7. (16 points) a toy boat sits on the surface of lake scappoose. on a calm day the boat would sit 10m above…

7. (16 points) a toy boat sits on the surface of lake scappoose. on a calm day the boat would sit 10m above the lake bottom. today, a stiff breeze is blowing causing the boat to rise and fall regularly with the waves. the total difference in the height from the top of a wave to the bottom is 4m. the boat bobs up and down 15 times every minute. assume the boat begins at the calm lake level. a) sketch a graph that models the position of the boat in reference to the lake bottom. only one period of the function is necessary.

7. (16 points) a toy boat sits on the surface of lake scappoose. on a calm day the boat would sit 10m above the lake bottom. today, a stiff breeze is blowing causing the boat to rise and fall regularly with the waves. the total difference in the height from the top of a wave to the bottom is 4m. the boat bobs up and down 15 times every minute. assume the boat begins at the calm lake level. a) sketch a graph that models the position of the boat in reference to the lake bottom. only one period of the function is necessary.

Answer

Explanation:

Step1: Determine amplitude

The total height difference from top - to - bottom of wave is 4m. Amplitude $A=\frac{4}{2}=2m$.

Step2: Determine period

The boat bobs 15 times per minute. Frequency $f = 15$ per minute. Period $T=\frac{60}{15}=4$ seconds.

Step3: Determine vertical shift

On a calm day, the boat is 10m above the lake bottom, so vertical shift $D = 10m$.

Step4: Sketch the graph

Since the boat starts at the calm - lake level, we can use a sine or cosine function. Let's use a sine function $y = A\sin(\omega t)+D$, where $\omega=\frac{2\pi}{T}=\frac{2\pi}{4}=\frac{\pi}{2}$. The graph will start at the point $(0, 10)$ (because when $t = 0$, $y=2\sin(0)+10 = 10$), reach a maximum of $10 + 2=12m$ at $t = 1$ second ($y = 2\sin(\frac{\pi}{2})+10=12$), return to 10m at $t = 2$ seconds ($y = 2\sin(\pi)+10 = 10$), reach a minimum of $10-2 = 8m$ at $t = 3$ seconds ($y=2\sin(\frac{3\pi}{2})+10 = 8$) and return to 10m at $t = 4$ seconds ($y = 2\sin(2\pi)+10 = 10$). Mark the x - axis as time in seconds and the y - axis as height above the lake bottom in meters. Draw a smooth sine - like curve for one period from $t = 0$ to $t = 4$ seconds.

Answer:

Sketch a sine - like curve starting at $(0,10)$, reaching a maximum of 12 at $t = 1$, returning to 10 at $t = 2$, reaching a minimum of 8 at $t = 3$ and back to 10 at $t = 4$ on a graph with x - axis as time (seconds) and y - axis as height above lake bottom (meters).