Imagine a cannonball, right, flying through the air and uh an amusement park trying to squeeze every last dime out of ticket sales.
Yeah. Two very, very different things.
Exactly. But what if I told you, listening right now, that you can find the exact sweet spot for both of those things using the exact same mathematical blueprint?
It sounds kind of wild. I know.
It really does. Yeah.
Welcome to your custom deep dive. Today we're tearing apart a well a seemingly dry college math final.
Right, and we're using it to extract some incredible aha moments about finding maximum potential in the real world.
Because on paper, I mean, algebra just feels like staring at abstract equations, you know, it's easy to tune out.
Oh, absolutely. But when you realize the underlying logic dictates both naval battles and business profits, it suddenly becomes this very practical treasure map.
So, let's start with the naval battle. This is question 59 on this exam. Admiral Boom fires a cannon at the HMS Endeavor.
Okay, setting the scene. Right. The flight path is a parabola and it's written out as h of x=.96x-.004x^2
which uh sounds like an instant headache.
It totally is. But I look at that and see basically the arc of a thrown baseball. What goes up must come down.
Exactly.
But how do we actually figure out how high it goes without, you know, deo and calculus?
Well, we use the natural symmetry of that arc. First, we need to know where it starts and where it hits the ground. Those are the zeros.
The zeros. Okay.
Yeah. So, since the equation is 0.96x - 0.004x^2, we can just like pull an x out of both parts to simplify it.
Okay. So if x is the distance traveled, the first zero is just zero. The cannonball hasn't gone anywhere yet.
Correct. And for the second zero, we figure out where the rest of that equation cancels itself out.
Right. Right.
So we set .004x to equal .96. If you do the division there, 96 divided by 4 is 240.
Wait, so the cannonball hits the water at 240 feet away.
You got it.
Okay. So, if it launches at zero and lands at 240, and because a parabola is perfectly symmetrical, do we just like chop that distance in half
to find the peak?
That is the hidden shortcut. Yes. The absolute highest point is exactly halfway. So, 120 ft away.
Oh, wow. So, we just take that 120, plug it back into the original equation as x.
Yep.
And I've got the math right here. It comes out to a maximum height of 57.6 ft.
Which brings us to why that actually matters in this scenario.
Yes. Because the main math of the HMS Endeavor is 150 ft tall
and boomshot maxes out at 57 ft.
He's going to hit way too low. He'll definitely have some explaining to do.
He really will. But look at what we just did. We bypassed all that complicated math just by finding the starting line, the finish line, and cutting the difference in half.
Okay, physical space makes sense to me, but how on earth does that same physical symmetry apply to human behavior, like pricing theme park tickets?
Uh, well, this is where we look at questions 63 and 64, which about Seabreeze Park. Human behavior in markets often forms the exact same parabola.
No way.
Yeah. The park currently charges $11 a ticket, which gets them 800 customers. But their consultants realize that for every single dollar they raise the price, they lose 50 customers.
Ah, so there's the dilemma. You want more money per ticket, but if you keep jacking up the price, eventually the park is totally empty.
Right? So to find the sweet spot, we build another set of zeros for this profit parabola.
Okay, how does that work?
First, what if it admission is free. At $0, the equation shows they hit their absolute maximum capacity of 1,350 people.
Got it. But how do we find the other zero? Like the price where absolutely nobody shows up.
Let's do the arithmetic. They start at 800 customers losing 50 for every dollar increase. Okay.
So 800 divided by 50 is 16. That means they can raise the price 16 times before they run out of customers entirely.
Ah, I see it now. So the base price is $11 plus those 16 $1 increases. My that's $27.
At $27, zero people buy a ticket.
So, we have our two zeros, $0 and $27.
Wait, so applying the Admiral Boon trick, the peak revenue is just halfway between 0 and 27.
You got it.
Which is $13.50.
Without any trial and error, we know 1350 maximizes their daily revenue.
That is mind blowing. The mechanism for pricing tickets is mathematically identical to a flying cannonball.
It really is.
Okay, so we found the peak of cannonballs and profits. But wait, in both of those, our starting point was basically zero,
Right, The launchpad or the free ticket.
Yeah. What if you're looking at a situation where the starting line isn't zero?
Well, then you're looking for a baseline. Question 21 illustrates this brilliantly with Peter Piper growing peppers.
Growing Peppers.
His crop yield is a function of how much fertilizer he uses on a graph. That starting point before any variables are added is called the vertical intercept.
Meaning the vertical intercept isn't just a dot on a piece of graph paper.
No, not at all.
If we map that to the real world, that's the natural yield of Peter's orchard if he literally does nothing. No fertilizer, no extra peppers.
That is the crucial takeaway here. In algebra, the vertical intercept is just where X equals zero. But in life, it's your baseline. Whether you're adding fertilizer, raising ticket prices, or loading cannon powder, you cannot measure success unless you know exactly what your output is when your input is nothing.
I love that. It grounds all these abstract formulas. So this math final isn't just dry numbers.
Definitely not.
It's really a blueprint for identifying your natural baselines and finding your maximum potential.
It gives us a framework to, you know, stop guessing and start mapping our actual trajectories
which leaves you, the listener, with something to ponder today as you go about your routine. Right.
We're all on some kind of trajectory, right?
Absolutely.
What other parabalas are you writing in your own life right now? Whether in your career, your finances or your daily habits are is where you haven't calculated your absolute peak yet. Are you about to clear the mast or do you need to adjust your angle?