the average daily temperature, in degrees fahrenheit (°f), in a certain city can be modeled by the function…

the average daily temperature, in degrees fahrenheit (°f), in a certain city can be modeled by the function f given by f(t)=63.6 + 21.9 sin((2π/365)(t - c)), where t is the number of days since the beginning of the year and c is a constant. january 1 corresponds to t = 1. on january 10 (t = 10), the average temperature was 41.7°f. based on the model, what is the predicted average temperature on february 28 (t = 59)?

the average daily temperature, in degrees fahrenheit (°f), in a certain city can be modeled by the function f given by f(t)=63.6 + 21.9 sin((2π/365)(t - c)), where t is the number of days since the beginning of the year and c is a constant. january 1 corresponds to t = 1. on january 10 (t = 10), the average temperature was 41.7°f. based on the model, what is the predicted average temperature on february 28 (t = 59)?

Answer

Explanation:

Step1: Find the value of c

We know that when (t = 10), (f(10)=41.7). Substitute into the function (f(t)=63.6 + 21.9\sin\left(\frac{2\pi}{365}(t - c)\right)): [41.7=63.6+21.9\sin\left(\frac{2\pi}{365}(10 - c)\right)] First, subtract 63.6 from both - sides: [41.7−63.6 = 21.9\sin\left(\frac{2\pi}{365}(10 - c)\right)] [-21.9=21.9\sin\left(\frac{2\pi}{365}(10 - c)\right)] Then divide both - sides by 21.9: [\sin\left(\frac{2\pi}{365}(10 - c)\right)=- 1] We know that (\sin\theta=-1) when (\theta=\frac{3\pi}{2}+2k\pi,k\in\mathbb{Z}). Let (k = 0), then (\frac{2\pi}{365}(10 - c)=\frac{3\pi}{2}). Cross - multiply: (4(10 - c)=3\times365). [40-4c = 1095] [4c=40 - 1095=-1055] [c=\frac{-1055}{4}=-263.75]

Step2: Predict the temperature at (t = 59)

Substitute (t = 59) and (c=-263.75) into the function (f(t)=63.6 + 21.9\sin\left(\frac{2\pi}{365}(t - c)\right)): [f(59)=63.6+21.9\sin\left(\frac{2\pi}{365}(59+263.75)\right)] [=63.6+21.9\sin\left(\frac{2\pi}{365}\times322.75\right)] [=63.6+21.9\sin\left(\frac{645.5\pi}{365}\right)] [=63.6+21.9\sin(1.77\pi)] Since (\sin(1.77\pi)\approx - 0.707) [f(59)=63.6+21.9\times(-0.707)] [=63.6-15.4833] [=48.1167\approx48.1^{\circ}F]

Answer:

(48.1^{\circ}F)