Minimum Definition and 1000 Threads

In triangle ABC, ∠C = 90 degrees, ∠A = 30 degrees and BC = 1. Find the minimum length of the longest side of a triangle inscribed in triangle ABC (that is, one such that each side of ABC contains a different vertex of the triangle).

(1) For ##x>a## ##f(x)=x^2+x-a+1 \rightarrow## minimum value obtained when ##x=-\frac{1}{2}## Minimum value of ##f(x)=\frac{3}{4} -a## (2) For ##x<a## ##f(x)=x^2-x+a+1 \rightarrow## minimum value obtained when ##x=\frac{1}{2}## Minimum value of ##f(x)=\frac{3}{4}+a## But the teacher said...

The Attempt at a Solution I know the answer is supposed to be ##(-1,0)##. However when I differentiate the above expression I get. $$ 2x+{\frac 5 2} $$ Then the shortest distance would be when the expression equates to 0. $$ 2x+{\frac 5 2}=0 $$ I should be getting ##x=-1## but solving for ##x##...

Now I've tried looking at the problem like this. Considering that a is the length off the vehicles that he is trying to jump over I would consider that to be s. The plane from which he starts (b) should be the h. So considering that he is jumping from a horizontal plane, gravity should also...

Let $a,\,b$ and $c$ be positive real numbers. Determine the minimum value of $\dfrac{a+3c}{a+2b+c}+\dfrac{4b}{a+b+2c}+\dfrac{8c}{a+b+3c}$.

I tried to do it by derivative but there are two variables, so I don't know how to proceed. Does anyone know how I can solve it? Remembering that you don't need to find the value of ##\theta##. I just need to find a relationship between ##\theta_1## and ##\theta_2##

f=ma=71.5*9.8 = 700.7 I know this is not right because he is also going up against gravity but I don't know what else to use for acceleration. I don't know the angle but I assume it is a 90 degree cliff. w = (Fcos90)1380 = 0. But zero is not the correct answer.

Let $a$ be an integer. Consider the function $y=\dfrac{12x^2-12ax}{x^2+36}$. For what integral values of $a$ the maximum and the minimum of the function $y=f(x)$ are integers?

Attempt at solution in attachment.

Given that f is the function on (−∞, ∞) and the graph is the derivative of f 1.) Find the critical point on the graph ? 2.) Find the interval of the increasing function on the graph ? 3.) Find the interval of the decreasing function on the graph ? 4.) Find the point which is the absolute...

I have a formula for cost calculation that contain x and y two variable. I have to find the value of (x,y) where that formula will gives minimum value as cost should not be equal to zero, it has some minimum value. I took 1st partial derivative with respect to x and then with y and found the...

After getting the values of ψ₀(x) and ψ₁(x), I put them in the expression of ϕ(x) to get: ϕ(x) = (mw/πℏ)^(1/4) * exp[-(mw/2ℏ)x^2] * [α + βx√(2mw/ℏ)] Now when attempting to find the value of <x> by ∫xϕ(x) dx, I am having trouble determining the limits, as I am getting nothing useful by...

When the lamina rotates about A, FA must act on B (because it is the farthest away) perpendicular to AB (so that all of FA contributes to rotation). Same argument is valid for rotation of lamina about B as well. Having noted that, I tried two approaches: Approach 1- If I assume that the...

I defined the angle ##\beta## as the angle from the right horizontal to the ball C, from B, and ##\alpha## as the angle from the left horizontal to the ball A, from B. I also work in the CM frame, which has a velocity downwards of magnitude ##\frac{v}{3}## w.r.t. the lab frame. The positions of...

E = (1/√2)^2(E1) + (1/√2)^2(E2) = (E1+E2)/2 Let ψ(x,t=0) = ψ0 So, ψ1 = ψ0*exp(-i*E*T1/ħ) and, ψ2 = ψ0*exp(-i*E*T2/ħ) Given, <ψ1|ψ0> = <ψ2|ψ0> = 0 So, <ψ0*exp(-i*E*T1/ħ)|ψ0> = 0 => exp(i*E*T1/ħ)<ψ0|ψ0> = 0 => exp(i*E*T1/ħ) = 0 Similarly, exp(i*E*T2/ħ) = 0 So, exp(i*E*T1/ħ) = exp(i*E*T2/ħ)...

I have found code to find simply the minimum numbers needed, but I need to do it without repetition given the nature of an electric circuit. I hope that is a sufficient enough explanation of the problem. Despite being an engineering project this aspect is more mathematical.

I thought in this equations f is the man's pull\ f + dm*g = T < 600 Where dm is equal to the mass of the string that pull the up part (15-x) after descending x meters. dm/(15-x) = m/15 And, to the man: W - f = Mx'' I can solve this, and i got ~8m/s Is this right?

So my work includes using the acceleration formula a=delta v/t (Vrtf-Vrti)/a -> (0-54)/(-0.31) -> t=174 seconds I plug in 174 seconds to find the acceleration of the left train. and got -0.22m/s^2 I then used the displacement equation x=(1/2)at^2+Vo+So coming out with Xrt=4703m and Xlt=...

The height can be determined by conservation of energy (ignoring all friction). The mechanical energy when the car is at rest, equals the mechanical energy when the car is in the middle of the loop (at the top of the loop): \begin{equation} E_{0} = E_{loop} \\ mgh_0 = \frac{1}{2}mv^2+mgh_{loop}...

I have 2 quadratic functions and I am interested in their root in the specific range. I use quadratic equation to get their roots and what I find that if their any real solution exist for both or any of the function that lie in it designated specific range, then the roots are maximum or minimum...

OK going to comtinue with these till I have more confidence with it $$\dfrac{dy}{dx}=2 (1+x) (1+y^2), \qquad y(0)=0$$ separate $$(1+y^2)\, dy=(2+2x)\, dx$$

Homework Statement:: ... Relevant Equations:: . What is the minimum mathematic requirement to the Lagrangian and hamiltonian mechanics? Maybe calc 3 and linear algebra?

First, I tried using the Archimedes principle and calculated the weight of the surrounding air displaced when taking off. ##W = 2500\times 1.29\times 9.81 = 31637.25 N## But then, I got stuck and do not know how to proceed from here on. I don't want the full solution yet but can I get some...

S8.3.7.6. What is the minimum vertical distance between the parabolas $$y = x^2+1 \textit{and } y = x- x^2$$ Ok I think what question is ... The vertical distance between vertex's

S8.3.7.3. Find two positive numbers whose product is 100 and whose sum is a minimum $x(100-x)=100x-x^2=100$ So far Looks like it's 10+10=20Doing all my lockdown homework here since I have no access to WiFi and a PC. and just a tablet where overkeaf does not work

I recognise that the normal force must alwayss act towards the centre of the circle loop, as the rail always has to be exertign a pushing force on the car/carriage in order for it to follow the trajectoryof the loop. However , I cannot understand why, the reaction force has to be greater than...

Problem: Suppose that the function $p : N \rightarrow [0, 1]$ satisfies $p >> n^{-1}ln(n)$ (i.e. $n^{-1}ln(n) = o(p)$). (a) Prove that as $n \rightarrow \infty$, the random graph $G(n, p)$ has minimum degree at least $\frac{np}{2}$ almost surely. Idea: Look at the degree of each individual...

When I tried using the equations the only thing I could see is that it is impossible for such point to be an anti-node. In this case, how do I find the frequency? The answer is not even with the form of v*n/2L which is very confusing to me, I thought that the frequency of a standing wave must...

Since in Debye aproximation Debye's frecuency is defined as the maximum frecueny , the corresponding wavelenght should be the minimum one, due to the inverse relation among those λ=v/f=v·2π/ω=5.9 Å , which is higher than the given result. I believe I should be using the information 'cubic...

Many, many years ago while in engineering graduate school I was studying calculus of variations. One classic problem was to determine the shape of a hanging cable supported at its two ends. After minimizing the integral, the catenary curve was the solution. The basic assumption in setting up...

Here's what I tried to do: f Continuous function at R, x1 local minimum point of f, x2 local maximum point of f. Existing f(x1)>f(x2). Let's look at the interval [x1,x2]⊆ℝ . f is continuous in R and therefore continuous in its partial segment. Therefore f continuous in [x1,x2]. Therefore, there...

Hello to all, In a short pulse laser emission setup, can a pulse be emmited with beam length shorter than one wavelenght? (can a pulse have a duration shorter than its period?) Lets say a laser emmiter shoots a quarter cycle pulse, what would happen to this short beam? (lets supose the...

I tried to solve this problem and this is what I could come through: When the object is moving, the force acting on object is the frictional force, so, it got to be μmg. So, F = ma and as F is μmg μmg = ma μg = a So, to find out the magnitude of the initial velocity v given to the smaller...

We have 27 balls in the container, some of which are white and some black. How many white balls in the container must be at least, so that the probability that two black balls were drawn at random without a return was less than 23/30?

I've noticed that x^x is a minimum for x = e^-1 I put it as a high school problem because I presume it's one of those simple differential proofs/identities, but I can't really see how to get to e^-1. Too long since I did any calculus. Can someone please show me how to arrive at that? How about...

Im having a really tough time with this problem, I am assuming that in order for q'(z) to be a maximum, e^(-az/L)sin (pi(z)/L) must be a maximum. I believe this occurs when the derivative with respect to with respect to z/L is zero, which gives me z = 0.322L, but I am not sure if this is correct...

For this problem, since the weight force on the "particle" (child) is not always aligned with the tangential circular path of the disks, I couldn't think of a way to use rotational kinematics equations. As such, I tried to solve the problem using work principles (namely, that the change in...

The farthest distance of two places in an area is 200 km. If someone wants to make a map of that area on a 1 m × 1 m paper, the possible scale to make it is ... a. 1 : 210 b. 1 : 2.100 c. 1 : 21.000 d. 1 : 210.000 Can you help? The 200 and 210 makes me think that the distance on map won't be an...

By working with the following definition of minimum of a quadratic form ##r(\textbf{x})##, ##\lambda_1=\underset{||\textbf{x}||=1}{\text{min}} \ r(\textbf{x})## where ##\lambda_1## denotes the smallest eigenvalue of ##r##, how would one tackle the above problem? It is clear that the diagonal...

15. What is the minimum length of a plane mirror in order for you to see a full view of yourself? A 1/2 your height B 1/4 your height C 3/4 your height D your full height Q Why is answer of A given is the correct one, I understand pictorially how it is, since visually if you were to draw a...

This is the figure for the problem: 1.) Solved for initial total EPE of the system EPE system = (kq2q3/a) + (kq2q1/b) + (kq1q3/√a^2 + b^2) 2.) Solved for final EPE of the system negating q1 as if it were off to infinity EPE system final = (kq2q3/a) 3.) Plugged values into equation W =...

Proton is going towards the ##\alpha## particle. So, I am thinking of using the conservation of energy as the initial kinetic energy of the proton is known and initial interaction potential energy is zero. But, we don't know the kinetic energies of proton and ##\alpha## particle when they are at...

Below is my solution, which I'm not sure is correct. Hopefully the picture shows clearly what I'm trying to find.

So I integrated the work done on the object by both planets. Work1 is until x, and Work2 is from x to d. Where x is the point where both gravitational forces are equal. ##W_1=\int_0^x \frac{GMm}{r^2}dr - \int_0^x \frac{GMm}{(3R+D-r)^2}dr ## ##W_2=\int_x^D \frac{GMm}{(3R+D-r)^2}dr - \int_x^D...

Following up on @A. Neumaier's excellent series of articles on virtual particles, I'm confused about one thing (well of of several). If you pop over to the discussion of virtual particles on Matt Strassler's page, he mentions that, for example, an excitation in the photon (em) field will also...

Problem Statement: How to calculate minumum angular velocity of a mass on a spinning plate Relevant Equations: f=mrw^2 Hi, here's the question: a) A rough horizontal plate rotates with a constant angular velocity of w about a fixed vertical axis. A particle of mass m lies on the plate at a...

Using Lagrange multiplier ##\lambda## (only one is needed) the integral to minimize becomes $$\int_{\tau_1}^{\tau_2} (y + \lambda) \sqrt{{x'}^2+{y'}^2} d \tau = \int_{\tau_1}^{\tau_2} F(x, x', y, y', \lambda, \tau) d\tau $$ Using E-L equations: $$\frac {\partial F}{\partial x} - \frac d {d \tau}...

Recently I've come across a question that seems very simple, but had puzzled me for a while. Suppose a point object with mass M is placed on a rough plane inclined at 30 degree to the horizontal and is subjected to the force of gravity acting down vertically (to make it simple, assume g = 10...

Sadly, that's what all I could do.

Problem Statement: What is the minim Force F necessary to make the crate start moving up the incline? Relevant Equations: F_push=mgsin(ø)+F_f F_f= µ_s mgcos(ø) My values m = 80kg ø = 20 Fø = 15 static friction = 0,5 constant friction = 0,4 F _push = 80kg * 9.8 m/s^2 *sin(20) + 0,5 * 80kg *...