i dont understand how to solve this- A quantity is growing at a constant 3% yearly rate. Whats the growth after 15 years?

Answer 1

e^.03(15) = e^.45 = about 1.56831 = 56.5683% growth over 15 years at 3% yearly interest rate continuously compounded

(1.03)^15 = 1.5580 = about 55.8% growth over 15 years compounded annually at 3% annual interest rate

copounded quarterly (1.03/4)^4(15) = (1,0075)^60 = 1.565681 = about 56.5681% growth over 15 years

growth over 15 years is between about 56 and 57% rounded to nearst 1% for 3% interest compounded yearly to continuously

general formula is (1+r/n)^nt where n= number of compounding periods per year, t= years, r= interest rate

the formula collapses to e^rt when n is infinite

56% is probably the answer you want

if you started with a quantity of 100, then after 15 years you would have about 156

growth is continuous in the natural world, such as growth rate of trees, rabbit population, or bacteria growth or pandemics

for money in a bank, it's often annual, quarterly, monthly or daily compounding

or if, for some reason, the problem has no compounding, then it's "simple interest" and the answer would be

.03 x 15 = .45 = 45% growth over 15 years. If your parents loaned you money, they might just go with that, and be nicer to you than banks. They wouldn't have late fees or penalties or repossess collateral if you were slow repaying. Maybe even forgive the loan if you're nice to them.

Answer 2

Suppose you start with an amount A of the quantity.

After one year you get extra 3%, which is A×0.03.

Thus after one year you will have in total A + A×0.03 = A×1.03.

Then you start a new year with this new amount A×1.03, so at the end of the second year you will have

A×1.03×1.03 = A×1.03².

And after another year you will have A×1.03³.

After 15 years you will have A×1.03¹⁵.