Math 111: Exam 02 Solutions - Blake Farman - University Of South Carolina

Math 111: Exam 02 Solutions - Blake Farman - University Of South Carolina - 2013 Page 6

BLAKE FARMAN UNIVERSITY OF SOUTH CAROLINA

(d) If the cost of the event is $400, then we know

400 = 4x + 300.

Subtracting 300 from both sides we have 4x = 400

300 = 100 and dividing both sides

by 100 gives us x = 25. Therefore if the cost of the event is $400, 25 guests attended.

10 (16 Points). A population of size 25 grows by 20% every day.

(a) Give the growth factor for this population.

(b) Give an exponential model for the size of the population after t days.

ratio, rather than a decimal, and this will be easy to compute.]

Solution. (a) We are given the growth rate, r = 20% =

= 15, so to ﬁnd the growth

100

factor we use the formula

a = 1 + r = 1 +

(b) We are given the initial population, 25, and we have calculated the growth factor a =

so our exponential model is

P (t) = 25

Note that our t value here is measured in days.

P (2) = 25

= 25

= 36.

11 (Bonus - 10 Points). Let f (x) = m

x + b

and g (x) = m

x + b

. Using f and g, derive

the general formula for the intersection of two lines. Use this to explain why two parallel

lines never intersect.

Solution. To compute the point of intersection for two lines we ﬁrst compute the x-coordinate

by solving the equation

x + b

= m

x + b

for x. Subtracting m

x from both sides we get the equation

x + b

= (m

)x + b

= b

Subtracting b

from both sides gives us

)x = b

Math 111: Exam 02 Solutions - Blake Farman - University Of South Carolina - 2013 Page 6