# Search results for query: *

1. ### Shoot a basketball with a minimum speed at some angle

Homework Statement You should shoot a basketball at the angle ##\theta## requiring minimum speed. Avoid line drives and rainbows. Shooting from (0, 0) with the basket at (a, b), minimize ##f(\theta)= 1/(a \sin (\theta) \cos (\theta) -b \cos^2 (\theta))##. (a) If b =0 you are level with the...
2. ### Pick a,b,c,d for y=ax^3+bx^2+cx+d that models path of plane.

You're absolutely right. I forgot the passengers in the plane. I fixed the solution and the end result equation will be ##y=\frac{2h}{L^3}x^3+\frac{3h}{L^2}x^2## and with this, the passenger will land safe and sound. :D https://www.desmos.com/calculator/5ah7z7cs11
3. ### Pick a,b,c,d for y=ax^3+bx^2+cx+d that models path of plane.

Since I'm considering only the ##[-L,0]## interval, I would say that ##dy\over dx##(the limit is also one-sided) is negative for that point because the plane is going down to land. I think you're suggesting that the path of the plane before ##x=-L## is a straight horizontal line, then it starts...
4. ### Pick a,b,c,d for y=ax^3+bx^2+cx+d that models path of plane.

The solution I typed however didn't assume ##dy\over dx## =0 at ##(-L, h)##. I'm only considering ##y =-\frac{h}{L^3}x^3## over the interval ##[-L, 0]## for the path of the plane.
5. ### Pick a,b,c,d for y=ax^3+bx^2+cx+d that models path of plane.

Homework Statement A plane starts its descent from height ##y =h## at ##x = -L## to land at ##(0,0)##. Choose ##a, b, c, d## so its landing path ##y =ax^3 + bx^2 + cx + d## is "smooth". With ##\frac{\mathrm {d}x}{\mathrm {d}t} = V =##constant, find ##\frac{\mathrm {d}y}{\mathrm {d}t}## and...
6. ### Interpret success-rate/time * $I just want to ask, even though my solution in the end was valid, was the way of thinking to obtain the solution justified? 7. ### Interpret success-rate/time *$

Sorry, I feel confused a bit, isn't ##p(x)## a probability function? Aren't they supposed to be dimensionless?
8. ### Interpret success-rate/time * $This one I understand as this is how ##p(x)## was defined. I don't understand this one. I understand since my time is worth 10$/hour and that I want to get the $1000 scholarship, I wouldn't want to spend more than 100 hour writing for the scholarship. If I spend an extra hour on that... 9. ### Interpret success-rate/time *$

##p(x)## is the rate of success function, which calculates the probability of being successful, you're referring to ##p'(x)##.
20. ### B Volume of a Cube: Definition & Explanation

Thank you for the answer. I was having doubts about how I defined volume and feared that this interpretation of stacking layers upon layers of the surfaces would break down once you start introducing 3d shapes that doesn't work well with it.
21. ### B Volume of a Cube: Definition & Explanation

Then is the way I looked at volume the wrong way?
22. ### B Volume of a Cube: Definition & Explanation

Suppose if we have a cube: The volume of the cube is the product of the length, width and the height. All this time, I've been looking at it as: To get the volume, multiply the area of the cross section of the cube by how many "layers" it has. To elaborate with the diagram given, one can see...
23. ### Prove ##5^n+9<6^n## for ##n\epsilon N|n\ge2## by induction

What are the general guidelines that I should follow when writing a proof like this? I'm still quite new to proofs.
24. ### Prove ##5^n+9<6^n## for ##n\epsilon N|n\ge2## by induction

Homework Statement Prove ##5^n+9<6^n## for ##n\epsilon \mathbb{N}|n\ge2## by induction. Homework Equations None The Attempt at a Solution The base case which is when ##n=2##: ##5^2+9<6^2## ##34<36## Thus, the base case is true. Now for the induction step. Induction hypothesis: Assume...
25. ### Finding an angle of a triangle as function of another angle

No doubt that Ray's proposed approach is much more better than the approach I was going with. But I felt much more intrigued on investigating the method I was going with. Ever since ehild had suggested me a different function to find angle ##\theta## with the ##\arccos## function and that using...
26. ### Finding an angle of a triangle as function of another angle

Ah. I was too invested in the method I'd chose to consider this. This was a very straightforward approach like you said!
27. ### Finding an angle of a triangle as function of another angle

During halfway of the problem, I was presented with a system of trigonometric equation that you presented to me which is: ## \begin{cases} A\cos{k}=c\cos{\beta}-z \\ A\sin{k}=c\sin{\beta} \end{cases} ## Initially, I found ##k##(or the angle ##\theta## that we spoke of) using the ##\arctan##...
28. ### Finding an angle of a triangle as function of another angle

Forget my answer because I noticed that ##\alpha## and ##\beta## always add up to ##\pi## which doesn't make any sense because two angles in triangle doesn't add up to ##\pi##. I had graphed ##\alpha(\beta)=\pi -\left(\arcsin \left(\frac{z\sin x}{\sqrt{c^2-2cz\cos x+z^2}}\right)+\arccos...
29. ### Finding an angle of a triangle as function of another angle

Oh no. I forgot that ##\alpha(\beta)## has to be defined for all values in ##(0, \pi)##. Which means the solution should be this:## \alpha(\beta)=\begin{cases} \arcsin \left(\frac{z\sin {\beta}}{\sqrt{c^2-2cz\cos \beta+z^2}}\right)-\arctan \left(\frac{c\sin \beta}{c\cos \beta-z}\right)+\pi &...
30. ### Hey I posted in homework help but now it's gone

I've seen a post where the OP doesn't show any attempt on solving the problem but manages to get some help from users. Like this one:https://www.physicsforums.com/threads/math-olympiad-number-theory.916126/ but this user has shown in history that he/she won't always leave the attempt blank. So...
31. ### Finding an angle of a triangle as function of another angle

Alright! It seems that I've figured it out thanks to you. So the solution should be a piecewise function: ##\alpha(\beta)=\begin{cases} \arcsin \left(\frac{z\sin {\beta}}{\sqrt{c^2-2cz\cos \beta+z^2}}\right)-\arctan \left(\frac{c\sin \beta}{c\cos \beta-z}\right)+\pi & \text{ if }...
32. ### Finding an angle of a triangle as function of another angle

My bad. I had edited the post above you by the way.
33. ### Finding an angle of a triangle as function of another angle

My bad. I had edited the post above you by the way.
34. ### Finding an angle of a triangle as function of another angle

Why did you define ##\theta=\frac{a}{b}##? Are you talking about the ##\arctan## function? Because I think if the ##\frac{a}{b}## you are talking about are the triangle side ratios then it should be ##\arctan## because it maps side ratios to the appropriate angle. After taking some time...
35. ### Finding an angle of a triangle as function of another angle

I don't understand. It's true that the inverse trig functions' range are ##(\frac{-\pi}{2}, \frac{\pi}{2})## but why do I need to choose k in the appropriate quarter? I know why ##\alpha## should be between ##(0, \pi)## because there's 180 degrees in total in a triangle. Also, I made an error...
36. ### Finding an angle of a triangle as function of another angle

How are you so sure that it's being multiplied by a constant? Or I guess to rephrase it differently, what's the justification behind it? Assuming it's true, I tried graphing the solution which is ##\alpha(\beta)=\arcsin \left(\frac{z\sin {\beta}}{\sqrt{c^2-cz\cos \beta+z^2}}\right)-\arctan...
37. ### Finding an angle of a triangle as function of another angle

1. Find α(β) given that the sum of the 2 sides= ##(x+y)## and its third, ##z## is a constant for 0<β<180. You can imagine that there's two pieces of string connected between two points. One string is as long as the distance between the two points while the other string is longer. If you...