Optimization

401

Appendix C: Solutions to Chapter Problems

Chapter 1

1. Let x = number of times fare is reduced by Rs. 300

Revenue = R = price × quantity = (15,000 − 300x) × (130 + 4x)

For maximization,

d

d

R

x

=

0

∴ x = 8.75

Now,

d

d

2

2

2400

R

x

= −

∴ R has a local maximum at x = 8.75

Best fare = (15,000 − 300x) = Rs. 12,375

Number of passengers = (130 + 4x) = 165

Revenue = Rs. 2,041,875

2. Let x = number of additional trees that need to be planted

Yield = y = (50 + x) × (300 − 3x)

For maximization,

d

d

y

x

= 0

∴ x = 25

Now,

d

d

2

2

6

y

x

= −

∴ R has a local maximum at x = 25

402 Appendix C

3. Let r = the radius of the circle; w and h are the width ...