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Answers by Richard P.

Circular Measure (answer)

There is a straightforward calculus approach to this problem.   Since 6,8,10 are the side lengths of a right triangle (double the 3,4,5 triangle) , there is probably a purely geometric approach as well.   The calculus approach is to note that the area bounded by a chord (length...

A trick answer to this is that all seven number are zero.    Zero is a whole number, the mean and median are both zero and the conditions on the greatest and least in terms of the median are satisfied.   I looked for other solutions but did not find any.  ...

Statistics (answer)

The difference between the mean  score (74) and a score of 99 is  25.   Dividing by the standard deviation (8), gives 3.125 .  This means that a score of 99 is more than three standard deviations above the mean.  Clearly, there will be few scores more than 99. ...

For part a)  To locate the zero of the electric field it is useful to consider three intervals of x:     (-∞ , -5)  ;   (-5, 0) ; (0, ∞)           For the first interval, the contribution from the positive charge...

The reason why the double filters reach a higher terminal velocity than a single one   is the same as the reason that a 120 kg parachutist falls faster (terminal speed) than does a 60 kg parachutist.  (Identical parachutes assumed).   At terminal speed, the air drag force...

I assume that the curve is  y =  x2   (a parabola)   This is an optimization problem.   The task is to minimize the distance, d, between a point on the parabola and the point (3,0).     It practice, it is somewhat easier to minimize...

This problem is a typical optimization problem.    There is a cost function , C(x,z)     and a constraint condition:   50 =  x2 z     C(x,z) = 2 x2 5  +  4 xz 2        The 2 is there...

The triple integral is    M = ∫dx ∫dy ∫dz  k z Since the boundary condition on x and y is the circle of radius a,   the easiest way to proceed is to change  ∫ dx ∫ dy  to  ∫ρ dρ ∫ dθ  That is - change to polar coordinates.   ...

The average rate of change is similar to an average velocity in physics related problems.  The average velocity is the change in displacement divided by the time that the change took.   So by analogy:     The average rate  is    Ravg = ...

The objective function, Q, is the potential profit that could be made.   From the problem statement  Q(x)  = p(x) x - 1 x  =   (8 - ln(x)) x -x   = 7x - x ln(x)   The calculus approach is to find the value of x that maximizes Q.    ...