What was my pulse rate? Exponential problem?

Question

My resting heart rate is 60 pulses/minute. After a strenuous run, I stop and immediately count my pulse. In the first 60 seconds, I counted 150 pulses, in the next 60 seconds, I counted 119 pulses. What was my pulse rate at the moment I started counting? Pulse rate drops exponentially to 60 pulses/minute anytime I stop exercising, starting from any higher pulse rate.

Thank you, Sheri. Maybe I just faked my numbers just to make this math problem work out?

redwing64, this is a real life practical problem. How to best measure peak pulse rate if it drops too fast to measure directly? Of course, one can take a quick 10 second count, but is there an easy, more accurate way?

Answer 1

170 bpm!!!

Mathematically your pulse per second as a function of time is

p(t) = 1 + a*exp(-b*t)

this gives you a pulse of 1 p/second (=60 pulse/min) when t is large.

The first reading of 150 pulse, after 60 second is equal to the integral of p(t) between t=0 and t=60s. The second reading of 119 pulse is the integral between t=60s and t=120s

∫p(t) dt = t - a/b*exp(-b*t)

This gives rise to two equations and two unknown:

-(a/b)*exp(-60b) + 60 + a/b = 150

and

-(a/b)*exp(-120b) + 60 + (a/b)*exp(-60b) = 119

Solving this numerically gives

a=1.84
b= 1/143 = 0.006993

So p(t) = 1 + 1.84*exp(-t/143)

and p(0) = 2.84 pulse per second = 170.4 pulse/minute

Answer 2

Interesting question. I'm sure there are simple ways to solve it but it's too long since I did integrals in a meaningful way. Is this from a math test/assignment or is it something that you simply want to calculate? Problems from real life are usually more interesting, but not always easy to solve!

Answer 3

See a doctor. Maybe you're just healthy after all.