Aditya's dog routinely eats Aditya's leftovers, which vary seasonally. As a result, his weight fluctuates throughout the year. The dog's weight W(t) (in kg ) as a function of time t (in days) over the course of a year can be modeled by a sinusoidal expression of the form a*cos(b*t)+d. At t=0, the start of the year, he is at his maximum weight of 9.1kg. One-quarter of the year later, when t=91.25, he is at his average weight of 8.2kg. Find W(t). t should be in radians. W(t)=

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Aditya's dog routinely eats Aditya's leftovers, which vary seasonally. As a result, his weight fluctuates throughout the year. The dog's weight  W(t) (in  kg ) as a function of time  t (in days) over the course of a year can be modeled by a sinusoidal expression of the form  a*cos(b*t)+d. At  t=0, the start of the year, he is at his maximum weight of  9.1kg. One-quarter of the year later, when  t=91.25, he is at his average weight of  8.2kg. Find  W(t).  t should be in radians.  W(t)=

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Identify Maximum and Average Weight: Identify the maximum weight and the average weight from the given information. The maximum weight is given as 9.1 kg at t=0, and the average weight is given as 8.2 kg at t=91.25 days.Determine Amplitude: Determine the amplitude of the sinusoidal function. The amplitude (a) is the difference between the maximum weight and the average weight. a = 9.1 kg - 8.2 kg = 0.9 kgDetermine Vertical Shift: Determine the vertical shift (d) of the sinusoidal function. The vertical shift is the average weight, which is given as 8.2 kg. d = 8.2 kgDetermine Period: Determine the period of the sinusoidal function. Since the weight fluctuates over the course of a year, the period is one year. In days, this is 365 days. However, we need to convert this period into radians since the problem asks for t in radians. The period (T) in radians is given by T = 2π/b, where b is the frequency.Calculate Frequency: Calculate the frequency (b) using the period in days. We know that one complete cycle occurs in 365 days, so we can find b by rearranging the period formula: T = 2π/b 365 = 2π/b b = 2π/365Write Sinusoidal Function: Write the sinusoidal function using the values for amplitude (a), frequency (b), and vertical shift (d). W(t) = a*cos(b*t) + d W(t) = 0.9*cos((2π/365)*t) + 8.2Verify Conditions: Verify that the function meets the given conditions. At t=0, W(0) should be the maximum weight: W(0) = 0.9*cos((2π/365)*0) + 8.2 = 0.9*1 + 8.2 = 9.1 kg At t=91.25, W(91.25) should be the average weight: W(91.25) = 0.9*cos((2π/365)*91.25) + 8.2 Since cos((2π/365)*91.25) = cos(π/2) = 0, we get: W(91.25) = 0.9*0 + 8.2 = 8.2 kg Both conditions are satisfied.

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