Question: Find the largest possible real part of\[(75+117i)z+\frac{96+144i}{z}\]where $z$ is a complex number with $|z|=4$.

Thinking:
I notice that this expression is a complex number, so I want to find its real part.<|end_of_thought|>

To do that, I need to simplify the expression first.<|end_of_thought|>

I can use the fact that $|z|=4$ to rewrite $z$ in polar form: $z=4e^{i\theta}$ where $\theta$ is some angle.<|end_of_thought|>

Then, I can use the distributive property and the fact that $i^2=-1$ to expand the expression: \[(75+117i)z+\frac{96+144i}{z}=(75+117i)(4e^{i\theta})+\frac{96+144i}{4e^{i\theta}}\] \[=300e^{i\theta}+468ie^{i\theta}+24e^{-i\theta}+36ie^{-i\theta}\]<|end_of_thought|>

Now, I can use the fact that $e^{ix}=\cos x+i\sin x$ and $e^{-ix}=\cos x-i\sin x$ to write the expression in terms of trigonometric functions: \[=300\cos\theta+468i\sin\theta+24\cos\theta-36i\sin\theta\] \[=(324\cos\theta+432\sin\theta)+i(432\sin\theta-324\cos\theta)\]<|end_of_thought|>

The real part of this expression is $324\cos\theta+432\sin\theta$, which is a linear combination of $\cos\theta$ and $\sin\theta$.<|end_of_thought|>

I know that the maximum value of a linear combination of $\cos\theta$ and $\sin\theta$ occurs when the coefficients are proportional to the direction of the vector $(\cos\theta,\sin\theta)$.<|end_of_thought|>

In other words, the maximum value occurs when the expression is equal to the length of the vector times the unit vector in the direction of the vector.<|end_of_thought|>

The length of the vector $(\cos\theta,\sin\theta)$ is $\sqrt{\cos^2\theta+\sin^2\theta}=1$, so the maximum value is just the length of the coefficients: $|324\cos\theta+432\sin\theta|$.<|end_of_thought|>

To find the maximum value, I need to find the angle that makes the expression equal to the length of the coefficients.<|end_of_thought|>

I can use the dot product formula to find this angle: \[(324\cos\theta+432\sin\theta)\cdot (1,1)=(324,432)\cdot (\cos\theta,\sin\theta)=324\cos\theta+432\sin\theta\]<|end_of_thought|>

The dot product of two vectors is equal to the product of their lengths and the cosine of the angle between them.<|end_of_thought|>

So, I have \[324\cos\theta+432\sin\theta=|324,432|\cos\theta=|324,432|\cos\phi\] where $\phi$ is the angle between the vectors $(324,432)$ and $(\cos\theta,\sin\theta)$.<|end_of_thought|>

I can use the inverse cosine function to solve for $\theta$: \[\theta=\cos^{-1}\left(\frac{324\cos\phi+432\sin\phi}{|324,432|}\right)\]<|end_of_thought|>

Now, I can plug this value of $\theta$ into the expression for the real part and simplify: \[324\cos\theta+432\sin\theta=324\cos\left(\cos^{-1}\left(\frac{324\cos\phi+432\sin\phi}{|324,432|}\right)\right)+432\sin\left(\cos^{-1}\left(\frac{324\cos\phi+432\sin\phi}{|324,432|}\right)\right)\] \[=324\frac{324\cos\phi+432\sin\phi}{|324,432|}+432\frac{-324\sin\phi+432\cos\phi}{|324,432|}\] \[=\frac{324^2\cos\phi+432^2\sin\phi+432^2\cos\phi-324^2\sin\phi}{|324,432|}\] \[=\frac{(324^2+432^2)\cos\phi+(432^2-324^2)\sin\phi}{|324,432|}\]<|end_of_thought|>

I can factor out the common factor of $324^2$ from the numerator and simplify the expression: \[=\frac{324^2(\cos\phi+\sin\phi)+432^2(\sin\phi-\cos\phi)}{|324,432|}\] \[=\frac{324^2\cos\phi+432^2\sin\phi}{|324,432|}\]<|end_of_thought|>

This is the maximum possible real part of the expression, and it occurs when $\phi$ is the angle between the vectors $(324,432)$ and $(\cos\theta,\sin\theta)$.<|end_of_thought|>

Final output:
# Answer

\frac{324^2\cos\phi+432^2\sin\phi}{|324,432|}