Group Work BC Calculus 3.17.15
1. [10-4 w/ calc] A roadway is to be constructed to connect an offshore oil well 4 miles from the shoreline and the oil
company’s field office 10 miles along the shore from the point closest to the well (see figure below). The roadway
along the shore costs 100 thousand dollars per mile to build. The roadway across the bridge costs 250 thousand
dollars per mile to build. Where should the bridge be constructed so that the cost is a minimum?
WELL
4 MI
OFFICE
2. Juana daily trips from the math building to the English building. She has three possible routes: Along the sidewalk all
the way; Straight across the grass; or Angle across the grass to the other sidewalk. She figures her speed is 6.2 ft/sec
on the sidewalk and 5.7 ft/sec across the grass. Which route takes the least time? Give details of this route.
Math building
grass
200 ft
700 ft
English building
3. Fran’s Optimal Study Time Problem: Fran forgot to study for her BC Calculus Test until late Monday night. She
knows she will score a zero if she doesn’t study at all and that her potential score will be
S
100t
t 1
if she studies
for time t, in hours. She also realizes that the longer she studies, the more fatigued she will become. So, her actual
score will be less than the potential score. Her “fatigue factor” is
F
9
.
t 9
This is the number she must
multiply by the potential score to find her actual grade, G. That is, G = S * F.
a) Sketch the graphs of S, F, and G versus time, t.
b) What is the optimal number of hours for Fran to study? That is, how long should she study to maximize her
estimated grade, G?
c) How many points less than the optimum will Fran expect to make if she studies for…
i.
1 hour more than the optimum
ii.
1 hour more than the optimum
4. A new subdivision is being built and pipes for water must be installed and connected to the city lines. The main water
connector to the subdivision is to be in a field 50-yards from the nearest street, and the main connector to the city
water line is 400-yards down the street. Installing water lines across the field costs $75/yard, while installing them
along the street is $50/yard. How should the water line be laid out in order to minimize its total cost?
5.
a)
Find a formula in terms of k for the average value of
y a x
on the interval
[0,k ] , where
k is a positive constant and a is also a constant.
b)
Use the formula you found in Problem a) to find the average value of
interval
[0,4] .
y 2 x
on the