Question

Type the correct answer in each box. use numerals instead of words. for these questions, non-integer answers should be entered as a decimal rounded to the hundredths place. samantha won a trip to europe in a contest she had entered. rather than take the trip, she opted to take the cash value of the prize and put it in a savings account she uses as a college fund. she plans to keep this money in the college fund and let it earn interest for 8 years. at the end of 8 years, she’ll use the total balance to help pay for tuition at law school. the equation below represents the total dollars, f(x), in her college fund after x years. f ⁡ ( x ) = 5 , 450 ⁢ ( 1.024 ) x the domain for the function that represents the amount of time the money will be in the college fund is ≤ x ≤ . the range for the function that represents the amount of money that is in the college fund is ≤ f(x) ≤.

Answer 1

The correct answer for the range is
`\( 5,450 \leq f(x) \leq 6,641.29 \)`.

The equation representing the total dollars,
`\( f(x) \)`, in Samantha's college fund after
`\( x \)` years is given by:

`\[ f(x) = 5,450 * (1.024)^x \]`

1. Domain:

- The domain is the set of all possible values for
`\( x \)`. Since Samantha plans to keep the money in the college fund for 8 years, the domain is
`\( 0 \leq x \leq 8 \)`.

So, the correct answer for the domain is:
`\( 0 \leq x \leq 8 \)`.

2. Range:

- The range is the set of all possible values for
`\( f(x) \)`. Since the initial amount is $5,450 and it's compounded annually at a rate of 2.4% for 8 years, the range is
`\( 5,450 \leq f(x) \leq \)` (the maximum value after 8 years).

To find the maximum value, plug in
`\( x = 8 \)` into the function:

`\[ f(8) = 5,450 * (1.024)^8 \]`

To calculate the value of
`\( f(8) \)`, substitute
`\( x = 8 \)` into the function:

`\[ f(8) = 5,450 * (1.024)^8 \]`

Calculating this:

`\[ f(8) = 5,450 * 1.21899 \]`

`\[ f(8) \approx 6,641.285 \]`

Therefore, the calculated value is approximately $6,641.29.

So, the correct answer for the range is
`\( 5,450 \leq f(x) \leq 6,641.29 \)`.