Paul, a dentist, determined that the number of cavities that develops in his patient's mouth each year varies inversely to the number of seconds spent brushing each night. His patient, Lori, had 4 cavities when brushing her teeth 30 seconds each night. Write the equation that relates the number of cavities, c, to the time, t, spent brushing. How many cavities would Paul expect Lori to have if she had brushed her teeth for 120 seconds each night?

Step-by-step explanation:

Inverse variation is written as

which, in words, says "y varies inversely with x". If cavities varies inversely with time brushing, then

We are given the initial condition for which we need to solve for k:

c = 4 when t = 30:

so

k = 120.

Now we will use that value of k to solve the problem of how many cavities, c, would she have if she brushed her teeth 120 seconds, t, each night:

(the 120 on top is the k value and the 120 on the bottom is the number of seconds she brushed) to get

c = 1