Question

Remember to write your name, date, instructor name, and section number in heading section on first page and center the lab number and title underneath this on the first page of your lab assignment Part A-Enter your work by hand into the answer sheet provided and turn into your instructor at the end of the lab period. Make sure to include UNITS in all of your answers. Exponential Growth 1. If you invested $30 in a single stock 40 years ago and it grew 25% annually, what would the stock be worth today? Use the exponential growth equation (also used to calculate continuing compounding interest) to answer this question. 2. It took all of human history up to the year 1800 for humans to reach a population of 1 billion. Human population increased by that same amount in the twelve years from 1987 to 1999 where we went from 5.0 x 10 to 6.0 x 10'. Calculate the average annual growth rate over that 12 year period. 6th billion 12 years 5th billion 12 years 4th billion 3rd billion 15 years 30 years 1st billion All of human history 2nd billion 130 years 1800 1930 1960 1975 1987 1999 3. Given an annual growth rate of 1.5%, how long would it take a population of 5 x 10° to grow to 10 x 10%? 4. Assuming the same constant annual growth rate of 1.5% and a current world population of 7.4 x 10°, determine when the year) the world population was only two people, the first two human beings? In a few sentences, discuss your results from the previous question (#4) and why they may differ from what you expected.

show all the work please!

Answer 1

Solutions:

1.

Exponential Growth Formula
Y=	A (1+r)^n
Y	To find	(Value at Year 40)
A	$30.00	(Value at Year 0)
r	25.00%	Annual Compound interest rate
n	40	number of years
(1+r)	1.2500
(1+r)^n	7523.16
A (1+r)^n	=$30*7523.16
Y	$225,694.92

Today's stock worth is $225.694.92

2.

Exponential Growth Formula
Y=	A (1+r)^n
Y	6,000,000,000.00	Future Value (6*10^9)
A	5,000,000,000.00	Present Value (5*10^9)
r	TO FIND	Annual Compound interest rate
n	12	number of years
(1+r)	1.0000

When r is not given
(1+r)^n	Y / A
	1.2000
Taking log on both sides
n log(1+r)^n	log( 1.2000 )
log(1+r)	0.0152
1+r	=exp(0.0152)
	1.0153
r	0.0153
	1.53%

Average growth rate over 12 years is 1.53%

3.

Exponential Growth Formula
Y=	A (1+r)^n
Y	6,000,000,000.00	Future Value (6*10^9)
A	5,000,000,000.00	Present Value (5*10^9)
r	1.50%	Annual Compound interest rate
n		number of years
(1+r)	1.0150

When n is not given
(1+i)^n	Y/ A
	1.2000
Taking log on both sides
n log(1+r)^n	log( 1.2000 )
n	log( 1.2000 ) / log(1+1.50% )
log( 1.2000 )	0.1823
log(1 + 1.50% )	0.0149
n	12.2457
	12.25 years

It takes 12.25 years for the population to grow from 5*10^9 to 6*10^9

4.

Exponential Growth Formula
Y=	A (1+r)^n
Y	$7,400,000,000.00	Future Value (6*10^9)
A	$2.00	Present Value (5*10^9)
r	1.50%	Annual Compound interest rate
n		number of years
(1+r)	1.0150

When n is not given
(1+i)^n	Y / A
	3700000000
Taking log on both sides
n log(1+r)^n	log( 3700000000 )
n	log( 3700000000 ) / log(1+1.50% )
log( 3700000000 )	22.0316
log(1 + 1.50% )	0.0149
n	1479.7617
	1479.76 years

It has taken 1480 years for the population to grow from 2 persons to 7.4*10^9. According to this given growth rate there were only two persons 1480 years ago!! It differs from what was expected, as the given growth rate may not be an accurate one.