William W. answered • 03/09/21
Experienced Tutor and Retired Engineer
An undamped spring would oscillate like D(t) = -9cos(26πt) assuming the equilibrium point is "0" and below the equilibrium point is negative. But damping the spring adds an "eλt" damping factor. To achieve the 16% decrease each second λ = -0.1744 so the equation becomes:
D(t) = -(e-0.1744t)(9cos(26πt))
For the rabbits, without oscillation the average is modeled by P = 170t + 700 (t in years) but we are asked for an equation in months so P = (170/12)t + 700.
The oscillation is required to initially be negative (January is a minimum) so we would need to add a negative cosine wave with a period of 12 months. That makes the equation:
P(t) = (170/12)t + 700 - 33cos((2π/12)t) or (simplifying)
P(t) = (85/6)t + 700 - 33cos((π/6)t)
William W.
You don't say which is wrong so I'll guess the 1st one. Try using the same equation without the negative. Distance can be thought of as just the magnitude (without the sign).03/09/21
William W.
You can also add more significant figures to the lambda. It is -0.1743533871503/09/21
Yazmine W.
The last one is saying its wrong03/11/21
William W.
It’s most likely a data entry error. I suggest you take the equation I wrote and copy/paste it into Desmos and look at the graph. You will see it does exactly what the word problem states. At t=0 (January), the function value is 667 which is 33 below the average of 700. After 6 months, the average grows by 85 (half of the 170 for the year) and the function models the population as 85 + 700 +33 (which is 33 rabbits above the average as stated in the problem). You can check other values to ensure to yourself that it is correct.03/11/21
William W.
You might try adding putting the "t" with the "pi" like this: P(t) = (85/6)t + 700 - 33cos(πt/6)03/11/21
Yazmine W.
It saying its wrongs03/09/21