When a person is swinging the amplitude (the swings"s distance above it's rest position) increases with each pump. According to Scientific magazine the amplitude increases by adding a constant when the person pumps sitting down, and by multiplying by a constant when the person pumps standing up. Suppose that when you get on a swing, its amplitude is 3cm (ie t(sub 0) =3). Assume that the amplitude increases by 2 cm each pump when you are sitting, and increases by multiplying by 1.2 each pump when you are standing.
a. what is your amplitude after 20 pumps if you are standing? If you are sitting?
b. What is the minimum number of pumps needed to make the amplitude equal 75 cm when you pump sitting down? Standing up?
C. What is the minimum number of pumps it takes to make the amplitude equal 25cm when you are sitting? Standing?
(a) after x pumps, we have
sit(x) = 3 + 2x
stand(x) = 3 * 1.2^x
Now just use those functions to solve for the needed values.
To solve these problems, we will use the formulas given in the problem and apply them iteratively based on the number of pumps given.
a) To find the amplitude after 20 pumps, we can use the formula:
Amplitude = Initial Amplitude * Multiplication Factor ^ Number of Pumps
i) If you are standing, the multiplication factor is 1.2 and the initial amplitude is 3cm.
Amplitude = 3 * 1.2 ^ 20
Calculating this equation will give you the amplitude after 20 pumps when you are standing.
ii) If you are sitting, the addition factor is 2 and the initial amplitude is 3cm.
Amplitude = 3 + (2 * 20)
Calculating this equation will give you the amplitude after 20 pumps when you are sitting.
b) To find the minimum number of pumps required to reach a specific amplitude, we can rearrange the formula as follows:
Number of Pumps = log (Final Amplitude / Initial Amplitude) / log (Multiplication Factor)
i) When pumping sitting down, the multiplication factor is 1 (since we're not multiplying, only adding), and the initial amplitude is 3cm while the final amplitude is 75cm.
Number of Pumps = log(75/3) / log(1 + 2)
Rounding the division up to the nearest whole number will give you the minimum number of pumps needed when sitting.
ii) When pumping standing up, the multiplication factor is 1.2, and the initial amplitude is 3cm while the final amplitude is 75cm.
Number of Pumps = log(75/3) / log(1.2)
Rounding the division up to the nearest whole number will give you the minimum number of pumps needed when standing.
c) To find the minimum number of pumps required to reach a specific amplitude, we will follow the same steps as in part b.
i) When pumping sitting down, the multiplication factor is 1 (since we're not multiplying, only adding), and the initial amplitude is 3cm while the final amplitude is 25cm.
Number of Pumps = log(25/3) / log(1 + 2)
Rounding the division up to the nearest whole number will give you the minimum number of pumps needed when sitting.
ii) When pumping standing up, the multiplication factor is 1.2, and the initial amplitude is 3cm while the final amplitude is 25cm.
Number of Pumps = log(25/3) / log(1.2)
Rounding the division up to the nearest whole number will give you the minimum number of pumps needed when standing.