Answer:
The maximum height a rider will experience is 55 feet.
Step-by-step explanation:
Let's start writing the function that defines the path of a seat on the new Ferris wheel. This function will depend of the variable ''t'' which is time.
In which are the coordinates of the seat (the x - coordinate and the y - coordinate) that depend from time.
and
are functions that depend from the variable ''t''.
For this exercise :
In order to find the maximum height a rider will experience we will study the behaviour of the y - component from the function .
The function to study is
To find its maximum, we will derivate this function and equalize it to 0. Doing this, we will find the ''critical points'' from the function.
⇒ ⇒
Now we equalize to 0 ⇒
⇒
In this case it is easier to look for the values of ''t'' that verify :
Now we need to find the values of ''t''. We know that :
Therefore we can write the following equivalent equations :
(I)
(II)
(III)
From (I) we obtain
From (II) we obtain
And finally from (III) we obtain
We found the three critical points of . To see if they are either maximum or minimum we will use the second derivative test. Let's calculate the second derivate of
:
⇒
Now given that we have an arbitrary critical point '''' ⇒
If then we will have a minimun at
If then we will have a maximum at
Using the second derivative test with and
⇒
⇒ We have a minimum for
⇒ We have a maximum for
⇒ We have a maximum for
The last step for this exercise will be to find the values of the maximums.
We can do this by replacing in the equation of the critical points
and
⇒
We found out that the maximum height a rider will experience is 55 feet.