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Answers by Richard P.

The expression can be factored giving:   (sin(x) +4)(4 sin(x) - 1)  = 0   Since sin(x) cannot equal 4,  this implies   4 sin(x) -1 = 0.    Thus  sin(x) =  1/4    x = arcsin(1/4) (1st quadrant)  x...

The standard form of the equation of  parabola with vertex at the origin is     y = a x2          In this case y = 8  when x = 36/2 = 18 .  Plugging in and solving for a gives a = 0.02469   The P value for a...

Using implicit differentiation, I get    dy/dx = -(2 x ey + y ex) /(x2 ey + ex)    Evaluating this for x = 2, y = 0  gives      m =  dy/dx =  -4/(4 + e2)     Using the point slope form, the equation...

The yield per tree depends on the number of trees.    Let y(N) be the yield per tree if N trees are planted in the lot. The formula for y(N)  can be seen to be  y(N) = 1500 - 20 (N-65).     The size of the crop,  S, is given by:    S(N)...

The law of cosines can be used as the input to the process of implicit differentiation.   Let D be the distance between the friends and θ be the angle between (the line joining the center of the circle and the standing friend ) and (the radius line joining the center of the circle and...

One way to analyze this problem is to change from a rectangular system (x,y) to a polar system (r , θ)  via x = r cos(θ) ,  y = r sin(θ).    The expression to be analyzed becomes:      [r3 cos3(θ) + r3 sin3(θ) ]/r2  =   r[cos3(θ)...

Yes.  In fact this one can be done by direct substitution.   cot(pi/2) = 0   and csc(pi/2) =1     0/1 is not an indeterminate form  , so the answer is 0.

One way to address this problem is to ask:   What is the lower limit, as x approaches zero from above, of     sqrt(x + sqrt(x + sqrt(x + sqrt(x + ...)    ))    I do not know how to rigorously evaluate this limit.   However,...

Find angle. (answer)

I will assume that point E is inside the square.    The length of diagonal AC is 10 sqrt(2)   The law of sines   gives:     sin(154) / (10 sqrt(2)) =  sin( ∠EAC)/ 6.   Solving for  ∠ EACV   gives  ...

The answer to this one is fairly simple, but the reasoning leading to the answer is not.   The factorization is   [ (a2+b2)/2 +ab- (a2-b2)/2 +1] [ (a2+b2)/2 -ab +(a2-b2) +1]   This can be checked by multiplying it all out.   The...

Since the length of the major axis is 10,   the semi-major axis, a, is a = 5 The distance from the center to a focus is the c value, c = sqrt( a2 - b2)  where b is the semi-minor axis.    Since c =  2,    b = sqrt(21)    The...