Tricky Damage Problem

[Guest]

A word problem of my own creation, based on the game HypixelSkyblock. Managed to get it 80% but couldn't finish it off. Interested in how YOU would attempt it

In a game, different Weapons do different amounts of FinalDamage based on many different factors. All Weapons have a BaseDamage stat and grant Strength to the player weilding the Weapon.

The MidasSword has 290 BaseDamage and gives 215 Strength to the player.

The EmeraldBlade has 130 BaseDamage and gives 60 Strength to the player.

The EmeraldBlade also has a special ability: It gains AdditionalDamage based on the amount of money the player has in their purse. The formula for this AdditionalDamage is: 4/9 * (1000 * coins)^(1/4)

Let D be the FinalDamage done by a Weapon
E be short for EmeraldBlade
M be short for MidasSword
d be the BaseDamage done by a Weapon
s be the Strength granted to the player by holding the Weapon
a be the AdditionalDamage granted to the EmeraldBlade based on how much money(x) the player has in their bank
x be the amount of Money in the player's bank
and y be all additional strength of the player that is NOT from the Weapon (ie from other sources)

(x axis be CoinsInPurse, y axis be Strength)

The formula to calculate D is: (d + a + (y + s)/5)(1 + (y + s)/100)
or (BaseDamage + AdditionalDamage + (Strength + AdditionalStrength)/5)(1 + (Strength + AdditionalStrength)/100)

Find the inequation for how much money(x) it takes for the EmeraldBlade to out-damage the MidasSword at various amounts of strength(y):

Go wild

P.S You cannot have negative damage, coins etc

[shyamjayakannan]

first we rearrange the damage equation: [tex]\left\{d+a+\frac{\left(y+s\right)}{5}\right\}\left\{1+\frac{\left(y+s\right)}{100}\right\}=\frac{\left(y+s\right)^2}{500}+\frac{\left(y+s\right)\left(20+d+a\right)}{100}+d+a[/tex]

For the emeraldblade, [tex]a=\frac{4}{9}\sqrt[4]{1000x}[/tex] and for the midassword, [tex]a=0[/tex]. Putting in the values for th inequality, we get,

[tex]\frac{\left(y+60\right)^2}{500}+\frac{\left(y+60\right)}{100}\left(20+130+\frac{4}{9}\sqrt[4]{1000x}\right)+130+\frac{4}{9}\sqrt[4]{1000x}\ge\frac{\left(y+215\right)^2}{500}+\frac{\left(y+215\right)}{100}\left(20+290+215\right)+290+215[/tex]

Now simplify from here