Calculus (Basic) WORD PROBLEM Why Calculus is so POWERFUL!

[@charlesmitchell5841]

No need to apologize for the length of the problem. Calculus is the real deal. Thanks for the lesson.

[@bandini22221]

This takes me back to my first Calculus course, in the 70s, finding minimums and maximums. I've forgot almost everything.

[@noferblatz]

No problem on the length. This is a practical application for calculus. I would never have known that calculus was a method of solving the problem.

[@Hobby.Turner]

the angle where the rope hangs is the same both sides. tan a = x÷12, tan a = (30-×)÷28; x÷12=(30-x)÷28 28x=12(30-x) 7x=3(30-×) 7x=90-3x 10x=90 ×=9

[@DaveRissik]

It is interesting to note that the solution results in a common ratio of 4/3 (pole height to distance of anchorage from their respective bases) of the two triangles. This makes sense since such ratio leads also to minimising simultaneously their hypotenuse lengths which is what was set out to be determined!

[@Kleermaker1000]

Indeed. This means that there is also another, much easier way to solve this problem: find the same proportions of both right triangles: 12/x = 28/y, the sum of x and y must of course be 30, so y must be 30 - x. Then 12/x = 28/(30 - x) => 360 - 12x = 28 x => 360 = 40 x => x = 360/40 = 9. :)

[@bpark10001]

You don't need calculus for this problem, only geometry & algebra. Flip one of the poles over, so it pokes down into the ground. Draw straight line to connect the poles. This line will cross the ground line. That's the point for the ground stake. You have 2 similar right tringles, in ratio of 12:28. 12X + 28X = 30. 40X = 30, X = 3/4. Distance from 12' pole = (12)(3/4) = 9'. Distance from 28' pole = (28)(3/4) = 21'. A better problem is to determine the ratio of height to diameter of soup can to determine the most volume to surface area, or how much beer to drink from a can to maximize stability (lowest center-of-mass).

[@riziqabulawi9627]

I thank you for this opportunity to watch and help

[@jameslof2451]

Can also think of shining light from the top on the one pole to the ground which acts as a mirror. Light will take the shortest path. Angle of incidence equals angle of reflection, creating 2 similar triangles. Simply solve as a ratio problem. Also is you look in the mirror and see one pole inside the mirror, you will have a large right triangle with base of 30 and height of 40. Therefore, the shortest distance will be 50.

[@oladelebamgboye2813]

Fabulous explanation takes me back, thank you!

[@adriancazan6957]

I think the problem may be solved by geometry/trigonometry by reflecting one pole against the horizontal line and joining the end of the other unchanged pole with the end of the reflected pole. From engineering point of view consider the horizontal line as a mirror , but reflecting only one pole. The length of the straight joining line is the shortest length of the cable and one can also calculate where the connecting line crosses the horizontal line (between the poles).

[@tarekraja7537]

Excellent programme and analysis. Thank you sir and salute to you sir

[@Tim-Kaa]

Thank you

[@gregorymagery8637]

Another way to solve the problem : If you turn the second pole downwards, the shortest path between the ends of the poles is a straight line. This creates 2 triangles of the same shape of different sizes. => x/12 = (30-x)/28 |______ 28x = 12*30 - 12x | 40x = 12/30 | x = 9

[@gregorymagery8637]

Typo -3. line should be 40x = 12*30

[@jigneshsoni9263]

@@gregorymagery8637 how did you conclude it forms a similar triangles? I understand it would look similar visually. I am wondering what is the proof?

[@gregorymagery8637]

.@jigneshsoni9263 (Sorry for my bad English. I have to use Google Translate) The triangles are similar because all the opposite angles of the triangles are equal. One angle is 90 degrees and the other angle, where the line connecting the ends of the pillars intersects the ground plane is equal as well. When there are 2 congruent angles, the third angle is also equal. So the shape of the triangles is similar. I hope you understod what i ment

[@gregorymagery8637]

@@jigneshsoni9263 (Sorry for my bad English. I have to use Google Translate) The triangles are similar because all the opposite angles of the triangles are equal. One angle is 90 degrees and the other angle, where the line connecting the ends of the pillars intersects the ground plane is equal as well. When there are 2 congruent angles, the third angle is also equal. So the shape of the triangles is similar. I hope you understod what i ment.

[@gregorymagery8637]

@jigneshsoni9263 - This is not a proof but you can also think of angles like this: Light takes the SHORTEST PATH from one point to another Imagine that the ground plane is a mirror and you reflect a thin laser beam from the tip of one pole so that it hits the tip of the other pole. Light reflects from mirrors according to a certain rule. This rule is called the law of reflection: the angle of incidence is always equal to the angle of reflection. => 90 degrees + angle of incidence = 90 degrees + angle of reflection => So the third angle in triangles is also equal. => So the shape of the triangles is similar.

[@winterstar4813]

i love these

[@thomasabuckley2188]

Flip LH Δ down on Hor axis. Shortest distance between pole tops = straight line. Inspection shows angles same at ground point. Tans same = 12/x = 28/(30-x). Solving x= 9. No calculus!

[@richardhole8429]

Dang! Wish I had seen that solution. Your algebra beats the calculus!

[@richardhole8429]

Everybody, do scroll down to @thomasabuckey2188 for a very simple solution using algebra. Solves this in seconds. Imagine moving the right pole downward from the x axis. For all values of x, the wire length is the same. Draw a straight line between the posts to get the shortest wire. Brilliant solution, Thomas!

[@black__panther108]

can you explain more

[@thomasabuckley2188]

⁠@@black__panther108 OK. Here is more detail. Draw left pole down from ground level. The top of this pole, now pointing downwards, is the same distance from ground level. Now looking at pole tops, the shortest distance between them is a straight line. Where this direct line crosses the ground line is the required point. Notice that opposite angles, where these straight lines cross are equal. So the Tangents are the same. Left Δ, Tan = Opposite/ Adjacent = 12/X. (Using same references as used in calculus form of solution) Right Δ, Tan = 28/(30-X) So 12/X=28/(30-X) 360 - 12Z = 28X 360 = 40X X = 360/40 = 9. Note this method confirms that the extensive calculus solution provided is without error.

[@thomasabuckley2188]

Typo error: 360 = 28Z + 12X, should be 360 = 28X + 12X. Did not affect calculation.

[@thebluegarden2060]

My problem is the bottom leg is 30 feet across. Where did you get 10 and 20? It could be 9 and 21. It could be an infinite amount of length that total 30 feet. From the drawing, it looks like it’s centerline which will make it 15 and 15 making each angle 45°. Please answer me this. Where did you get 10 and 20 for the bottom leg?

[@Anu55203]

Thats why he took x and 30-x.

[@peterweusten4251]

Can do by reflection of 12 ft pole in ground level and join top of posts. Good to check answer

[@benavidezf]

Congratulations, you just proved that the shortest distance between two points is a straight line. You can get the same result by projecting the 12 foot pole into the ground and drawing a straight line between the tips of both poles. This gives you 12 + 28 = 40 on the vertical side and 30 on the horizontal which is a 3, 4, 5 triangle. The ratios of 28 and 40 is 7/10 so 30 x .7 is 21. Or 21 feet from the 28 foot pole and 9 feet from the 12 foot pole.

[@Jedermeister]

basic optimisation problem. fun one. My further mathematics students will like this

[@ralphhardie7492]

😂😂😂 Catcher in the rye teacher

[@kel-in5gi]

I don't understand @ 32:00 why you don't distribute the negative sign across the denominator.

[@tomtke7351]

T H A N K Y O U -- for others... review the "chain rule" before doing this problem. I'd completely forgotten it.

[@gpil2000]

Jeepers Batman! LOL I will replay at least 5 more times. Thanks for the exercises.

[@barryclements62]

Some of your solution was fuzzy owing to how many years ago it was taught. Is the length of wire then 50 ft?

[@JohnBerry-q1h]

For decades, high school Physics classes have been presenting Calculus equations to students who haven’t taken Calculus. Why apologize now? To do so would be to mess with a time-honored teaching tradition.

[@chamberizer]

It's like he explained how to use calculus, but solved as a quadratic. It might be a combination of both? I think - I may have to look at again. My wife wants to talk to me at the same time, so it is hard to concentrate.

[@kurtsopa]

The length of wire is 50. Please correct me if I’m wrong.

[@StevenStyczinski-sy8cj]

So the slopes are 4/3. 12 high by 9 long: 28 high by 21 long: 12/9=4/3: 28/21=4/3! O K , next set of poles.

[@glogenuniversal6047]

Say I want the maximum length of wire?

[@DancingPony1966-kp1zr]

Beautiful and accurate, of course. But you lost me way back there with all tose steps and big numbers. The trigonometric approach above seems quicker and more intuitive.

[@russelllomando8460]

Wow...

[@chriswright9819]

its pythagorus's

[@carloheinz6465]

Distance on left....12/40×28=9 or distance on right...... 28/40×12=21. (Test: 21+9 =30.). No need for such an elaborate approach 😬

[@chriswright9819]

it.s