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Answers by Richard P.

The easiest way to analyze problems like this one is via the concept of closing speed.   The doctor and Clara are closing the distance between then at the rate of  1.5 m/s + 2 m/s =  3.5 m/s.     The distance between them is decreasing at this rate.   ...

There are a number of approaches to this problem.   The approach using the least sophisticated methods is to multiply both top and bottom by √(1+t) + √(1-t).     The numerator becomes  (1+t) - (1-t)     =  2 t         ...

The series as you have written it diverges  (it is the harmonic series).   However a modification of this series converges conditionally to ln(2).   The modification is to consider the alternating sign version :      1 - 1/2 + 1/3 - 1/4 +...

Riemaan integral (answer)

To a mathematician, the   notation log(x) means the natural log function.    To scientists and engineers,  the natural log function is denoted  ln(x).   To avoid confusion, the ln(x) notation will be used here.   If one plots ln(x) over the...

The dilution equation is helpful here.    It is:     M1V1  =  M2V2       where the Mi  are the molar strengths of the two solutions and  Vi  are the volumes.     For the problem at hand  V2 = ...

This problem can be solved by setting up two free body diagrams:  one for mass A, the other for mass B. The cord connecting the two masses will have some tension T, which is the same everywhere in the cord.  Let the coefficient of friction be μ,the acceleration of gravity be g, ...

NH3  is a base,   so the conjugate acid must be the structure such that removal of an H+   results in NH3 .   That structure is the ion NH4 +  .       Conjugate acid/base pairs differ by one  H+  .   &nbs...

random variable (answer)

I used the TI-84 calculator for this.   Into L1   I put  0,1,2,3,4,5 Into L2    I put  .125, .25, .125, .25, .125, .125   Then Stat / Calc /1   with List : L1   ; FreqList  : L2    Calculate  ...

This system can be solved by using substitution multiple times, by Cramer's rule or matrix inversion.   However, the rref (reduced row echelon formation) function can be found in the catalog of of the TI-84 calculator.      Make  matrix A a 5x4 and enter the...

The most straightforward way of solving this problem is to cast it as an optimization problem in differential calculus.   Since the path must touch the x axis, the path will include a point with coordinates (x,0) for some value of x.   To be the shortest path it will touch...

The ratio of 490 to 350 is   7 to 5.   Since resonators closed at one end and open at the other have only odd harmonics, the 7th harmonic  (490 Hz) is the one that is next in sequence after the 5th (350 Hz).   Thus the fundamental is 350/5 = 70 Hz.  ...