Answer:
152 days
Step-by-step explanation:
Given:
w(t) = 15 cos ((2π / 365)t) + 43
To find first time after the hottest day of the year that the water levels is 30 cm, substitute with w(t) = 30 and solve for t
30 = 15 cos ((2π / 365)t) + 43
30 - 43 = 15 cos ((2π / 365)t)
15 cos ((2π / 365)t) = -13
cos ((2π / 365)t) = -13/15
(2π / 365)t = cos⁻¹ (-13/15) = 150.07° ≈ (5/6) π
t = (365/2) * (5/6) ≈ 152 days
So, the first time after the hottest day of the year that the water levels is 30 cm is after 152 days
Functions can be used to model real situations.
The first time after the hottest day of the year that the water levels is 30 cm is after 151 days
The function is given as:
[tex]\mathbf{W(t) = 15cos((2\pi/365)t) + 43}[/tex]
When water level is 30, we have:
W(t) = 30
So, the equation becomes
[tex]\mathbf{30 = 15cos((2\pi/365)t) + 43}[/tex]
Subtract 43 from both sides
[tex]\mathbf{-13 = 15cos((2\pi/365)t) }[/tex]
Divide both sides by 15
[tex]\mathbf{-\frac{13}{15} = cos((2\pi/365)t) }[/tex]
Take arccos of both sides
[tex]\mathbf{cos^{-1}(-\frac{13}{15} )= (2\pi/365)t }[/tex]
The arccos of -13/15 is approximately 150 degrees.
So, we have:
[tex]\mathbf{(2\pi/365)t = 150}[/tex]
Express as radians
[tex]\mathbf{(2\pi/365)t = 150 \times \frac{\pi}{180}}[/tex]
[tex]\mathbf{(2\pi/365)t = \frac{5\pi}{6}}[/tex]
Cancel out [tex]\pi[/tex]
[tex]\mathbf{(2/365)t = \frac{5}{6}}[/tex]
Multiply both sides by 365/2
[tex]\mathbf{t = \frac{5}{6} \times \frac{365}{2}}[/tex]
[tex]\mathbf{t = 152.1}[/tex]
Hence, the first time is 151 days.
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