# Math:Dragon Platebodies

The gp rate, $R_{gp}$ was calculated by finding the profit, $P$ from performing 1 platebody smithing action, and dividing by the total time, $T_{tot}$, required to perform that action. Note that this is the per second rate, and must be multiplied by 3600 to obtain the hourly rate.

$R_{gp} = \frac {P}{T_{tot}}$

The total time can be found by combining the time from mining ores($T_{mine}$), smelting bars ($T_{smelt}$), smithing the bars into platebodies ($T_{smith}$), and making diamond luck potions ($T_{luck}$).

$T_{tot} = T_{smith} + T_{smelt} + T_{mine} + T_{luck}$

## Contents

#### Smithing Time

The smithing time is simply the time it takes to complete one smithing action. $T_{smith} = 2s$.

#### Bonuses

$B_s = \begin{cases} 1 & \text {if signet is not equipped} \\ 1.1 & \text {if signet is equipped} \end{cases}$

$B_m = \begin{cases} 1 & \text {if mining gloves are not equipped} \\ 2 & \text {if mining gloves is equipped} \end{cases}$

$B_g = \begin{cases} 0.01 & \text {if gem gloves are not equipped} \\ 1 & \text {if gem gloves are equipped} \end{cases}$

#### Smelting Time

First, the number of smelting actions required, $N_{smelt}$ is calculated by dividing the average number of ingredients to perform one platebody action, $I_{smith}$, by the average quantity of bars produced by one smelting action, $Q_{smelt}$.

$N_{smelt} = \frac {5I_{smith}} {Q_{smelt}}$

This is then multiplied by the time it takes to smelt an item, which gives:

$T_{smelt} = \frac {10I_{smith}}{Q_{smelt}} \text{s}$

The average ingredient cost is dependent on the Mastery level of dragon platebodies,$M_{smith}$ ,and is reduced by 0.1 for every 20 mastery levels.

$I_{smith} = 1 - 0.1 \left \lfloor \frac {M_{smith}}{20} \right \rfloor$

The average quantity of bars is dependent on the Mastery level of dragon bars, $M_{smelt}$, and is increased by 0.1 for every 20 mastery levels above -10. The signet ring multiplies this amount by 1.1.

$Q_{smelt} = B_s \left ( 1 + 0.1 \left \lfloor \frac {M_{smelt}+10}{20} \right \rfloor \right )$

#### Mining Time

First, the amount of dragonite ore ($O_{do}$), runite ore ($O_{ro}$), and coal ore ($O_{co}$), are calculated by multiplying the number of smelting actions, $N_{smelt}$ by the ingredient cost to perform one smelting action, $I_{smelt}$ and the number quantity of the ore in the recipe.

$O_{do} = N_{smelt}I_{smelt}$

$O_{ro} = 2N_{smelt}I_{smelt}$

$O_{co} = 6N_{smelt}I_{smelt}$

The average ingredient cost to perform one smelting action depends on the Mastery level of dragon bars, $M_{smelt}$, and is reduced by 0.1 for every 20 mastery levels.

$I_{smelt} = 1 - 0.1 \left \lfloor \frac {M_{smelt}}{20} \right \rfloor$

Next, the number of mining actions for each ore is calculated. For dragonite, $N_{do}$ ,and runite, $N_{ro}$ ,this is calculated as:

$N_{do} = \frac {O_{do}}{Q_{do}}$

$N_{ro} = \frac {O_{ro}}{Q_{ro}}$

Where $Q_{do}$ and $Q_{ro}$ are the average quantity of ore produced per mining action for dragonite and runite respectively. These are dependent on the mastery of the ores,$M_{do}, M_{ro}$ ,which gives a 0.01 increase in the average quantity per 10 mastery levels, the ore bonus from the pickaxe used,$P_{ob}$, (0.07 for dragon), and if the signet ring is worn or not.

$Q_{do} = B_mB_s \left ( 1 + P_{ob} + 0.01 \left \lfloor \frac {M_{do}}{10} \right \rfloor \right)$

$Q_{ro} = B_mB_s\left ( 1 + P_{ob} + 0.01 \left \lfloor \frac {M_{ro}}{10} \right \rfloor \right)$

For coal ore the number of mining actions, $N_{co}$ , is reduced due to the coal generated from the mining skillcape while mining dragonite and runite ores. It is given by:

$N_{co} = \frac {O_{co} - N_{do} - N_{ro}}{Q_{co}}$

Additionally, the quantity of ore per mining action is increased by one:

$Q_{co} = B_mB_s \left ( 1 + P_{ob} + 0.01 \left \lfloor \frac {M_{co}}{10} \right \rfloor \right) + 1$

Finally the average time to perform a mining action is calculated. This is dependent on the respawn time of the ore, R_{o}, the effective ore health, $H_o$, and the pickaxe bonus speed (0.5 for dragon), $P_{bs}$.

$T_o = \frac {3(1-P_{bs})H_o+R_o}{H_o}$

The respawn times for dragonite, runite and coal ore are 120s, 60s and 10s respectively.

The effective ore health is dependent on the mastery of the ore, $M_o$ and the probability to not consume health provided by perfect swing potions, $P_{ps}$.

$H_o = \frac {\left \lfloor M_o+5 \right \rfloor}{1-P_{ps}}$

Finally, to obtain the total time spent mining the sum of mining actions multiplied by time per mining actions is calculated:

$T_{mine} = T_{do}N_{do} + T_{ro}N_{ro} + T_{co}N_{co}$

#### Diamond Luck Time

The time to make diamond luck potions, $T_{luck}$ is dependent the number of diamonds mined, $Q_{diam}$, and the average number of diamonds required to craft a potion $I_{luck}$ as follows:

$T_{luck} = \frac {2Q_{diam}}{I_{luck}} \text{s}$

The quantity of diamonds mined depends on the number of mining actions performed and if gem gloves are worn. First we define the average number of gems mined as:

$Q_{gem} = B_g \left ( N_{do} + N_{ro} + N_{co} \right )$

Then the average number of diamonds follows as:

$Q_{diam} = 0.05Q_{gem}$

The average number of diamonds to craft a potion depends on the mastery, $M_{luck}$ and is given by:

$I_{luck} = 1 - 0.0025M_{luck}+0.002$

#### Calculating Profit

To calculate the profit per platebody the average value of a gem, $V_{gem}$ must first be calculated. Without making diamond luck potions, this is simply the sum of the gems sell price, $S_i$ multiplied by their probabilities, $P_i$ (these can be found on the Mining page).

$V_{gem} = \sum S_iP_i$

This results in a value of 381.25 gp per gem.

When making diamond luck potions the sell price of a diamond is replaced by the sale price of the potions, $S_{luck}$, multiplied by the quantity made:

$S_{diam} = B_s \frac {2S_{luck}}{I_{luck}}$

The profit is then calculated by adding the value of items made and subtracting the cost of gem/mining gloves:

$P = 3450Q_{smith}+V_{gem}Q_{gem} - (N_{do} + N_{ro} + N_{co})C_g$

Where $C_g$ is the cost per glove charge. Gem gloves are 250 gp per charge, while mining gloves are 150 gp per charge.

$Q_{smith}$ is the average number of platebodies made per smithing action and is dependent on platebody mastery, $M_{smith}$, as per:

$Q_{smith} = B_s \left ( 1 + 0.1 \left \lfloor \frac {M_{smith}+10}{20} \right \rfloor \right )$