Juliette,   The orthocenter of a triangle is the point at which the three altitudes of a triangle intersect.  This involves creating 3 lines, one from each vertex (K, L, and M) perpendicular to the line segment opposite it, so K⊥LM, L⊥KM, and M⊥KL. Because these three...

Zakiya,   In this situation the problem is the double counting.  Suppose no one played both sports, there would be 44 just football players and 35 just basketball players or 79 students playing sports, but we are told that only 55 students playing sports, so to reduce the...

Emily, the keys are (1) translating the words into algebra and (2) remembering that everyone ages at the same rate.  So if we define F=Grandfather's age now and D=Granddaughter's age now, the first sentence translates as F is 5 times as old as D.  The "is" is the equal...

Cashlin, these kind of problems involve converting the ratio or percentage into a number of people.  Check your numbers though, my math is giving me fractional people, which is always suspect in these kinds of problems).  There are often faster ways to do these kinds of problems for standardized...

Edward,   1st, recognize that csc=1/sin and sec=1/cos.  So we can simplify the equation to sin(6b+pi/8)=cos(2b-pi/8).  Generally trig problems are easier if you convert everything to sines and cosines.   2nd, sin & cos are 90 degrees out of phase or pi/2...

Brenda, the key is to recognize that you need to convert the percentages into amounts.   You start with 2 quarts of liquid that is 25% antifreeze, or you have 2 x .25 = .5 quarts of antifreeze to start with.   You are going to add another amount of liquid that is 100% antifreeze...

Todd, the key is to translate the sentences into equations. Let's define L=Length of the rectangle and W=Width of the rectangle   The word "is" tells you where the equals sign is.   Your first sentence translates: (a rectangles length) [is] 8 cm more than...

Ivy,   In general, the formula for compound interest involves:   P(0)=Principal amount in the account at t=0 P(t)=Principal in the account at time = t t = time in years n= Number of times the account compounds per year r= annual interest rate (expressed...

Emily, the key is to turn this geometry problem into a 2-equation / 2-unknown algebra problem. A = Angle S = Supplement   A + S = 180 degrees (definition of supplementary angles) S = A - 78 (The supplement (S) has a measure (=) of the angle (A) less 78)   Substituting...

Emily,   First we need to calculate the painting "rate."  A rate is something like miles per hour, that is to say an amount of something (i.e. miles) done in a unit of time (i.e. minutes).  In this case our generic rate is painted-walls per minute per person. ...

Anna,   There is a good explanation of the model I'm using below at http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCsQFjAA&url=http%3A%2F%2Fwww.csus.edu%2Findiv%2Fb%2Fblakeh%2Fmgmt%2Fdocuments%2FOPM101SupplC...

Esli,   First, the area of a rectangle is length (l) time width (w).  Now its not very useful to use l and w as the variables, but recognize that we can always swap l and w, that one will normally be larger and the other smaller.  So lets assume for the moment that the length...

Esli,   The consecutive numbers must both be positive (given), so representing the two numbers as (n) and (n+1), if we multiply n by 3 we have 3*n*(n+1)=6, or if we multiply (n+1) by 3 we get n*3*(n+1)=6 and the left hand side is the same for both equations 3n^2+3n=6...

Jeff, I think you have an error in your transcription   Take a look at this link   http://www.numberempire.com/graphingcalculator.php?functions=%281%2Bsin%282*x%29%29%2F%281-sin%282*x%29%29%2C%28cot%282*x%29^2%29&xmin=-3&xmax=3&ymin=-3&ymax=3&var=x   I've...

Heather,   Drexel has a good webpage describing the calculations using matrix algebra at https://www.math.drexel.edu/~jwd25/LM_SPRING_07/lectures/Markov.html   If its a little to technical, I'll summarize as needed for your problem   Given a transition...