3.5: Modeling with Sinusoids | AP Precalculus

How it works

Choose a sine or cosine model from the range, repeat time, and starting position. The coefficient of time uses radians per time unit. Include the physical domain and check whether the initial value and direction agree with the context.

H(t)=D+A\cos\left(\frac{2\pi}{T}t\right)

Worked examples

Example 1

A wheel seat ranges from $\displaystyle 2$ to $\displaystyle 18$ m and repeats every $\displaystyle 40$ s, starting at the top. Model height.

Midline $\displaystyle 10$ , amplitude $\displaystyle 8$ , period $\displaystyle 40$ : $\displaystyle H(t)=10+8\cos(\pi t/20)$ for $\displaystyle t\ge0$ .

Example 2

Using that model, find height at $\displaystyle t=20$ s.

$\displaystyle H(20)=10+8\cos\pi=2$ m, the bottom.

Problems

Problem 3.5.1

Water depth is $\displaystyle D(t)=5+2\cos(\pi t/6)$ meters with t in hours. Find maximum, minimum, and repeat time.

Depth oscillates between 3 and 7 meters over a 12-hour repeat time. — Time is in hours and depth in meters.

Show worked solution

The maximum is $\displaystyle 7$ m, minimum $\displaystyle 3$ m, and period $\displaystyle 2\pi/(\pi/6)=12$ hours.

Depth oscillates between 3 and 7 meters over a 12-hour repeat time. The solution adds labeled relationships. — Maximum depth = 7 m, minimum = 3 m, period = 12 h.

Problem 3.5.2

For the depth model, find depth at $\displaystyle t=3$ and $\displaystyle t=6$ hours.

Show worked solution

$\displaystyle D(3)=5+2\cos(\pi/2)=5$ m; $\displaystyle D(6)=5+2\cos\pi=3$ m.

Problem 3.5.3

A temperature cycles from $\displaystyle 12$ to $\displaystyle 28$ degrees every $\displaystyle 24$ hours and is maximal at $\displaystyle t=0$ . Give a model.

Show worked solution

$\displaystyle T(t)=20+8\cos(\pi t/12)$ , with t in hours.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 3.5.1

Problem 3.5.2

Problem 3.5.3

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