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Willow is in her fourth year working for the same company. Her starting salary was $48,000. The next year, her salary was $50,400. The following year, her salary was $52,920. This year, her salary is $55,566. Willow’s boss says her salary will continue
to increase by the same percentage each year.
A. Write the values given in the scenario as a sequence. already done.
Year 1: $48,000
Year 2: $50,400
Year 3: $52,920
Year 4: $55,566
$48,000, $50,400, $52,920, $55,566.
B. Is this an example of an arithmetic sequence, a geometric sequence, or
another type of sequence? Explain. already done.
This is a geometric sequence because Willow's salary is increasing by a certain percentage each year, not by adding the same amount. In simpler terms, every year her salary gets bigger by multiplying it by a specific number. For example, from Year 1 to Year 2, her salary went up by 5% (which is a little bit more than $2,400), and that same idea happens each year. So, it's not just adding a certain number each time; it's multiplying by a factor that makes it grow faster each year.
C. Create a function that models this sequence/scenario. already done.
To find Willow's salary each year, we can use a function that shows how it grows. Let's call her starting salary s, which is $48,000. The factor we multiply by each year is 1 plus the percentage increase (in decimal form). If her salary increases by 5% every year, the factor is 1.05. So, we can write the function like this:
Salary=sX(1.05)^(n-1)
Here, n is the year number. So, each year we take her starting salary and multiply it by 1.05 for every year after the first. This way, we can find out how much she makes each year!
D. By what percentage is Willow’s salary increasing each year? Show your work. already done.
To find out by what percentage Willow's salary is increasing each year, we can compare her salaries from one year to the next. First, let's look at Year 1 to Year 2:
From $48,000 to $50,400, the increase is $50,400 - $48,000 = $2,400. To find the percentage, we take the increase and divide it by the starting salary, then multiply by 100. So, ($2,400 / $48,000) × 100 = 5%. Next, from Year 2 to Year 3: From $50,400 to $52,920, the increase is $52,920 - $50,400 = $2,520. So, ($2,520 / $50,400) × 100 = 5%. Then, from Year 3 to Year 4: From $52,920 to $55,566, the increase is $55,566 - $52,920 = $2,646. So, ($2,646 / $52,920) × 100 = 5%. In all the comparisons, we see that Willow’s salary increases by the same percentage: 5%. So, her salary goes up by 5% each year!
E. Willow wants to use your function to predict her salary in the future. She wants to make sure her predictions are made to a reasonable level of accuracy. How can she do this?
make it simple in a small paragraph and like a kid wrote it.
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Willow is in her fourth year working for the same company. Her starting salary was $48,000. The next year, her salary was $50,400. The following year, her salary was $52,920. This year, her salary is $55,566. Willow’s boss says her salary will continue
to increase by the same percentage each year.
A. Write the values given in the scenario as a sequence. already done.
Year 1: $48,000
Year 2: $50,400
Year 3: $52,920
Year 4: $55,566
$48,000, $50,400, $52,920, $55,566.
B. Is this an example of an arithmetic sequence, a geometric sequence, or
another type of sequence? Explain. already done.
This is a geometric sequence because Willow's salary is increasing by a certain percentage each year, not by adding the same amount. In simpler terms, every year her salary gets bigger by multiplying it by a specific number. For example, from Year 1 to Year 2, her salary went up by 5% (which is a little bit more than $2,400), and that same idea happens each year. So, it's not just adding a certain number each time; it's multiplying by a factor that makes it grow faster each year.
C. Create a function that models this sequence/scenario. already done.
To find Willow's salary each year, we can use a function that shows how it grows. Let's call her starting salary s, which is $48,000. The factor we multiply by each year is 1 plus the percentage increase (in decimal form). If her salary increases by 5% every year, the factor is 1.05. So, we can write the function like this:
Salary=sX(1.05)^(n-1)
Here, n is the year number. So, each year we take her starting salary and multiply it by 1.05 for every year after the first. This way, we can find out how much she makes each year!
D. By what percentage is Willow’s salary increasing each year? Show your work. already done.
To find out by what percentage Willow's salary is increasing each year, we can compare her salaries from one year to the next. First, let's look at Year 1 to Year 2:
From $48,000 to $50,400, the increase is $50,400 - $48,000 = $2,400. To find the percentage, we take the increase and divide it by the starting salary, then multiply by 100. So, ($2,400 / $48,000) × 100 = 5%. Next, from Year 2 to Year 3: From $50,400 to $52,920, the increase is $52,920 - $50,400 = $2,520. So, ($2,520 / $50,400) × 100 = 5%. Then, from Year 3 to Year 4: From $52,920 to $55,566, the increase is $55,566 - $52,920 = $2,646. So, ($2,646 / $52,920) × 100 = 5%. In all the comparisons, we see that Willow’s salary increases by the same percentage: 5%. So, her salary goes up by 5% each year!
E. Willow wants to use your function to predict her salary in the future. She wants to make sure her predictions are made to a reasonable level of accuracy. How can she do this?
make it simple in a small paragraph and like a kid wrote it.
Answers
Answered by
GPT-4o mini
Willow can use her salary function to guess how much money she'll make in the future by just plugging in the number of years she wants to look ahead. For example, if she wants to see her salary in Year 5, she can replace n in the function with 5. It’s important for her to remember that this is just a guess since things can change at work, like if she gets a big promotion or if the company decides to cut salaries. But using the function will give her a good idea of what her salary might be if it keeps going up by that same 5% each year!