July 2016

04 Largest Right Triangle of Given Hypotenuse

Problem
Find the area of the largest right triangle whose hypotenuse is fixed at c.
 

03-largest-right-triangle-given-hypotenuse.gif           03-largest-right-triangle-given-hypotenuse-theta.gif

 

Manoj Karmakar's picture

Diagonaliseation a square matrix of any order.

Submitted by Manoj Karmakar on July 3, 2016 - 3:42pm

Let A be a square matrix of order n. Then (A - kI)X = 0, where k is called eigen value and I is called Identity matrix of order n, X is called eigen vector, is called characterstic equation. The equation A - kI = 0 gives n values of k and for each value of k, n column vectors. P be the square matrix of order n formed by the n column vectors.

The diagonal matrix of A = (inverse of P)AP.

Sydney Sales's picture

HOMOGENEOUS DE: $(x - y \ln y + y \ln x) dx + x(\ln y - \ln x) dy = 0$

Submitted by Sydney Sales on July 4, 2016 - 9:16am

1. (x - ylny + ylnx) dx + x(lny - lnx) dy= 0
2. (x csc y/x - y) dx + xdy=0
3. (x^2 + 2xy - 4y^2) dx - ( x^2 - 8xy - 4 y^2)=0
4. x^y ' = 4x^2 + 7xy + 2 y^2

Adrian T. Laurente's picture

Series, Patterns (Help please)

Submitted by Adrian T. Laurente on July 5, 2016 - 5:05pm

Hello mga Engrs/ Sirs,

Pwede po ba patulong ng correct answer and solution for this question:

A series of square figures are made with match sticks. If the first three figures are the following, how many matchsticks will be needed to form the 6th figure? see link po, thank you in advance po.

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Elimination of arbitrary constants: $y = Ae^{ax} \cos (bx) + Be^{ax} \sin (bx)$

Submitted by wackadoodle on July 6, 2016 - 8:41pm

Im having a difficulty solving this problem y = Aeaxcos(bx)+Beaxsin(bx) where a and b are parameters. Can anyone show me how to solve this ?

Rubie Ponce Imboy's picture

diffirential equations: how to eliminate arbitrary constant where h is a parameter; y=mx+h/m

Submitted by Rubie Ponce Imboy on July 8, 2016 - 9:01pm

please help
how to eliminate arbitrary constant where h is a parameter;
y=mx+h/m

Example 8 | Area bounded by arcs of quarter circles

Problem
Arcs of quarter circles are drawn inside the square. The center of each circle is at each corner of the square. If the radius of each arc is equal to 20 cm and the sides of the square are also 20 cm. Find the area common to the four circular quadrants. See figure below.
 

018-arcs-of-quarter-circles-common-area.gif

 

14 - Area the goat can graze inside a right triangular lot

Problem
A 30° right triangular lot has the long leg measuring 67 m. On the long leg and 15 m from the short leg, is a peg to which a goat is tied such that the farthest distance its mouth can reach is 30 m from the peg. Find the area inside the lot from which the goat can graze.
 

036-area-graze-by-goat-inside-triangular-lot.gif

 

Differential Equations: $(6x-3y+2)dx - (2x-y-1)dy = 0$

Submitted by agentcollins on July 10, 2016 - 5:02pm

Help a stranger please. This is for my homework and I'm having a hard time solving these equations. Show the complete solutions and final answers please. It will be a great help. Thank you so much.

This covers Additional Topics on Equations of Order One, Coefficient Linear in Two Variables.

1. (6x-3y+2)dx-(2x-y-1)dy=0

2. (x+2y-1)dx-(2x+y-5)dy=0

Sydney Sales's picture

Differential Equations: $[x \csc (y/x) - y] dx + x \, dy = 0$

Submitted by Sydney Sales on July 10, 2016 - 9:57pm

2. (x csc y/x - y) dx + xdy=0
3. (x^2 + 2xy - 4y^2) dx - ( x^2 - 8xy - 4 y^2)=0
4. x^y ' = 4x^2 + 7xy + 2 y^2

Differential Calculus: Cylinder of largest lateral area inscribed in a sphere

Submitted by Elainne on July 13, 2016 - 8:58pm

Find the ratio of the altitude h to the base radius r of a right circular cylinder having the largest lateral surface area S, if the right circular cylinder is to be inscribed in a sphere of radius R.

Differential Calculus: largest cone inscribed in a sphere

Submitted by Elainne on July 13, 2016 - 8:59pm

Find the dimensions of the largest circular cone that can be inscribed in a sphere of radius R.

Sydney Sales's picture

statics

Submitted by Sydney Sales on July 14, 2016 - 7:07pm

determine the required length of the cord AC so that the 8 kg lamp is sudpended. The undeformed length of the spring AB is I' ab = 0.4m, and the spring has a stiffness of Kab = 300 N/m.

Sydney Sales's picture

DE: $x \, dx + [ sin^2 (y/x) ](y \, dx - x \, dy) = 0$

Submitted by Sydney Sales on July 16, 2016 - 3:08pm

xdx + sin^2 ( y/x ) [ ydx - xdy ] = 0

Differential Equations: $(x - 2y - 1) dy = (2x - 4y - 5) dx$

Submitted by agentcollins on July 16, 2016 - 11:12pm

Coefficient Linear in Two Variables? Help please.

(x-2y-1)dy=(2x-4y-5)dx

Differential Equations

Submitted by agentcollins on July 16, 2016 - 11:14pm

Help me with this please.

The topic is Additional Topics in Ordinary DE of the first order.

y(3x^3 -x +y)dx + x^2(1-x^2)dy =0

Differential Equations: $(r - 3s - 7) dr = (2r - 4s - 10) ds$

Submitted by agentcollins on July 16, 2016 - 11:17pm

The topic is Additional Topics in Ordinary DE of the first order.

(r-3s-7)dr=(2r-4s-10)ds

Differential Equation: $ye^{xy} dx + xe^{xy} dy = 0$

Submitted by qwerty on July 17, 2016 - 8:28am

Please help me to solve this differential equation

yexydx+xexydy=0

Differential Equations: (3y - 2yx^2)[ 1 + ln^2 (2x^3 / 3y^2) ] dx - 2x dy = 0

Submitted by agentcollins on July 17, 2016 - 10:22am

The topic is Additional Topics in Ordinary DE of the first order. Thank you so much.

3y-2yx^2 [ 1 + (ln (2x^3 / 3y^2))^2 ]dx - 2xdy =0

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