Download MAX and MIN APPLICATIONS Steps for Solving Extrema Problems

MAX and MIN APPLICATIONS
Steps for Solving Extrema Problems:
• read problem and make sketch, if possible;
• decide on variable to be maximized or minimized;
• find critical points and classify local extrema;
• use domain information to find global extrema.
Examples
1. Maximization:
a) maximize area for rectangular field bordering a river,
using 1200m of fence;
MAX/MIN APPS CONTINUED
b) % of population with disease for time 0 ≤ t ≤ 20 is
20t3 − t4
;
p(t) =
1000
find t for maximum.
c) % of population with disease for time 0 ≤ t ≤ 40 is
p(t) = 10te−t;
find t for maximum.
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MAX/MIN APPS CONTINUED
d) Biomechanics: maximize shotput distance d(θ)
using angle θ if
s
d(θ) = 5.1 sin(2θ)(1 + 1 + .41/ sin2(θ)).
Matlab
d=@(t)5.1*sin(2*t).*(1+sqrt(1+.41./sin(t).^2));
t = [ 0: .01: pi/2 ]; plot(t,d(t))
[D j] = max(d(t)); disp([D t(j)])
12.112
0.7
title(’Biomechanics: Shotput Distance’)
Biomechanics: Shotput Distance
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MAX/MIN APPS CONTINUED
2. Minimization:
a) minimize cost of fence to enclose 15625m2
rectangular area with one open side if fence
costs $2/m for ends and $4/m for closed side;
b) minimize amount of material for cylindrical can
with volume of 1000cm3;
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MAX/MIN APPS CONTINUED
c) minimize pigeon flight energy if pigeon released
1 mile from shore needs to reach point on shore
2 miles from closest shore point and pigeon needs
4/3 more energy to fly over water.
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