Question

Aurora is planning to participate in an event at her school's field day requires her to complete tasks various stations in the fastest time possible.

Modeling
ANSWERED
asked 2021-08-14
Aurora is planning to participate in an event at her school's field day that requires her to complete tasks at various stations in the fastest time possible. To prepare for the event, she is practicing and keeping track of her time to complete each station. The x-coordinate is the station number, and the y-coordinate is the time in minutes since the start of the race that she completed the task. \(\displaystyle{\left({1},\ {3}\right)},\ {\left({2},\ {6}\right)},\ {\left({3},\ {12}\right)},\ {\left({4},\ {24}\right)}\)
Part A: Is this data modeling an algebraic sequence or a geometric sequence? Explain your answer.
Part B: Use a recursive formula to determine the time she will complete station 5. Show your work.
Part C: Use an explicit formula to find the time she will complete the 9th station. Show your work.

Answers (1)

2021-08-15
Step 1
We have, Aurora is practicing and keeping track of her time to complete each station. The x-coordinate is the station number, and the y-coordinate is the time in minutes since the start of the race that she completed the task. \(\displaystyle{\left({1},\ {3}\right)},\ {\left({2},\ {6}\right)},\ {\left({3},\ {12}\right)},\ {\left({4},\ {24}\right)}.\)
(A) Here, y-coordinate have values as
\(\displaystyle{a}_{{{1}}}={3},\ {a}_{{{2}}}={6},\ {a}_{{{3}}}={12},\ {a}_{{{4}}}={24},\ {a}_{{{5}}}\cdots{a}_{{{n}}}\)
In the above sequence,
\(\displaystyle{\frac{{{a}_{{{2}}}}}{{{a}_{{{1}}}}}}={\frac{{{6}}}{{{3}}}}={2}\)
\(\displaystyle{\frac{{{a}_{{{3}}}}}{{{a}_{{{2}}}}}}={\frac{{{12}}}{{{6}}}}={2}\)
Hence, above sequence data is modeling a geometric sequence because it follow the geometric sequence pattern having same common ratio.
Step 2
(B) We know that for a geometric sequence, \(\displaystyle{n}^{{{t}{h}}}\) term of sequence having first term a and common ratio r is given as
1) \(\displaystyle{a}_{{{n}}}={a}{r}^{{{n}-{1}}}\)
From the given data, x-coordinate corresponds to station number and y-coordinate corresponds to time t.
For station5, we need to calculate \(\displaystyle{5}^{{{t}{h}}}\) term of the sequence given by equation (1).
On substituting \(\displaystyle{a}={3},\ {r}={2},\) and \(\displaystyle{n}={5}\) in equation (1) and using explicit formula to determine the time she will complete station 5 as
\(\displaystyle{a}_{{{5}}}={\left({3}\right)}\cdot{\left({2}\right)}^{{{5}-{1}}}\)
\(\displaystyle={\left({3}\right)}\cdot{\left({2}\right)}^{{{4}}}\)
\(\displaystyle={\left({3}\right)}\cdot{\left({16}\right)}\)
\(\displaystyle={48}\) minutes
Hence, the time she will complete station 5 is 48 minutes.
(C) For station 9, we need to calculate \(\displaystyle{9}^{{{t}{h}}}\) term of the sequence given by equation (1).
On substituting \(\displaystyle{a}={3},\ {r}={2},\) and \(\displaystyle{n}={5}\) in equation (1) and using explicit formula to determine the time she will complete station 9 as
\(\displaystyle{a}_{{{9}}}={\left({3}\right)}\cdot{\left({2}\right)}^{{{9}-{1}}}\)
\(\displaystyle={\left({3}\right)}\cdot{\left({2}\right)}^{{{2}}}\)
\(\displaystyle={\left({3}\right)}\cdot{\left({256}\right)}\)
\(\displaystyle={768}\) minutes
Hence, the time she will complete station 9 is 768 minutes.
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