Volumes of Tori and Pyramids

The volume of some solids can be found intuitively (e.g. for a right rectangular prism, V = lbh). However, some more exotic solids will require calculus for their volumes to be found.

Two such solids are the torus (donut) and rectangular pyramid. The former's volume can be found, possibly among others, using the technique of volume by rotation, and the latter's with volume by cross-section. These techniques will be used for deriving the formulae for the volumes of the respective solids. In summary:

• Volume of torus = 2cπ²r²
• Volume of rectangular pyramid = ¹/₃abh

WoW Mathematics: Fatally Attracted to You

Back in the Burning Crusade era, Mother Shahraz was notorious for the encounter's high Shadow Resistance requirement and her femininity. She would also, at random times, cast Fatal Attraction on three random non-tank players.

Despite having Shadow Resistance at the cap, the damage is still great, and the unlucky trio need to separate promptly if they are to survive. It may be useful to know, given a certain set of vectors and using calculus, how fast they are moving away from each other...

Don't slack off during the Thaddius encounter!

Aim: To calculate the drop in DPS of a single death in Phase 2 of the Heroic Thaddius encounter, subject to certain assumptions. To observe the change in total raid DPS as additional dps die.

Summary:

• The minimum personal DPS needed to down Thaddius in time is 2186 DPS.
• The total personal DPS is 100% average DPS + 115% bonus DPS from charge buff OR 215% of average DPS.
• A 1-stack of the charge buff grants 218.6 DPS.
• A single death of a healer drops total raid DPS by 1748.8 DPS, while a single death of a dps drops total raid DPS by 6230.2 DPS.
• While DPS continues dropping as additional dps die, its rate of decline is decreasing.

The given:

• Raid composition: 2 tanks, 7 healers, 16 dps.
• Raid positioning: 12.5 players on each side of Thaddius, each side having 8 dps.
• Thaddius' health: 27 600 000 HP.
• Enrage timer: 360 seconds (6 minutes)
• Average size of stacking group: 12.5 players.
• Average stack of charge buff among dps: 11.5 stacks. This gives a damage boost of 115% of average personal DPS.
• Healers do not dps, tanks do minimal damage (in this scenario, VERY minimal).

Detail:

The figure that will be used as the average personal DPS, fully buffed, is the minimum personal DPS needed to down Thaddius before the enrage. The total raid DPS is:

27 600 000/360 = 76 666.6666... DPS

The total personal DPS is:

7666.6666.../16 = 4791.6666... ~ 4700 DPS (attribute the extra 91.6666... DPS to the tanks :))

Total personal DPS is:

100% average DPS + 115% bonus DPS from charge buff (decimal 1.15) = 215% of average personal DPS (decimal 2.15)

The average personal DPS is:

4700/2.15 ~ 2186 DPS

That means the bonus DPS from the charge buff is:

4700 - 2186 = 2514 DPS

A 1-stack of the charge buff grants:

2514/11.5 ~ 218.6 DPS

Consider the death of a healer. That removes a 1-stack of the charge buff off the 8 dps on their side. So, the drop in DPS from that death is:

218.6 x 8 = 1748.8 DPS. That's about 524.6k damage over 5 minutes lost!

Consider the death of a dps. That removes the dps' total personal DPS plus a 1-stack of the charge buff from each of the 7 dps on their side. So, the drop in DPS from that death is:

7 x 218.6 + 4700 = 1530.2 + 4700 = 6230.2 DPS. That's about 1.87M damage over 5 minutes lost!

To observe the change in DPS using calculus, additional deaths will need to be considered in pairs (one dps from each group will drop total raid DPS by equal amounts) and we will assume that only the dps die. The change in total raid DPS from the first pair of deaths is:

-6230.2 x 2 = -12 460.4 DPS

Calculating the incremental changes in DPS from additional pairs of deaths, we get the following progression of values:

-12 460.4, -12 023.2, -11 586, -11 148.8, ...

This is in fact an arithmetic progression with common difference of 437.2 and first term of -12 460.4. Total DPS lost from n pairs of deaths will be the following sum of values:

S(n) = 218.6n^2 - 12 679n

Differentiating S(n) with respect to n:

dS(n)/dn = 437.2n - 12679. Since the first derivative is negative over the domain 0 < n < 8, S(n) will continue dropping.

Differentiating dS(n)/dn with respect to n:

d2S(n)/dn2 = 437.2. Since the second derivative is always positive, S(n) is decreasing at a decreasing rate.

We have the following trend:


Originally posted here.