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)= ◻
Q. 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)= ◻
Identify Maximum and Average Weight: Identify the maximum weight and the average weight from the given information.The maximum weight is given as 9.1kg at t=0, which is the start of the year. This means that the value of d (the vertical shift) is the average weight, and the amplitude a is the difference between the maximum weight and the average weight.
Calculate Amplitude: Calculate the amplitude ( extit{a}) of the sinusoidal function.The amplitude is the difference between the maximum weight and the average weight.a=maximum weight−average weighta=9.1kg−8.2kga=0.9kg
Determine Value of d: Determine the value of d, which is the average weight.Since the dog's weight is at its average at t=91.25 days, and this is one-quarter of the year, the average weight is also the vertical shift of the sinusoidal function.d=average weightd=8.2kg
Calculate Value of b: Calculate the value of b, which is related to the period of the sinusoidal function.Since the dog's weight fluctuates over the course of a year, the period of the sinusoidal function is one year. In days, this is 365 days. Since t=91.25 corresponds to one-quarter of the year, the cosine function should complete one-quarter of its cycle by t=91.25.The period T of a cosine function is given by T=b2π.Therefore, b=T2π.Since one-quarter of the period corresponds to t=91.25, we have:4b2π=91.25b=4×91.252π3650
Write Sinusoidal Function: Write the equation of the sinusoidal function W(t).We have determined that a=0.9, b=182.5π, and d=8.2. The function is a cosine function, which starts at its maximum value at t=0. Therefore, the phase shift is 0.The equation of the sinusoidal function is:W(t)=acos(b⋅t)+dW(t)=0.9cos(182.5π⋅t)+8.2
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