You need to convert time and speed into distance. Distance=(speed)*(time). Man1 starts at 1 pm and by 4 pm he has walked for 3 hours, and his distance is 3*3=9 miles. Man2 starts at 2 pm and by 4pm he has walked for 2 hours, and his distance is 2*4=8 miles. So we have a triangle and the included angle of 60 degrees. At 4 pm they are separated by sqrt(9^2+8^2-2*8*9cos60)=sqrt(81+64-144/2)=sqrt(145-72)=sqrt(73)=8.544 miles (cosine rule, and cos60=1/2).
To work out how fast they are separating we need to work out the rate of change of the distance between them. Let's suppose that they each walk for another h hours, where h is a very small fraction. Man1 walks a further 3h miles and Man2 a further 4h miles. Their separation is again given by the cosine rule:
sqrt((9+3h)^2+(8+4h)^2-2(9+3h)(8+4h)cos60)=
sqrt(81+54h+9h^2+64+64h+16h^2-72-60h-12h^2)=
sqrt(73+58h+13h^2).
[As an example, put h=1 hr, which is not a small value. The separation is then sqrt(144)=12. So the rate of change in separation is 12-sqrt(73)=3.456 mph.]
This means that their separation has increased by sqrt(73+58h+13h^2)-sqrt(73). Because h is very small we can ignore h^2 terms and just consider sqrt(73+58h)-sqrt(73)=sqrt(73)*sqrt(1+58h/73)-sqrt(73) or sqrt(73)(sqrt(1+58h/73)-1). We can now use the Binomial Theorem to expand (1+58h/73)^(1/2)=1+29h/73 ignoring terms with h^2 and beyond as insignificant. So now we have: sqrt(73)(1+29h/73-1)=29hsqrt(73)/73. This is the separation just after 4 pm, h hours after 4 pm in fact. So the speed of separation is this expression divided by the time, h=29sqrt(73)/73 or 29/sqrt(73)=3.3942 mph. [Compare this to the value when h=1 hr, when the average rate of change was 3.456 mph.]
What we have just done is calculated the results that calculus would have given us if we had formally applied it. By taking h as very small, the result we obtained applies to a vanishingly small h, or h approaches zero.
