Download BONUS: Find all values of x such that 2 log 2(x) − log (3x − 4) = 1

Midterm 2
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BONUS: Find all values of x such that 2 log2 (x) − log2 (3x − 4) = 1
Answer: x = 2 and x = 4
1. Find all possible solutions to each of the following equations. Round to two decimal places if necessary.
2x − 15
=x
4−x
Answer: x = −3, x = 5
(b) 4 32x−1 + 11 = 39
(a)
Answer: x ≈ 1.47
7 − log4 (x + 19)
(c) 1 =
5
Answer: x = −3
2. Sandy recently inherited $50 000 and she wants to invest it in a college fund for her daughter. She estimates that her
daughter will be ready to go to college 15 years after the investment and her goal is to have $100 000 saved by then.
(Round to two decimal places when necessary.)
(a) The first bank that Sandy visits agrees to invest her money in an account on which interest is compounded yearly at
an annual rate of 4 %. If she invests with this bank, how much will the fund be worth after 15 years?
Answer: $90 047.18
(b) A second bank is willing to give Sandy an account on which interest is compounded quarterly (i.e. four times per
year). What annual interest rate does she need to get on this account if she wants to reach her goal? (You may write
your answer as a decimal or as a percentage. If you write it as a decimal then round to four decimal places.)
Answer: 0.047 or 4.648 %
(c) A third bank is willing to give her an annual interest rate of 4.5 % compounded continuously. If she invests with this
bank, how long will it take for the account to reach $100 000?
Answer: 15.403 years
3. Solid-fuel rockets achieve their maximum speed shortly after take off; this is called the launch velocity. A rocket’s launch
velocity is related to its mass at take off (which changes depending on the amount of fuel it carries). An engineer is
working with a large solid-fuel rocket which weighs 75 kg when it carries no fuel and at that weight its launch velocity
is 0 m/s (because it can’t take off without fuel). Any time the engineer wants to increase the rocket’s launch velocity by
25 m/s, he needs to increase the rocket’s total mass by 12 %. Let M(v) be the mass (in kg) of the rocket required at take
off in order to achieve a launch velocity of v (in m/s). Note: launch velocity is the INPUT and mass is the OUTPUT. (Round to
two decimal places when necessary.)
(a) Find and simplify an equation for M(v).
t
Answer: M(v) = 75(1.12) 25
(b) The speed of sound is approximately 340 meters per second. What should the mass of the rocket be at take off in
order for the launch velocity to reach the speed of sound?
Answer: 319.93 kg
(c) What is the rocket’s launch velocity if it weighs 150 kg at take off?
Answer: 152.91 m/s
4. A small hotel rents out its ballroom for special events and they charge different rates depending on the number of people
in attendance. If x people attend the event then the hotel makes R(x) thousand dollars per hour in revenue and it costs
the hotel C(x) thousand dollars per hour where
R(x) =
November 23, 2015
x
x + 10
and
C(x) =
3x + 5
.
4x + 20
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Midterm 2
Let P be the profit function; if x people attend the event then the hotel makes a profit of P(x) thousand dollars per hour.
(a) Find an equation for P(x). Write it as a single rational expression (i.e. one fraction) and simplify as much as possible.
Answer: P(x) =
x2 − 15x − 50
4x2 + 60x + 200
(b) Find the long-term behavior of P and interpret what it means in this model.
Answer: As x goes to ∞, P(x) goes to 0.25. This means that as the number of people attending an event becomes
large, the profit approaches $250.
(c) How many people need to attend an event for the hotel to break even? Round to two decimal places. (Normally
a decimal number of people doesn’t make sense but decimals are allowed in this question. If you get a decimal,
assume that it represents an average number of people over all of the special events that the hotel hosts.)
Answer: 18 people
November 23, 2015
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