calculus- optimization - need desperately help on #5 - please show the right mat

Question

calculus- optimization
- need desperately help on #5
- please show the right math work how to find the answers to the question
- thank you

4.7 Practice 3 Health when defending 1. In League of Legends, a Effective Hao0 where His player's against physical damage is given by: E 100 health and A is armor. Health costs 2.5 gold per unit, and Armor costs 18 gold per unit. You have 3600 gold, and you need to optimize the effectiveness E of your health and armor to survive as long as possible against the enemy team's attacks. How much of each should you buy? 2 1 that are closest

Explanation / Answer

here we need to maximize H(100+A) ; let this be Z

also as we have 3600 gold

hence 2.5H+18A =3600

therefore A =(3600-2.5H)/18 =200 -2.5H/18

Z =100H+HA =100H +H(200-2.5H/18) =100H+200H -2.5H2/18 =300H-2.5H2/18

for dZ/dH =300-5H/18

putting dZ/dH =0

hence 300-5H/18 =0

H =18*300/5 =1080

also as A =200-2.5H/18 =200-2.5*1080/18 =50

therefore at A =50 and H =1080 ; effectiveness E will be maximum.

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