A $12-$cm long elastic resistive sensor with a $6-$mm diameter has a resistance of $1\mathrm{.}2$ k$.$

$($a$)$ Calculate its resistivity.$$

$($b$)$ Determine the resistance when it is wrapped around a patient's chest that has a circumference of $1\mathrm{.}2$m$.$ Assume that its cross$-$section remains circular in shape.$$

$($c$)$ Determine the change in voltage across the sensor terminals if the breathing increases the chest circumference by $15\%$ and a constant current of $0\mathrm{.}3$ mA is passed through the sensor$.$