March 31, 2016
Aim #81: How do we determine the number of years it will take an exponential
function to reach a value?
Homework: Handout
Do Now: If $400 is invested into a bank account that earns 3.2% interest
annually, how much will there be in the account after 15 years?
1) a. If a person takes a given dosage (d) of a particular medication, then the
formula
f(t) = d (0.8)t represents the concentration of the medication in the bloodstream t
hours later. If Charlotte takes 200 mg of the medication at 6:00 a.m., how much
remains in her bloodstream at 10:00 a.m.?
b. How long does it take for the concentration to drop below 1 mg?
2) a. When you breathe normally, about 12% of the air in your lungs is replaced
with each breath. Write an explicit formula for the sequence that models the
amount of the original air left in your lungs, given that the initial volume of air is
500 mL. Use your model to determine how much of the original 500 mL remains
after 50 breaths.
b. After how many breaths will 200 mL remain?
3) a. The Booster Club raised $30,000 for a sports fund. No more money will be
placed into the fund. Each year the fund will decrease by 4½%. Determine the
amount of money, to the nearest cent, that will be left in the sports fund after 4
years.
b. After how many years will the fund drop below $10,000?
March 31, 2016
4) A tennis ball is dropped from a height of 12 feet. Each time the ball bounces
back to 80% of the height from which it fell.
a. Write a formula that models the height of the ball after b bounces.
b. What is the height of the ball after 3 bounces?
c. Graph the points (b, f(b)) for integer values of 0 ≤ b ≤ 9 . Label and scale both
axes.
d. When will the height of the
ball fall below 3 feet?
5) a. A piece of machinery that costs $8,000 depreciates each year by an amount
equal to 1/10 of its value of the beginning of the year. To the nearest dollar, how
much will the machine be worth at the end of the 5th year?
b. When will the value be less than $1000?
March 31, 2016
6) A pool holds a maximum of 20,500 gallons of water. It evaporates at a rate of
0.5% per hour. The pool currently contains 19,000 gallons of water.
a. Write an exponential function w(t) to express the amount of water remaining in
the pool after time t where t is the number of hours after the pool reached
19,000 gallons.
b. At the same time, a hose is turned on to refill the pool at a net rate of 300
gallons per hour. Write a function p(t) where t is the time in hours the hose is
running to express the amount of water that is pumped into the pool.
c. Find C(t) = p(t) + w(t). What does this new function represent?
d. Use the graph or table of values for C(t) to determine after how many hours the
pool will reach its maximum capacity.
7) Erika and Jennifer are growing bacteria in a laboratory. Erika uses the growth
function f(t) = n2t while Jennifer uses the function g(t) = n6t, where n represents
the initial number of bacteria and t is the time in hours. If Erika starts with 27
bacteria, how many bacteria should Jennifer start with to achieve the same
growth over time?
8) a. At the end of last year, the population of Jason's hometown was
approximately 75,000 people. The population is growing at the rate of 2.4% each
year. Which equation models the growth of this city?
[1] y=
[2] y=
[3] y=
[4] y=
b. Approximately how many years will it take for the population to reach 100,000
people?
March 31, 2016