Question

[50 Points] The function
`y = 4.2√(0.28 x - 1)` models the annual number of visitors to an amusement park, where y is the number of visitors in millions and x is the number of years since 1975. Find the first year in which the amusement park had 12 million visitors?

Answer 1

In 2008.

Explanation:

The function:

`y=4.2√(0.28x-1)`

Models the annual number of visitors to an amusement park, where y is the number of visitors (in millions) and x is the number of years since 1975.

We want to determine the first year in which the amusement park had 12 million visitors.

So, we will set y equal to 12 and solve for x. This yields:

`12=4.2√(0.28x-1)`

By dividing both sides by 4.2, we acquire:

`\displaystyle (12)/(4.2)=(120)/(42)=(20)/(7)=√(0.28x-1)`

And by squaring both sides:

`\displaystyle (400)/(49)=0.28x-1`

Adding 1 to both sides produces:

`\displaystyle (400)/(49)+1=(449)/(49)=0.28x`

And finally, dividing both sides by 0.28 gives us the approximation:

`\displaystyle x=(449)/(49)\cdot (1)/(.28)=(11225)/(343)\approx32.72\approx33`

So, 33 years after 1975, the amusement park will have 12 million annual visitors.

Therefore, the first year in which the amusement park will have 12 million annual visitors is in the year 2008.