Question

Vanessa collects Barbie dolls. She began with two dollars and added the same amount of dollars to her collection each year. In the 24th year, Vanessa had $98. Which function, d(n), can be used to determine the number of dollars Vanessa had in a year?

a) d(n) = 2n
b) d(n) = 24n
c) d(n) = 2 + 24n
d) d(n) = 2 + n

Answer 1

The correct function to determine the number of dollars Vanessa had in the nth year is (d_n = 2 + n).

Step-by-step explanation:

Vanessa's initial amount was $2, and each year she added the same amount of dollars to her collection. The function \(d_n = 2 + n\) accurately represents this scenario. The constant term '2' corresponds to her initial amount, and the variable term 'n' represents the number of years. In the 24th year, plugging in \(n = 24\) into the function gives \(d_{24} = 2 + 24 = 26\), which matches the given information that Vanessa had $98 in the 24th year.

The other options (\(d_n = 2n\), \(d_n = 24n\), and \(d_n = 2 + 24n\)) do not align with the problem's description. For instance, \(d_n = 2n\) would result in Vanessa having $48 in the 24th year, not $98. Therefore, the correct function is \(d_n = 2 + n\), which accurately represents the progression of Vanessa's collection over the years.

In conclusion, the function \(d_n = 2 + n\) is the appropriate model to determine the number of dollars Vanessa had in the nth year. The constant initial amount and the linear increase with each year make this function consistent with the given information about Vanessa's Barbie doll collection.