### Video Transcript

The timing device in an automobileโs intermittent wiper system is based on an RC time constant and utilizes a 0.500-microfarad capacitor and a variable resistor. Over what range must ๐
be made to vary to achieve time constants from 2.00 to 15.0 seconds?

We can call the 0.500-microfarad capacitor ๐ถ. And we want to solve for the range of ๐
, the resistance. Weโll call the lower end of that range ๐
sub one and the upper end ๐
sub two that must be made so that the total time constants of our circuit vary from 2.00 to 15.0 seconds.

Weโll call the lower end of that 2.00 seconds ๐ก sub one and the upper end weโll call ๐ก sub two. And RC time constant ๐ก is simply the product of the resistance times the capacitance in a circuit. So we can write that ๐ก one equals ๐
one times ๐ถ and ๐ก two equals ๐
two times ๐ถ.

Rearranging these equations to solve for the resistances, ๐
one is ๐ก one over ๐ถ and ๐
two is ๐ก two over ๐ถ. Weโre giving the capacitance ๐ถ as well as the time range in our problem statement. So weโre ready to plug in and solve for the range of ๐
.

When we do and calculate these two fractions, we find that to three significant figures ๐
one is 4.00 million ohms and ๐
two is 30.0 million ohms. This is the lower and upper end of the range over which our resistor must vary.