf as it varies with the tides at various times during the morning.

$t$ (time)	midnight	$2$ a.m.	$4$ a.m.	$6$ a.m.	$8$ a.m.	$10$ a.m.	noon
$d$ (depth)	$8.16$	$12.16$	$14.08$	$12.16$	$8.16$	$5.76$	$7.26$

5.

Sketch the line that shows the average depth.

6.

Find the amplitude. $A = \frac{1}{2} (high - low)$

7.

Find the period. $p = 2 | low time - high time |$ . Since a normal period for sine is $2 π$ , the new period for our model will be $\frac{2 π}{p}$ , so $b = \frac{2 π}{p}$ .

8.

High tide occurred $4$ hours after midnight. The formula for the displacement is $4 = \frac{c}{b}$ . Use $b$ and solve for $c$ .

9.

Now that you have your values for $A$ , $b$ , $c$ , and $d$ , put them into an equation. $y = A \cos (b t - c) + d$

10.

Use your model to calculate the depth at 9 a.m. and 3 p.m.

a.

$9 a.m. =$

b.

$3 p.m. =$

11.

A boat needs at least $10$ feet of water to dock at the wharf. During what interval of time in the afternoon can it safely dock?

Go

Use the given solutions of a quadratic function and the $y$ -intercept to find the original equation.

12.

$x = - 2 \pm i$ with $y$ -intercept $(0, 5)$

13.

$x = - 3 \pm i \sqrt{6}$ with $y$ -intercept $(0, 30)$

14.

$x = - 5 \pm 5 i \sqrt{2}$ with $y$ -intercept $(0, 75)$

15.

$x = - 1 \pm 4 i$ with $y$ -intercept $(0, 51)$