Question

An iron ball is bobbing up and down on the end of a spring. The maximum height of the ball is 46 inches and its minimum height is 18 inches. It takes the ball 2 seconds to go from its maximum height to its minimum height. Which model best represents the height, h, of the ball after t seconds?

Answer 1

Required model is
`h=32\sin((\pi)/(2)t)+14`

Explanation:

Given : An iron ball is bobbing up and down on the end of a spring. The maximum height of the ball is 46 inches and its minimum height is 18 inches. It takes the ball 2 seconds to go from its maximum height to its minimum height.

To find : Which model best represents the height, h, of the ball after t seconds?

Solution :

According to question,

The height function must be trigonometric in nature.

So, we can use formula,

`h=A\sin(Bt)+D`

Now, We calculate A,B and D

1) Maximum height 46 inch

Minimum height =18 inch

Average height is A

`A=(46+18)/(2)=32`

2) It takes the ball 2 seconds to go from its maximum height to its minimum height

So, half of time period is 2 sec

i.e,
`(T)/(2)=2`

`T=4`

Period is B

`T=(2\pi)/(B)`

`4=(2\pi)/(B)`

`B=(2\pi)/(4)`

`B=(\pi)/(2)`

3) D is the midline

So, Max=46 and min=18

`D=(46-18)/(2)`

`D=14`

Substituting all the values,

A=32 , D=14 ,
`B=(\pi)/(2)`

`h=A\sin(Bt)+D`

`h=32\sin((\pi)/(2)t)+14`

Therefore, Required model is
`h=32\sin((\pi)/(2)t)+14`

Answer 2

The equation for height is

`h=32sin((\pi)/(2)t)+14`

Explanation:

we are given

An iron ball is bobbing up and down on the end of a spring

So, height function must be trigonometric in nature

So, we can use formula

`h=Asin(Bt)+D`

now, we can find A , B and D

Calculation of A:

maximum height 46 inch

minimum height =18 inch

so,

`A=(46+18)/(2)=32`

Calculation of B:

It takes the ball 2 seconds to go from its maximum height to its minimum height

So, half of time period is 2 sec

`(T)/(2)=2`

`T=4`

now, we can use period formula

`T=(2\pi)/(B)`

we can find B

`4=(2\pi)/(B)`

`B=(2\pi)/(4)`

`B=(\pi)/(2)`

Calculation of D:

Max=46

min=18

`D=(46-18)/(2)`

`D=14`

now, we can plug these values into formula

and we get

`h=32sin((\pi)/(2)t)+14`