Question

Introduction In the popular computer game League of Legends, a player’s Effective Health, E, when defending against physical damage is given by E = H(1 + 0.02A) where H and A denote the player’s chosen units of health and armor, respectively. Furthermore, a unit of health costs 4 gold pieces and a unit of armor costs 24 gold pieces. If you have 3,600 gold pieces to spend and you need to maximise the effectiveness E of your health and armor to survive as long as possible against enemy attacks, how much of each should you buy?

A1. (1 mark) Write down the Lagrange function associated with this optimisation problem.

A2. (2 marks) Solve the problem using the Lagrange Multiplier method (show your work).

A3. (2 marks) Thirty minutes into the game, you have 2,000 health and 40 armor. You have 1,800 gold pieces to spend, and again health costs 4 gold per unit and armor costs 24 gold per unit. If the goal is still to maximise the effectiveness E of your resulting health and armor (after purchase), how much of each should you buy with your 1,800 gold pieces (show your work).

Answer 1

1) Given that :

In the popular computer game League of Legends, a player’s Effective Health, E, when defending against physical damage is given by E = H(1 + 0.02A) where H and A denote the player’s chosen units of health and armor, respectively. Furthermore, a unit of health costs 4 gold pieces and a unit of armor costs 24 gold pieces. If you have 3,600 gold pieces to spend and you need to maximise the effectiveness E of your health and armor to survive as long as possible against enemy attacks

ans:

Let us assume that you buy  units of health.

Therefore you can buy

24 (3600-4x) = 150-6

(assuming you would like to spend your entire money).

And the effective health is :

0.02(150 - r/6))

We want to maximize this function. Since the coefficient of is negative, the point which we have a zero derivative, we have a maxima. We find that the derivative is 0 at the point .

So you should buy 600 units of health and 50 units of armor.

2) Given that :

Thirty minutes into the game, you have 2,000 health and 40 armor. You have 1,800 gold pieces to spend, and again health costs 4 gold per unit and armor costs 24 gold per unit. If the goal is still to maximise the effectiveness E of your resulting health and armor (after purchase)

ans :

Again if  denotes the number of health units bought, we have

units of armor.

75 - r/6

The effective health in this case is :

(2000 + x) (1 0.02(40 +75 - x/6))

Again at the point at which derivative is zero is the point of maxima. But this time we have a derivative 0 at point
475
. Since have to non negative units of health, we have to maxima when .

But after the point   the quadratic is strictly decreasing. Hence the maximum is at point

475

. Hence we should buy 0 unit of health and 75 units of armor.