Question Type 15: Formulating sinusoidal models given descriptions Exercises

Find a model $y=A\sin(Bx)+D$ with amplitude $3$, period $180^\circ$, and midline $y=2$, assuming no horizontal shift.

Formulate a sinusoidal model of the form $y = A\\sin(Bx)+D$ that has amplitude $5$, period $90^\circ$, and principal axis $y=11$, with no phase shift.

Determine the sinusoidal model $y= A\sin\bigl(B(x - C)\bigr)+D$ with amplitude $4$, period $120^\circ$, midline $y=-1$, and phase shift $30^\circ$ to the right.

Write a model $y=A\sin\bigl(Bx + C\bigr)+D$ for a sinusoid with amplitude $2$, period $60^\circ$, midline $y=0$, and shifted $15^\circ$ to the left.

In radians, write $y=A\sin\bigl(B(x - C)\bigr)+D$ for a sinusoid with amplitude $5$, period $\tfrac{2\pi}{3}$, shifted $\tfrac{\pi}{4}$ to the left, and midline $y=1$.

Formulate $y=A\sin\bigl(B(x - C)\bigr)+D$ for a sinusoid with amplitude $6$, period $100^\circ$, midline $y=-2$, phase shift $20^\circ$ right, and reflected across its midline.

Tidal height $h(t)$ (in metres) has amplitude $2$, mean level $5$, high tide at $t=6$ h, low tide at $t=18$ h, with period $12$ h. Find $h(t)=2\sin\bigl(B(t - C)\bigr)+5$.

A sinusoid has maximum $8$ at $x=10^\circ$ and minimum $2$ at $x=100^\circ$, repeating every $180^\circ$. Find its equation in the form $y=A\sin\bigl(B(x - C)\bigr)+D$.

A daily temperature oscillation has amplitude $10^\circ$C, average $20^\circ$C, and peaks at 3 pm. Model the temperature $T(t)$ in the form $T=10\sin\bigl(B(t - C)\bigr)+20$, where $t$ is hours after midnight.

Model the average daylight hours $D(d)$ over a year with amplitude $2$, mean $12$, maximum on day $172$, and period $365$ days, in the form $D=2\sin\bigl(\tfrac{2\pi}{365}(d - C)\bigr)+12$.

A Ferris wheel has radius $20$ m, centre $22$ m above ground, and completes one revolution per minute. A passenger starts at the lowest point at $t=0$. Write the height $H(t)$ in metres as $H(t)=20\sin(Bt+\phi)+22$.

A mass on a spring undergoes simple harmonic motion described by $d(t)=A\sin(\omega t+\phi)$. If the amplitude is $3$ cm, period $4$ s, and it passes upward through equilibrium at $t=1$ s, find $d(t)$.