SOLUTION: Jessica and Matthew work for different companies. Jessica has a base salary of $36,000 and will receive an increase of salary at a rate of 3.1% per year. Matthew has a base salary

Question 1109448: Jessica and Matthew work for different companies. Jessica has a base salary of $36,000 and will receive an increase of salary at a rate of 3.1% per year. Matthew has a base salary of $38,000 and will receive an increase of salary at 2.5% per year. Jessica and Matthew are calculating the number of years, t, it will take each of them to earn over $45,000 at their respective companies. Determine the inequality could be used to determine the number of years for each person to reach $45,000. Then, decide which statement is true for each person.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
jessica receives 36000 and gets an increase of 3.1% per year.

matthew has a base salary of 38000 and gets an increase of 2.5% per year.

inequality to use is f < p * (1 + r) ^ t.

f is the future value.
p is the present value.
r is the annual interest rate, which is the annual interest rate percent divided by 100.
t is the number of years.

for jessica, the inequality becomes 45000 < 36000 * (1 + .031) ^ t

for matthew, the inequality becomes 45000 < 38000 * (1 + .025) ^ t

solving for jessica:

start with 45000 < 36000 * (1 + .031) ^ t

divide both sides of the inequality by 36000 to get:

45000 / 36000 < (1 + .031) ^ t

take the log of both sides of the inequality to get:

log (45000 / 36000) < log((1 + .031) ^ t)

since log ((1 + .031) ^ t) is equal to t * log(1 + .031), the equation becomes:

log(45000 / 36000) < t * log(1 + .031)

divide both sides of the equation by log(1 + .031) to get:

log(45000 / 36000) / log(1 + .031) < t

solve for t to get 7.30918316 < t.

this is the same as t > 7.30918316.

when t is equal to that, you get f = 36000 * (1 + .031) ^ 7.30918316.

this results in f = 45000.

when t is greater than that, like when t = 7.302 years, you get f = 36000 * (1 + .031) ^ 7.302 = 45074.91875.

since 45000 < 45074.91875, the inequality is true when t is greater than 7.30918316.

when you do the same process for matthew, you get:

45000 < 38000 * (1 + .015) ^ t when t > 6.847243459

when t is equal to that, you get f = 38000 * (1 + .015) ^ 6.847243459 which results in f = 45000.

when t is greater than that, like when t = 6.848, you get f = 38000 * (1 + .025) ^ 6.848 = 45000.84065.

since 45000 < 45000.84065, the inequality is true when t is greater than 6.847243459.

your solution is that:

the inequality for jessica of 45000 < 36000 * 1.031 ^ t is true when t > 7.30918336.

the inequality for matthew of 45000 < 38000 * 1.025 ^ t is true when t > 6.847243459.

round to as many decimal places as required.

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