Answer:
a) $975
b) growth
c) 0.3% per month
d) 6 months: $992.68; 12 months: $1010.68
e) 8.5 months
Explanation:
Given the formula for the monthly balance in a savings account is b = 975(1.003^m), you want the initial deposit, the monthly growth rate, the balance after 6 and 12 months, and the time until the account has a value of $1000.
Exponential growth
The general form of an equation for exponential growth is ...
value = (initial value) × (growth factor)^(time periods)
where (growth factor) = 1 + growth rate per time period
Application
Comparing this form to the given equation, we can identify the parts:
value = b, the monthly balance
initial value = 975, the amount Michael opened the account with
growth factor = 1.003 per month; time periods are months
growth rate = 0.003 = 0.3% per month
a) Opening deposit
From our analysis of the equation, we see that the opening deposit is $975.
b) Growth
The growth rate is positive, so this is a case of exponential growth.
c) Monthly interest rate
The growth rate per month is 0.3%. The monthly interest rate is 0.3%.
d) Balances
We can find the balance after 6 and 12 months by evaluating the formula for those values of m.
- 6 months: $992.68
- 12 months: $1010.68
e) Time to $1000
To find the time it takes for the balance to reach 1000, we must solve the equation for m. Since m is an exponent, logarithms are involved.
1000 = 975(1.003^m)
1000/975 = 1.003^m . . . . . . divide by 975
log(1000/975) = m·log(1.003) . . . . take logarithms
m = log(1000/975)/log(1.003) . . . . . divide by the coefficient of m
m ≈ 8.5 . . . . months
It will take about 8.5 months for the account balance to be $1000.
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Additional comment
The relevant rule of logarithms is ...
log(a^b) = b·log(a)
Logarithms were invented for the purpose of transforming multiplication problems into addition problems. When exponents are involved, that means exponential problems are transformed to linear problems. That's what we're doing here.
We have left log(1000/975) as a log expression. We could have replaced it with a number. That might better help you see the simplicity of the equation for m:
0.0109954 = 0.00130093·m
As with all such equations, you find the value of the variable by dividing both sides by its coefficient.
You need to be careful with precision. These numbers are rounded to 6 significant figures. Their ratio is good to 4 significant figures. We let the calculator figure the value from the log expressions so we get the best possible accuracy.