$($a$)$ What is the best coefficient of performance for a heat pump that has a hot reservoir temperature of $55\mathrm{.}3$C and a cold reservoir temperature of

$12\mathrm{.}2$C$?$

$($b$)$ How much heat in kilocalories would it pump into the warm environment if $3\mathrm{.}60107$ J of work $(10\mathrm{.}0$ kW $$ h$)$ is put into it$?$

kcal

$($c$)$ Assume the cost of this work input is $10/$kW $$ h$.$ Also assume that the cost of direct production of heat by burning natural gas is $89\mathrm{.}0$ per therm $($a common unit of energy for natural gas$),$ where a therm equals $1\mathrm{.}055108$ J$.$ Compare the cost of producing the same amount of heat by each method.

cost of heat pumpcost of natural gas

$=$