Question

Khan academy an object is attached by a string to the end of a spring. fang throws the object upwards and starts a stopwatch at t = 0 t=0t, equals, 0 seconds. the object starts oscillating vertically in a periodic way that can be modeled by a trigonometric function. the object's average height is − 20 cm −20 cmminus, 20, start text, space, c, m, end text (measured from the top of the spring). it first achieves that average height on the way up at t = 0.2 t=0.2t, equals, 0, point, 2 seconds, and then again every 2 22 seconds. the object's maximum and minimum heights are each 5 cm 5 cm5, start text, space, c, m, end text from its average height. find the formula of the trigonometric function that models the height h hh of the weight t tt seconds after fang started the stopwatch. define the function using radians.

Answer 1

Answer:

$H(t)= 5cos(\pi(t-0.2))-20$

Step-by-step explanation:

Let the function be a cosine function

H(t)= A cos(B(t+C)) + D

Average height = -20 cm

The object's maximum and minimum heights are each 5 cm from its average height. Therefore:

Maximum Height, $H_{max}=-20+5=-15cm$

Minimum Height ,  $H_{min}=-20-5=-25cm$

The amplitude, A is denoted by :

$A=\dfrac{H_{max}-H_{min}}{2} \\=\dfrac{-15-(-25)}{2}=(-15+25)\div 2 =5\\A=5$

The vertical shift, D is denoted by :

$D=\dfrac{H_{max}+H_{min}}{2} \\=\dfrac{-15+(-25)}{2}=(-15-25)\div 2 =-20\\D=-20$

The period T is given by,

T=2 seconds

$B=\dfrac{2\pi}{T}\\=\dfrac{2\pi}{2}=\pi$

The phase shift , C is given by:

C=-0.2 (since it first achieves its average height to the right at t=0.2 sec)

Therefore, our function is defined as:

$H(t)= 5cos(\pi(t-0.2))-20$