Question

The curve of best fit to model the data is given below.

y=18,582(0.90)^x

Use the exponential model to complete the following statements. Round to the nearest whole number, if necessary.

There were ______ visitors on the opening weekend.

There were approximately _____ visitors on the 9th weekend.

The number of visitors was about 15,051 after ______ weekends.

Answer 1

1) There were 18582 visitors on the opening weekend.

2) There were approximately 7199 visitors on the 9th weekend.

3) The number of visitors was about 15,051 after 2 weekends.

Explanation:

Given : The curve of best fit to model the data is given below.

`y=18582(0.90)^x`

To find : Complete the statements,

1) There were _____ visitors on the opening weekend.

2) There were approximately _____ visitors on the 9th weekend.

3) The number of visitors was about 15,051 after ______ weekends.

Solution :

`y=18582(0.90)^x` is the exponential equation:

where, y is the number of visitors and x are the weekends since opening

1. Since the opening weekend is the day of the the park's opening.

So, replace x=0 in our exponential equation to get the number of visitors on the opening weekend:

`y=18582(0.90)^x\\y=18582(0.90)^0\\y=18582(1)\\y=18582`

Therefore, There were 18582 visitors on the opening weekend.

2. To find the approximate number of visitors on the ninth weekend.

We replace x=9 in our exponential equation:

`y=18582(0.90)^x\\y=18582(0.90)^9\\y=7199.0475`

Rounded to the nearest integer : y=7199

Therefore, There were approximately 7199 visitors on the 9th weekend.

3. Here, we know that the number of visitors is 15051

Since y represents the number of visitors.

So, replace y=15051 in our exponential equation and solve for x:

`y=18582(0.90)^x\\\\15051=18582(0.90)^x\\\\ (15051)/(18582) =0.90^x\\\\0.90^x= (5017)/(6194) \\\\ln(0.90^x)=ln((5017)/(6194))\\\\xln(0.90)=ln((5017)/(6194))\\\\x= (ln((5017)/(6194)))/(ln(0.90)) \\\\x=2.0002`

Rounded to the nearest whole number : x=2

Therefore, The number of visitors was about 15,051 after 2 weekends.