Answer:
6. 14.9 years
7. 4.8 years
8. 28.91 years
Step-by-step explanation:
Exponential growth or decay is described by the function ...
y = a·b^t
where 'a' is the initial value, and 'b' is the growth (or decay) factor. The growth (or decay) factor is one added to the growth rate.
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This equation can be solved for t to get ...
y/a = b^t . . . . . . . divide by a
log(y/a) = t·log(b) . . . . take logarithms
t = log(y/a)/log(b) . . . . divide by the coefficient of t
Note that the time period is in the same units as the growth rate. (If the growth rate is per quarter, then the value of t will be in quarters.)
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<h3>6.</h3>
The initial value is a=8000; the final value is y=35000; and the growth rate is 10%/4 = 0.025 per quarter. The above equation will give the time period in quarters.
t = log(35000/8000)/log(1+0.025) ≈ 59.77 . . . quarters
Anisha will have $35,000 after 59.77/4 = 14.9 years.
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<h3>7.</h3>
No loan payment period is given, so we assume that the $43000 represents the repayment of the entire loan. The growth rate is 6% per year. The time required for the loaned amount to accumulate to a value of $43,000 is ...
t = log(43,000/32,500)/log(1 +0.06) ≈ 4.80 . . . years
If the loan period is 4.80 years, Kevin will have paid 43,000 to pay it off.
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<h3>8.</h3>
The growth rate is 0.024/12 = 0.002 per month. The number of months to achieve the desired balance is ...
t = log(3200/1600)/log(1 +0.024/12) ≈ 346.92 . . . . months
The balance in the account will be $3200 after 346.92/12 = 28.91 years.
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<em>Additional comment</em>
As a crude check on the answers, the "rule of 72" can be used. It says the doubling time is approximately 72 divided by the growth rate in percent. That is, a 10% growth rate will cause the value to double in approximately 72/10 = 7.2 years.
In the first problem, the value grows from 8000 to 35000, which represents a little more than 2 doublings (8000 to 16000 and 16000 to 32000). So, it is reasonable to expect the time period to be near 2 times 7.2 years, or 14.4 years. The value of 14.9 years is not unreasonable.
