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...
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
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...
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.
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...
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 $1000 scholarship, I wouldn't want to spend more than 100 hour writing for the scholarship. If I spend an extra hour on that...
Homework Statement
You are applying for a ##\$1000## scholarship and your time is worth ##\$10## an hour. If the chance of success is ##1 -(1/x)## from ##x## hours of writing, when should you stop?
Homework Equations
Let ##p(x)=1 -(1/x)## be the rate of success as a function of time, ##x##...
Forgive me for being inept at this but I don't understand what you said. Are you saying an example is enough to justify the current truth table of the conditional?
Why do we need to consider in the case(or even all of the cases from the truth table) where the antecedent and consequent of the statement are false to be a true statement so that our mathematical logic checks out? I've been seeing a lot of people trying to justify this by providing examples...
So if one knows the definition(truth values) of the conditional, one can deduce that it is true for any values of ##x##. But the author asserted this before he had completely determined the truth values of the conditional. How would one interpret what he meant given that the truth values hadn't...
I'm reading Velleman's book titled "How to Prove it" and I'm very confused when I'm reading about conditional statements. I understand that there exists some issue with the conditional connective and I accept that because that's the cost of espousing a truth-functional view. I came here to ask...
Thanks for the lengthy answer. There some parts where I don't understand but my immediate question is that why did you set the condition that ##n>1## and not ##n>0##? What's the reason behind it?
Homework Statement
The constant ##e## can be defined in many ways. My first exposure to this number involved compound interests. Specifically, if you decide to continuously compound 1$ at a 100% interest rate for a certain period of time(year, month, etc), you'll end up with ##e##$ where it is...
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.
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...
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...
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...
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##...
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...
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 &...
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...
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 }...
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...
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...
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...
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...