David starts off at a salary of $10,000 per yer. At the end of each year, he will earn a 4% raise on his previous salary. What will be the first year that he makes $15,000? If Valerie starts off at a salary of $10,000 per year and receives an option of a constant raise of $500 then how many years until she first makes $15,000?
Could somebody walk me through on how to do this?
10,000(1.04)^n = 15000
1.04^n = 1.5
take logs of both sides
log 1.04^n = log 1.5
n log 1.04 = log 1.5
n = log 1.5/log1.04 = 10.338..
10 years will not be enough , so she makes 15,000 or more in the 11 th year
for the second question you have an arithmetic sequence with
a = 10000
d = 500
n = ?
a + (n-1)d = 15000
10000 + 500(n-1) = 15000
500(n-1) = 5000
n-1 = 10
n = 11
Sure! I'd be happy to walk you through the process of solving these types of problems.
To find the first year that David makes $15,000, we need to determine how many years it will take for his salary to increase from $10,000 to $15,000, given a 4% raise at the end of each year.
Here's how to calculate it step by step:
1. Start with David's initial salary: $10,000.
2. Calculate the raise amount for each year by multiplying his previous salary by 0.04 (4%).
Year 1: $10,000 * 0.04 = $400
Year 2: ($10,000 + $400) * 0.04 = $416
Year 3: ($10,400 + $416) * 0.04 = $417.44
Year 4: ($10,816.44 + $417.44) * 0.04 = $433.26
And so on...
3. Keep adding the raise amount to his previous salary each year until it reaches or exceeds $15,000.
Year 1: $10,000 + $400 = $10,400
Year 2: $10,400 + $416 = $10,816
Year 3: $10,816 + $417.44 = $11,233.44
Year 4: $11,233.44 + $433.26 = $11,666.70
Year 5: $11,666.70 + $466.67 = $12,133.37
Year 6: $12,133.37 + $485.33 = $12,618.70
Year 7: $12,618.70 + $504.75 = $13,123.45
Year 8: $13,123.45 + $524.94 = $13,648.39
Year 9: $13,648.39 + $545.94 = $14,194.33
Year 10: $14,194.33 + $567.77 = $14,762.10
Year 11: $14,762.10 + $590.48 = $15,352.58
Therefore, it will take 11 years for David to first make $15,000.
Now let's solve the same problem for Valerie, who receives a constant raise of $500 each year.
1. Start with Valerie's initial salary: $10,000.
2. Calculate how many times her raise needs to occur to reach or exceed $15,000. Divide the difference ($15,000 - $10,000 = $5,000) by the constant raise amount ($500).
$5,000 / $500 = 10
Valerie needs 10 raises to reach or exceed $15,000.
3. Since Valerie receives one raise per year, it will take her 10 years to first make $15,000.
I hope this explanation helps you understand how to solve these types of problems! Let me know if you have any further questions.