the depth of the water at the end of a pier changes periodically along with the movement of tides. on a…

the depth of the water at the end of a pier changes periodically along with the movement of tides. on a particular day, low tides occur at 12:00 am and 12:30 pm, with a depth of 2.5 m, while high tides occur at 6:15 am and 6:45 pm, with a depth of 5.5 m. let t = 0 be 12:00 am.\ngraph the equation (d=-1.5cos(\frac{4pi}{25}t)+4) that models the situation using a graphing calculator and use it to answer the following questions.\nhow many times during this day is the depth at the end of the pier equal to 4 meters?\n2 times\n3 times\n4 times\n5 times\ndone

the depth of the water at the end of a pier changes periodically along with the movement of tides. on a particular day, low tides occur at 12:00 am and 12:30 pm, with a depth of 2.5 m, while high tides occur at 6:15 am and 6:45 pm, with a depth of 5.5 m. let t = 0 be 12:00 am.\ngraph the equation (d=-1.5cos(\frac{4pi}{25}t)+4) that models the situation using a graphing calculator and use it to answer the following questions.\nhow many times during this day is the depth at the end of the pier equal to 4 meters?\n2 times\n3 times\n4 times\n5 times\ndone

Answer

Explanation:

Step1: Set up the equation

Set $d = 4$ in the equation $d=-1.5\cos(\frac{4\pi}{25}t)+4$. So we get $4=-1.5\cos(\frac{4\pi}{25}t)+4$.

Step2: Simplify the equation

Subtract 4 from both sides: $0=-1.5\cos(\frac{4\pi}{25}t)$. Then $\cos(\frac{4\pi}{25}t)=0$.

Step3: Solve for $t$

We know that $\cos x = 0$ when $x=(2n + 1)\frac{\pi}{2}$, $n\in\mathbb{Z}$. So $\frac{4\pi}{25}t=(2n + 1)\frac{\pi}{2}$. Cross - multiply to get $8t = 25(2n + 1)$, then $t=\frac{25(2n + 1)}{8}$.

Step4: Find the number of solutions in a day

A day has 24 hours. We need to find non - negative values of $t$ less than 24. When $n = 0$, $t=\frac{25}{8}=3.125$. When $n = 1$, $t=\frac{25\times3}{8}=9.375$. When $n = 2$, $t=\frac{25\times5}{8}=15.625$. When $n = 3$, $t=\frac{25\times7}{8}=21.875$. When $n = 4$, $t=\frac{25\times9}{8}=28.125>24$.

Answer:

4 times