May, 2008

Three forces 20 N, 30 N, and 40 N are in equilibrium. Find the angle between the 20 N and 40 N forces.

Find the eigenvalue of the matrix $\left[\matrix{1&1 \cr -2&4}\right]$

Determine the period and amplitude of y = 4 cos 2x.

Problem
Find the coordinates of the centroid of the area bounded by y² = 9x and the line y = x.

Answer: (3.6, 4.5)

Problem:
An investor wants to make 14% nominal interest compounded semi-annually on a bond investment. How much should the investor be willing to pay now for a 12% - P100,000 bond that will mature in 10 years and pays interest semi-annually.

Answer: P89,406

Problem:
A right cylindrical tank 2 m in diameter and 5 m long has its axis horizontal. It is filled with fuel to a depth of 1.5 m. Find the volume of fuel in the tank in liters.

Answer: 12,637 liters

There was a one hour interview on CNBC with Warren Buffet, the second richest man at that time who has donated $31 billion to charity. He is now the world's no. 1 richest man according to the news from forbes.com: Gates No Longer World's Richest Man

Here are some very interesting aspects of his life:

1. He bought his first share at age 11 and he now regrets that he started too late!
2. He bought a small farm at age 14 with savings from delivering newspapers.

Strictly speaking, do you know that the word antiderivative is far different from the word integral???

Do you know that antiderivatives are functions whereas integrals are real numbers??

Do you know that if $F'(x)=f(x)$ for all $x\in (a,b)$, then $\displaystyle\int_{a}^{b}{f(x)dx}$ and $F(b)-F(a)$ are two different things??? It just so happened they are equal....

We are blinded by so many books (specially those of Filipino Authors) and never know these are actually errors...

A line $\mathcal{L}$ consists of infinite number of points.

A segment is a "connected" subset of a line.

Do you believe that any two segments have the same number of points???

If yes, prove it! If no, disprove it.

A 5' 8" ft basketball player will shoot from a 10 ft high ring and from a distance of 18m from the base of the ring.

If the angle with reference to the horizontal is 55 degrees, find the initial velocity of the ball upon release ( up to 4 decimals)

There are 3 independent servers A, B and C.

The probability of failure are 10%, 5% and 3% respectively.

Find the probability that at most 2 will not fail???

2 circles with radii of the same length intersect each other's center.

Find the side of the largest possible square that can be put inside their overlapped/common region!

Go LJ!!!

If Nickson is just online, I am 100% this is just a piece of cake for him!

Are you now a member of EMC?

If cosα-cosβ=sinα+sinβ
Tan(3α/2-3β/2)=...

Go Lj!!!!

Find the general equation of all families of circles whose centers is on line y + x = 0 and tangent to both axes

A quarter circle has a radius = 4 units.

If two identtical circles are put in it, find the maximum length of either radii

Go Lj, I know you like this kind of problems!!!!

Find the volume generated when the area bounded by the following curves y = 2(x + 4)^2 + 3, 4x - y + 19 = 0 is revolved about

a.) x = 1

b.) y = 12

A Friend from Statistics

A variable $Z$ is said to be standard normal if and only if it is normal itself and if $E(Z)=0$ and $\text{Var}(Z)=1$.

It can be shown that its density function is given by

$f(x)=\dfrac{1}{\sqrt{2\pi}}\exp{\bigg(-\frac{1}{2}x^{2}\bigg)}$.

Questions:

1. For which value(s) of $x$ will $(x,f(x))$ be an inflection point of $f$?

2. What is/are the slope(s) of $f$ at the point(s) of inflection???

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Hi, Sta. Maria,

Philippine Mathematical Olympiad Sample Problem as published in Inquirer.net

Problem:
What is the probability of having four boys and four girls in a family of eight children?

Answer: 70/256

The answer above was also shown in the news. I made some attempt to solve this problem but my solution is too lengthy to post. This is my progress: the total number of possibilities is 2⁸ = 256, but I'm really having some hard time to get that 70 in the numerator.

Problem 49

The force between two magnetic poles at a distance r apart is F = k/r². Find the rate at which F changes with respect to r, when r = 3.

Solution 49

Problem 43

Find the points on y = x⁴ – 4x³ + 16x + 3 where the tangent is parallel to OX.

Solution 43

Forum Philosophy

Release no. 1: May 26, 2008

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The saying above is the underlying philosophy of our forums. We encourage all participants to uphold our philosophy.

Problem 39

Find the points on y = x³ + 6x² + 9x + 7 where the tangent is parallel to OX.

Solution 38