A Trigonometric Model In The Park

Geometry Level 4

Michaelle decided to go to the playground. She saw two boys at the swing set, composed of a board and two seats.

Michaelle obtained the following mathematical model for height $h$ above the ground of a seat under which she put a location sensor (in dark brown, picture below):

$h(t)=\displaystyle\dfrac{1}{2}\cdot\left[\sin\left(\dfrac{\pi}{4}t-\dfrac{\pi}{4}\right)\right]+\dfrac{4}{5}$

where $h$ is expressed in metres and time $t$ in seconds.

If the period of this motion is $A$ , the initial distance from the ground to the seat is $B$ , and the center of oscillation is $C$ , what is $A+B+C$ (rounded to the nearest hundredth)?

The answer is 9.25.

1 solution

Chew-Seong Cheong
Apr 25, 2016

Relevant wiki: Graphs of Trigonometric Functions - Problem Solving - Intermediate

The period of the motion $A$ is given by: $\dfrac{\pi}{4} t = 2 \pi \implies A = t = 8$ s.

The initial distance $B = h(0) = \dfrac{1}{2}\sin \left(-\dfrac{\pi}{4} \right) + \dfrac{4}{5} \approx 0.4464$

The extreme positions are given when $\sin \left(\dfrac{\pi}{4}t-\dfrac{\pi}{4} \right) = \pm 1$ , therefore the center of oscillation $C = \dfrac{4}{5} = 0.8$

Therefore, $A+B+C \approx \boxed{9.25}$

Extreme positions would be $\pm \frac {1}{2}$ instead of $\pm {1}$

M Abdul Muiz - 5 years, 1 month ago

Thanks, I have mistakenly included $\frac{1}{2}$ .

Chew-Seong Cheong - 5 years, 1 month ago

A Trigonometric Model In The Park

The answer is 9.25.

1 solution

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