Pre calculus question

Question

In a certain city the number of hours of daylight on day t (where t is the number of days after January 1) is modeled by the function

L(t)= 13 + 2.83 sin(2pi/365(t-80))

(a) Which days of the year have about 11 h of daylight? (Enter your answers as a comma-separated list.)

(b) How.many days of the year have more than 11 h of daylight?

Answer 1

L(t)= 13 + 2.83 sin((2π/365)(t - 80))

(a) Which days of the year have about 11 h of daylight? (Enter your answers as a comma-separated list.)

We set L(t) = 11 and solve for the angle in terms of t:

11 = 13 + 2.83sin(x); where x = (2π/365)(t - 80)

2.83sin(x) = -2

sin(x) = -0.7067

x = -0.7848 (reference angle in Q4)

Sine is negative in Q3 & Q4

x = π + 0.7848 (for Q3, add reference angle to π radians)

x = 2π - 0.7848 (for Q4, subtract reference angle from 2π radians)

x = 3.9264 rad. (Q3), 5.4983 rad. (Q4)

Replacing x with expression and solving for t:

(2π/365)(t- 80) = 3.9264 or (2π/365)(t - 80) = 5.4983

t – 80 = 228.1 or t – 80 = 319.4

x = 308.1 or 399.4

Since 399 is past one year subtract one year

x = 308; x = 399 - 365=34

Therefore day 34 and day 308 have approx. 11 hours of daylight

x (day of year) = 34, 308 (February 3, November 4)

(b) How many days of the year have more than 11 h of daylight?

Days between these days have more than 11 hours of daylight

No. of days = 308 – 34 - 1 = 273 days