Solve the following mathematical word problem.

Two servers, X and Y, operate at 52% and 67% less capacity than the maximum capacity of a data center, respectively. Server X operates at 120 units above the standard threshold capacity, whereas Server Y operates at 30 units below the standard threshold capacity. If Server Z operates at 54% of the maximum capacity, by how many units does Z exceed the standard threshold capacity?

The correct answer is 180.

Step 1: Express the capacities of Server X and Server Y in terms of maximum capacity.

Let $M$ represent the maximum capacity of the data center, and let $T$ represent the standard threshold capacity.

Server X operates at 52% less capacity than the maximum capacity $M$:

$C_{\text{X}}=(1-0.52)M=0.48M$

Server Y operates at 67% less capacity than the maximum capacity $M$:

$C_{\text{Y}}=(1-0.67)M=0.33M$

Step 2: Set up equations relating server capacities to the standard threshold capacity $T$.

We are given that Server X operates at 120 units above the standard threshold capacity $T$:

$0.48M=T+120$   (Equation 1)

We are also given that Server Y operates at 30 units below the standard threshold capacity $T$:

$0.33M=T-30$   (Equation 2)

Step 3: Solve for the maximum capacity $M$.

Subtract Equation 2 from Equation 1 to eliminate $T$:

$(0.48M-0.33M)=(T+120)-(T-30)$

$0.15M=150$

$M=\frac{150}{0.15}=1000$

The maximum capacity of the data center is 1000 units.

Step 4: Find the standard threshold capacity $T$.

Substitute $M=1000$ back into Equation 1:

$0.48\times 1000=T+120$

$480=T+120$

$T=480-120=360$

Step 5: Calculate the capacity of Server Z and how much it exceeds $T$.

Server Z operates at 54% of the maximum capacity $M$:

$C_{\text{Z}}=0.54\times 1000=540$

Finally, find by how many units Server Z exceeds the standard threshold capacity $T$:

$\text{Exceeding Units}=C_{\text{Z}}-T=540-360=180$

Server Z exceeds the standard threshold capacity by 180 units.