Intro to Exponentials
Is the following function exponential? (2*3^t)^3
Yes
A population has size 5000 at time t = 0, with t in years. a) If the population decreases by 100 people per year, find a formula for the population, P, at time t. b) If the population decreases by 8% per year, find a formula for the population, P, at time t.
a) P = 5000 - 100t b) P = 5000(.92)^t
Make a table of values for f(x) = 2^x for x = -3, -2, -1, 0, 1, 2, 3.
x -3 -2 -1 0 1 2 3 f(x) 1/8 1/4 1/2 1 2 4 8
Suppose 1000 dollars is deposited into an account paying interest at a nominal rate of 8% per year. Find the balance three years later if the interest is compounded annually.
$1259.71
A computer purchased for $2000 depreciates at a rate of 9% per year. What will be the decay factor?
.91
What is the growth/decay factor if a diamond mine is depleted by 1% per day?
.99
In an environment with unlimited resources and no predators, a population tends to grow by the same percentage each year. Should a linear or exponential function be used to model such a population? Why?
An exponential because it is changing at a constant rate. The same percentage does not mean it will be changing by the same number each year, which is what a linear model suggests.
Solve y = 46(1.1)^x graphically for x if y = 91.
x = 7.158
Suppose 1000 dollars is deposited into an account paying interest at a nominal rate of 9% per year. Find the balance five years later if the interest is compounded annually.
$1538.62
A population is 25,000 in year t = 0 and decreases at a continuous rate of 7.5% per year. Find a formula for P(t).
P(t) = 25000(.925)^t
Give the starting value a, the growth factor b, and the growth rate r if Q = 1750(1.593)^t.
a = 1750 b = 1.593 r = 59.3%
If your x values are 0, 1, 2, 3, 4 and your g(x) values are 0, 2, 4, 6, 8 respectively, is g(x) linear or exponential? Give a formula.
g(x) = 2x, linear
Suppose 5000 dollars is deposited into an account paying interest at a nominal rate of 7.5% per year. Find the balance 8 years later if the interest is compounded annually.
$8917.39
A boat purchased for $20000 depreciates at a rate of 5% per year. What will the value of the boat be in 10 years.
$11974.74
Write a formula for Q as a function of t and evaluate at Q(10) if the initial amount is 2000 and it is increasing by 5% per year.
Q = 2000(1.05)^t Q(10) = 3257.789
Suppose that f(x) is exponential and that f(-3) = 54 and f(2) = 2/9. Find a formula for f(x).
f(x) = 2(1/3)^x
An investment of $4000 grows by 5% per year for 20 years. What will the value be of the investment at the end of the 20-year period?
$10613.19
The current population of polar bears is 10000 and is decreasing at a rate of 2% per year. How many polar bears will there be in 10 years.
In 10 year there will be approximately 8171 polar bears.
Polluted water is passed through a series of filters. Each filter removes 85% of the remaining impurities. Initially, the untreated water contains impurities at a level of 420 parts per million (ppm). Find a formula for L, the remaining level of impurities, after the water has been passed through a series of n filters.
L = 420(.15)^n
Find a possible formula for the exponential function g given that the points (2.3, 0.4) and (3.5, 0.1) are on its graph.
g(t) = 5.7(0.315)^t
The population of a colony of rabbits grows exponentially. The colony begins with 10 rabbits; five years later there are 340 rabbits. Use a graph to estimate how long it takes for the population of the colony to reach 1000 rabbits.
t = 6.53 years
Without making any calculations, briefly describe in words what V = 500 (1 + 0.04)^12 tells you about the value of the investment it describes.
Initial investment is $500, growing at a rate of 4% annually, for 12 years.
A radioactive substance decays at a continuous rate of 14% per year, and 50 mg of the substance is present in the year 2009. How much will be present in the year 2019?
11.065 mg