Answer:
The equation for height is
[tex]h=32sin(\frac{\pi}{2}t)+14[/tex]
Step-by-step 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
[tex]h=Asin(Bt)+D[/tex]
now, we can find A , B and D
Calculation of A:
maximum height 46 inch
minimum height =18 inch
so,
[tex]A=\frac{46+18}{2}=32[/tex]
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
[tex]\frac{T}{2}=2[/tex]
[tex]T=4[/tex]
now, we can use period formula
[tex]T=\frac{2\pi}{B}[/tex]
we can find B
[tex]4=\frac{2\pi}{B}[/tex]
[tex]B=\frac{2\pi}{4}[/tex]
[tex]B=\frac{\pi}{2}[/tex]
Calculation of D:
Max=46
min=18
[tex]D=\frac{46-18}{2}[/tex]
[tex]D=14[/tex]
now, we can plug these values into formula
and we get
[tex]h=32sin(\frac{\pi}{2}t)+14[/tex]
Answer:
Required model is [tex]h=32\sin(\frac{\pi}{2}t)+14[/tex]
Step-by-step 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,
[tex]h=A\sin(Bt)+D[/tex]
Now, We calculate A,B and D
1) Maximum height 46 inch
Minimum height =18 inch
Average height is A
[tex]A=\frac{46+18}{2}=32[/tex]
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, [tex]\frac{T}{2}=2[/tex]
[tex]T=4[/tex]
Period is B
[tex]T=\frac{2\pi}{B}[/tex]
[tex]4=\frac{2\pi}{B}[/tex]
[tex]B=\frac{2\pi}{4}[/tex]
[tex]B=\frac{\pi}{2}[/tex]
3) D is the midline
So, Max=46 and min=18
[tex]D=\frac{46-18}{2}[/tex]
[tex]D=14[/tex]
Substituting all the values,
A=32 , D=14 , [tex]B=\frac{\pi}{2}[/tex]
[tex]h=A\sin(Bt)+D[/tex]
[tex]h=32\sin(\frac{\pi}{2}t)+14[/tex]
Therefore, Required model is [tex]h=32\sin(\frac{\pi}{2}t)+14[/tex]
