Question

W(t)W, left parenthesis, t, right parenthesis models the daily water level (in \text{cm}cmstart text, c, m, end text) at a pond in Arizona, ttt days after the hottest day of the year. Here, ttt is entered in radians. W(t) = 15\cos\left(\dfrac{2\pi}{365}t\right) + 43W(t)=15cos( 365 2π ​ t)+43W, left parenthesis, t, right parenthesis, equals, 15, cosine, left parenthesis, start fraction, 2, pi, divided by, 365, end fraction, t, right parenthesis, plus, 43 What is the first time after the hottest day of the year that the water level is 30 \text{ cm}30 cm30, start text, space, c, m, end text? Round your final answer to the nearest whole day.

Answer 1

Answer:

t=152 days (in radians)

Step-by-step explanation:

W(t)models the daily water level at a pond in Arizona, t days after the hottest day of the year. (t is entered in radian)

$W(t) = 15\cos\left(\dfrac{2\pi}{365}t\right) + 43$

We want to determine the first time,t at which the water level is 30cm.

When W(t)=30

$30 = 15\cos\left(\dfrac{2\pi}{365}t\right) + 43\\30-43=15\cos\left(\dfrac{2\pi}{365}t\right)\\-13=15\cos\left(\dfrac{2\pi}{365}t\right)\\-\dfrac{13}{15} =\cos\left(\dfrac{2\pi}{365}t\right)\\cos^{-1}(-\dfrac{13}{15})=\dfrac{2\pi}{365}t\right)\\cos^{-1}(\dfrac{13}{15})=\dfrac{2\pi}{365}t\right)$

$cos^{-1}(\dfrac{13}{15})=\dfrac{2\pi}{365}t\right)\\t=\dfrac{365}{2\pi}\cdot cos^{-1}(\dfrac{13}{15})\\t=152.16\\\approx 152 days \text{ (to the nearest whole day)}$

Answer 2

Answer:

216 days

Step-by-step explanation: