Question

A weight is attached by a string to the end of a spring and is thrown upwards while a stopwatch is started at t=0t=0t, equals, 0 seconds. The weight starts oscillating vertically in a periodic way that can be modeled by a trigonometric function. The weight reaches a maximum height of 12 \text{ cm}12 cm12, start text, space, c, m, end text at t=1.5t=1.5t, equals, 1, point, 5 seconds and falls to a minimum height of 4 \text{ cm}4 cm4, start text, space, c, m, end text before returning to its maximum height at t=6.5t=6.5t, equals, 6, point, 5 seconds. Find the formula of the trigonometric function that models the height HHH of the weight ttt seconds after it was thrown upwards. Define the function using radians.

Answer 1

4 cos((
`(2)/(5)`
`\pi`)(t-1.5))+8

Explanation:

I got it on khan academy:

Answer 2

The function is defined as H (t) = 4 cos (2π/5 ( t - 1.5)) + 8

Explanation:

Solution

Let the function be a cosine function

H(t) a cos(b(t+c)) + d

Now,

The maximum height,is H max =12

The minimum height , is H min = 4

The amplitude, a is denoted by :

a= H max - H min/2

= 12 - 4/2 = 8/2 = 4

Thus,

The vertical shift , d is given by:

d = H max + H min/2

= 12 + 4 /2 = 16/2 = 8

The period T is given by,

T=6.5-1.5=5

So,

b is given by ,

b= 2π /T = 2π/5

The phase shift , c is given by :

since maximum height occur at 1.5 we get, c=-1.5

Therefore, our function is defined as:

H (t) = 4 cos (2π/5 ( t - 1.5)) + 8