Another great Andy Math video. In the real world, one would also like to know whether it is more cost efficient over time to build the "cheapest" route, or to build the shortest route, based on how many trucks per day run between the buildings and how many days it takes to recoup the ~33K by building the shortest route. That might make for an even more exciting result.
I think in real life it's quite a bit more than double for the same length of bridge vs road Is assume so that another reason why they usually take the shortest way
I saw this before! More or less... Speedrunners wanted to find out what the optimal angle was to hit a wall at in Paper Mario TTYD, because certain vectors of movement caused Mario to move faster if pressed against a wall. So substituting time for cost, they used calculus to find more or less what you came up with here. It was very cool.
The amazing part is that this is essentially a physics problem. It’s the equivalent of the critical angle in total internal reflection. You just correspond the cost to indices of refraction (or light speed) and you apply Fermat‘s principle for the brachistochrone. (Not even kidding, look it up).
I love your videos but I do wish you had gone over the calculus version instead of using the tool to tell you where the minimum was. I clicked on the video hoping to be reminded of how to do optimization problems but alas. Regardless I watched the whole thing and gave a like because I still love the channel!
Solve for x where the first derivative of the cost function equals zero to find a min/max. If the value of 2nd derivative at x is greater than zero, that tells you it is a local min. Solving cost values for all x at mins tells you the global minimum.
Loved this one! I solved it using derivation and imposing the result equal to 0 to find a local minimum. This kind of things have real application in design, which make them even more exciting.
That’s a long bridge. If it’s not a cable-stay bridge, there will be piers. If the piers are not properly protected, ships may collide with them and the bridge may collapse, as recently happened in Baltimore. Cost = giant mess. Therefore I vote for the shortest possible bridge. You have to plan for the life cycle of the project.
That's great for right now, but maintanence costs for the bridge (per mile) are certainly going to be higher than for the road. So over time, spending a bit more up front for the shortest bridge would eventually be cheaper in the long run.
True, but it doesn't always go like that. Sometimes to get the project funded you need to have a lowest capital cost and eat the higher operating cost against your profits.
would have preferred the derivative function being showcased more than the graphical for this channel but i do appreciate that nod to "other solutions". just can't really graph stuff in our head but we can do derivatives w pen/paper
Not always, depends on the costs of both variables. This is called linear programming and taking both extremes (all bridge or most possible road) as well as somewhere in between to make an accurate guess. Summary, it's not always best to cut the base, you have to try everything
If you're in a hurry, have the formula prepared ahead of time in such a way that all you have to do is plug in the width of the river and distance downriver from one factory to the other, and it automatically spits out an optimized value generalized to *any* situation. Don't wait until your computer is on fire to look up the "fastest method to back up your hard drive."
A similar problem is the lifeguard with the drowning swimmer. He can run fast along the shore but is slower once swimming in the water. How far along the shore should they run before diving into the water to reach the swimmer in the shortest time. Surprising, these problems can also be studied with Snell's Law regarding refraction of light passing through different media.
He's talking about the thumbnail, not the title. The thumbnail says find the least expensive bridge, which ignoring the road, and going straight across the river, would be $240,000
Basically, it's just a couple tens of thousands $ cheaper for a bridge + road so practical answer is just build the bridge for shortest distance since: time = $ and making shorter trips will help society more
"find least expensive bridge" on thumbnail is misleading, as the least expensive bridge will always be the route straight across the body of water @ 6 mi x 40 k$/mi = 240 k$
"Find least expensive bridge"? (as per your video tile) Zero calculations needed. It is a bridge built across the river in any spot as long as it is perpendicular to the banks. No matter where you build it. That will be your "least expensive bridge". Sometimes people overanalyze a question and get into unnecessarily complicated solutions.
Such a gem of a channel, fr fr
Another great Andy Math video.
In the real world, one would also like to know whether it is more cost efficient over time to build the "cheapest" route, or to build the shortest route, based on how many trucks per day run between the buildings and how many days it takes to recoup the ~33K by building the shortest route. That might make for an even more exciting result.
Another factor would be the maintenance cost of a mile of the bridge vs. a mile of the highway over the years.
I think in real life it's quite a bit more than double for the same length of bridge vs road Is assume so that another reason why they usually take the shortest way
This is a math question, so we don't think every possible variable
HOW EXCITING!!!
Indeed, HOW EXCITING!
Quite exciting!
I love it when he says that 😊
In every one of his videos atleast one guy just comments: How Exciting
Your comment are important so put a box on it.
The Engineering application is awesome! Love this one
I saw this before! More or less...
Speedrunners wanted to find out what the optimal angle was to hit a wall at in Paper Mario TTYD, because certain vectors of movement caused Mario to move faster if pressed against a wall. So substituting time for cost, they used calculus to find more or less what you came up with here.
It was very cool.
The amazing part is that this is essentially a physics problem. It’s the equivalent of the critical angle in total internal reflection. You just correspond the cost to indices of refraction (or light speed) and you apply Fermat‘s principle for the brachistochrone. (Not even kidding, look it up).
I love your videos but I do wish you had gone over the calculus version instead of using the tool to tell you where the minimum was. I clicked on the video hoping to be reminded of how to do optimization problems but alas. Regardless I watched the whole thing and gave a like because I still love the channel!
Solve for x where the first derivative of the cost function equals zero to find a min/max. If the value of 2nd derivative at x is greater than zero, that tells you it is a local min. Solving cost values for all x at mins tells you the global minimum.
Once you have the function F(x), find the derivative F'(x), set F'(x) = 0, then solve for x.
@@Abion47 Yes, but finding the derivative F'(x) is the only bit I didn't fully understand. I can't see where the 2x came from.
@@simonharris4873that’s the derivative of the inner function, x^2 + 36. We multiply by it according to the chain rule.
First time I've seen a practical use for a function of x. How exciting
Economics are littered with practical uses like this for production, cost minimalisation, profit maximalisation etc.
I did it differentiate with angles,and it was fairly easy: the angle the bridge makes to the vertical is exactly 30º.
Loved this one! I solved it using derivation and imposing the result equal to 0 to find a local minimum.
This kind of things have real application in design, which make them even more exciting.
Very appealing calculation😊
That’s a long bridge. If it’s not a cable-stay bridge, there will be piers. If the piers are not properly protected, ships may collide with them and the bridge may collapse, as recently happened in Baltimore. Cost = giant mess. Therefore I vote for the shortest possible bridge. You have to plan for the life cycle of the project.
That's great for right now, but maintanence costs for the bridge (per mile) are certainly going to be higher than for the road. So over time, spending a bit more up front for the shortest bridge would eventually be cheaper in the long run.
True, but it doesn't always go like that. Sometimes to get the project funded you need to have a lowest capital cost and eat the higher operating cost against your profits.
maintenance costs may change the final answer but the method of solution is still the same, you’re just working with different values for the costs
would have preferred the derivative function being showcased more than the graphical for this channel but i do appreciate that nod to "other solutions". just can't really graph stuff in our head but we can do derivatives w pen/paper
Lets say I'm in a hurry, will it always be somewhat effective to just cut the base length in half and then proceed with pythagoras theorem?
Not always, depends on the costs of both variables. This is called linear programming and taking both extremes (all bridge or most possible road) as well as somewhere in between to make an accurate guess. Summary, it's not always best to cut the base, you have to try everything
If you're in a hurry, have the formula prepared ahead of time in such a way that all you have to do is plug in the width of the river and distance downriver from one factory to the other, and it automatically spits out an optimized value generalized to *any* situation.
Don't wait until your computer is on fire to look up the "fastest method to back up your hard drive."
So, how much would two docks cost? Only diff then would be boats instead of trucks.
A similar problem is the lifeguard with the drowning swimmer. He can run fast along the shore but is slower once swimming in the water. How far along the shore should they run before diving into the water to reach the swimmer in the shortest time.
Surprising, these problems can also be studied with Snell's Law regarding refraction of light passing through different media.
Re-upload?
The title says "Find Least Expensive BRIDGE". The least expensive bridge is the shortest which would be 6 miles @ $240,000.
No it doesn’t.
@@rbbl_ Does on my PC. The thumb nail clearly says "Find least expensive Bridge". Click on 'Andy Math", then 'Videos' and scroll down to it. So 😛.
@@bradl7439 no 😜
He's talking about the thumbnail, not the title. The thumbnail says find the least expensive bridge, which ignoring the road, and going straight across the river, would be $240,000
@@neurathal0n534 thumbnail isn't a title 🤦♂
You could also derive the trigonometric expression, finding out that the angle of the bridge with the edge of the river is exactly 60°! Brilliantly
Basically, it's just a couple tens of thousands $ cheaper for a bridge + road so practical answer is just build the bridge for shortest distance since: time = $ and making shorter trips will help society more
Same thing I was thinking! Who needs a diagonal bridge LOL
Honestly, unlike other youtube matematician, you really make math simpler and more understandable. Thank you for your videos!
Cost Engineering at its finest.
It's actually cheapest to move one of the factories to right next to the other.
Would you mind making a video on the calculus version?
This question is a fun one
How did you made the derivation? I got lost there?
Stationary points? He made the derivative = 0 and then found the point
what would cost be if only built a road around with no bridge?
Could you make videos about applied calculus and differential equations
Can you do it in :
Mojolang
Julia
Clojure
?
I intend to try
Would have liked to see the calculus worked out
Exiting !
can we use snell's law here??
Snell's law!
this is SO cool
Why not say that is 2*sqrt(3)? instead 3.464?
I feel motivated to go build a bridge for $367 k
Would you like a road with that bridge?
video on the tech free method?
Snell's law?
"find least expensive bridge" on thumbnail is misleading, as the least expensive bridge will always be the route straight across the body of water @ 6 mi x 40 k$/mi = 240 k$
Or a route around the water. No bridge is the cheapest bridge.
Could you use snell's law to solve it?
Solve this question
Square ABCD has area 100
E is mid point of AB and F is midpoint of BC. AF and DE meet at G . Find the area of triangle DFG .
Как называется программа где ты строишь графики?
Thats insane
This is why I went tech school.
How exciting
!! A 3-4-5 triangle. Did you see it?
maxima minima!!
Man plz solve any one JEE Advance Question
The lesson here is that if you are good at math you can afford to own two factories!
How rigorous!
It costs $400 000 to build this bridge for twelve seconds
C=10 (Pythagoras!)
i thought graphing was cheating an used this opportunity to lear more about calculus :D i had to google a lot but i got there in the end
Have a golden retriever swim across to get a ball while recording. Then build to that path.
Pst... this is real btw
and that, my friends, is applied mathematics.
Basically, it's just a couple tens of thousands cheaper...
I love how you look like a jock
Cost of doing this much Maths will be 50000$
😮😮😮😮😮😮
Show the calculation next time! Ty!
That’s a cuberatic or whatever that shi called. We still at quadratics they easy asl when the hard stuff coming 😂
anyone else think of the brachistochrone curve
How confusing
Where is he 💀💀💀
didn't see the vid yet but its gonna b the same if u take the hypotenuse or if u use the sides
im the 1000th like
Why not take the derivative?
just use logistics bots duh
You're so cute
You have 2 factories
You have a lot of money
Pay 32153.90 more and make it straight
lol
"Find least expensive bridge"? (as per your video tile) Zero calculations needed. It is a bridge built across the river in any spot as long as it is perpendicular to the banks. No matter where you build it. That will be your "least expensive bridge".
Sometimes people overanalyze a question and get into unnecessarily complicated solutions.