MATH 190 Midterm: MATH 190 Louisville Exam 2 160303 Solution
MATH 190–03 Exam #2 Solutions
1. (15 points) Answer the following questions about growth and decay.
(a) (5 points) A metal ingot is removed from a furnace into a warm metalworking studio; its
temperature tminutes after removal is given by the function f(t) = 90 + 410e−0.05t. How
long will it take the ingot to cool to 200◦F?
We want a solution to the exponential equation 90 + 410e−0.05t= 200. Performing ap-
propriate arithmetic on both sides of the equality, we can eventually liberate the value of
t:
90 + 410e−0.05t= 200
410e−0.05t= 110
e−0.05t=110
410
ln e−0.05t= ln 110
410
−0.05t= ln 11
41
t=ln 11
41
−0.05
For reference, this value is about 26 minutes, although you could not reasonably determine
that without a calculator.
(b) (4 points) You have $500 invested in an account which bears 2% annual interest, com-
pounding quarterly. Produce a function describing the value in your account after tyears.
After tyears, your account will have earned 2%
4interest 4ttimes, for a total value of
f(t) = 500 (1 + 2%
4)4t= 500(1.005)4t.
(c) (6 points) You hope to cash the account from the previous question out when it reaches
a balance of $600. Based on the function from the previous question, how many years do
you need to wait until you can do so?
We want a solution to the equation 500(1.005)4t= 600. Performing appropriate arithmetic
on both sides of the equality, we can eventually liberate the value of t:
500(1.005)4t= 600
(1.005)4t=600
500
ln(1.0054t) = ln 6
5
4tln 1.005 = ln 6
5
t=ln 6
5
4 ln 1.005
For reference, this answer is about nine years.
2. (20 points) Solve the inequality s+3
(s−1)2≤1
s+2 .
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Document Summary
We want a solution to the exponential equation 90 + 410e 0. 05t = 200. Performing ap- propriate arithmetic on both sides of the equality, we can eventually liberate the value of t: For reference, this value is about 26 minutes, although you could not reasonably determine that without a calculator. (b) (4 points) you have invested in an account which bears 2% annual interest, com- pounding quarterly. Produce a function describing the value in your account after t years. After t years, your account will have earned 2% interest 4t times, for a total value of. 4 f (t) = 500 (1 + 2% 4 )4t (c) (6 points) you hope to cash the account from the previous question out when it reaches a balance of . We want a solution to the equation 500(1. 005)4t = 600. Performing appropriate arithmetic on both sides of the equality, we can eventually liberate the value of t: (1. 005)4t =
