Question

A toy boat is bobbing in the water.

Its distance D(t) (in m) from the floor of the lake as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form a×sin(b×t) + d.

At t=0, when the boat is exactly in the middle of its oscillation, it is 1 m above the water's floor. The boat reaches its maximum height of 1.2 m after pi/4 seconds.

Find D(t).

Answer 1

Answer:

$D(t) = 1\,m + (0.2\,m)\cdot \sin \left[\right(2\,\frac{rad}{s} \left)\cdot t\right]$

Step-by-step explanation:

The sinusoidal expression has the following form:

$D(t) = D+A\cdot \sin (B\cdot t)$

Where:

$D$ - Initial distance from the floor of the lake, in meters.

$A$ - Amplitude of oscillation, in meters.

$B$ - Angular frequency, in radians.

Now, each coefficient is derived as follows:

Initial distance from the floor of the lake

$D = 1\,m$

Amplitude of oscillation

$A = 1.2\,m - 1\,m$

$A = 0.2\,m$

Angular frequency

From the statement it is known that boat reaches its maximum height in a quarter of its oscillation. Then, the angular frequency is:

$B = \frac{\frac{1}{2}\pi \,rad}{\frac{1}{4}\pi \,s}$

$B = 2\,\frac{rad}{s}$

The expression is:

$D(t) = 1\,m + (0.2\,m)\cdot \sin \left[\right(2\,\frac{rad}{s} \left)\cdot t\right]$