Hiro is riding a carousel that is next to a wall. His horizontal distance C(t) (in m ) away from the wall as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form a*cos(b*t)+d. At t=0, when he starts, he is closest to the wall, a distance of 2m away. After 7pi seconds he reaches his mid-way point from the wall, which is 7m away. Find C(t). t should be in radians. C(t)=

Start at Maximum Value: Since Hiro is closest to the wall at t=0, the cosine function starts at its maximum value. The amplitude a is the maximum distance from the average value, which is 2m in this case.Average Value Calculation: The mid-way point after 7pi seconds is 7m away from the wall. This is the average value of the function, so d equals 7m.Finding Cycle Time: To find the value of b, we use the fact that the cosine function completes a full cycle in 2pi/b seconds. Since Hiro reaches his mid-way point after 7pi seconds, this must be half of the full cycle time. So, 2pi/b = 2 * 7pi, which means b = 1/7.

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Hiro is riding a carousel that is next to a wall.
His horizontal distance
C(t) (in
m ) away from the wall as a function of time
t (in seconds) can be modeled by a sinusoidal expression of the form
a*cos(b*t)+d.
At
t=0, when he starts, he is closest to the wall, a distance of
2m away. After
7pi seconds he reaches his mid-way point from the wall, which is
7m away.
Find
C(t).

t should be in radians.

C(t)=

Hiro is riding a carousel that is next to a wall. $\newline$ His horizontal distance $C(t)$ (in $\mathrm{m}$ ) away from the wall as a function of time $t$ (in seconds) can be modeled by a sinusoidal expression of the form $a \cdot \cos (b \cdot t)+d$ . $\newline$ At $t=0$ , when he starts, he is closest to the wall, a distance of $2 \mathrm{~m}$ away. After $7 \pi$ seconds he reaches his mid-way point from the wall, which is $7 \mathrm{~m}$ away. $\newline$ Find $C(t)$ . $\newline$ $t$ should be in radians. $\newline$ $C(t)=$

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Q. Hiro is riding a carousel that is next to a wall.

\newline

His horizontal distance

C(t)

(in

\mathrm{m}

) away from the wall as a function of time

t

(in seconds) can be modeled by a sinusoidal expression of the form

a \cdot \cos (b \cdot t)+d

\newline

t=0

, when he starts, he is closest to the wall, a distance of

2 \mathrm{~m}

away. After

7 \pi

seconds he reaches his mid-way point from the wall, which is

7 \mathrm{~m}

away.

\newline

Find

C(t)

\newline

t

should be in radians.

\newline

C(t)=

Start at Maximum Value: Since Hiro is closest to the wall at $t=0$ , the cosine function starts at its maximum value. The amplitude $a$ is the maximum distance from the average value, which is $2\,\text{m}$ in this case.
Average Value Calculation: The mid-way point after $7\pi$ seconds is $7m$ away from the wall. This is the average value of the function, so $d$ equals $7m$ .
Finding Cycle Time: To find the value of $b$ , we use the fact that the cosine function completes a full cycle in $\frac{2\pi}{b}$ seconds. Since Hiro reaches his mid-way point after $7\pi$ seconds, this must be half of the full cycle time. So, $\frac{2\pi}{b} = 2 \times 7\pi$ , which means $b = \frac{1}{7}$ .