Answer
$1310 saved per year
About 11 years
$11,200 over 20 years
Work Step by Step
The $\$2000$ worth of heat that the electric heater provides is $Q_H$. The amount of energy W required for the heat pump to deliver that amount of heat is $Q_H$ divided by the coefficient of performance.
$COP=\frac{Q_H}{W}$
$W=\frac{Q_H}{COP}$
Therefore, the cost of providing $Q_H$ from the electric heater should be divided by the COP to calculate the cost of running the heat pump. The annual savings is $\$2000$ minus that.
Savings = $\$2000-\frac{\$2000}{COP}=\$2000-\frac{\$2000}{2.9}=\$1310$
Calculate the payback time in years.
$$\frac{\$15000}{\$1310/year}\approx 11 years$$
The savings over 20 years is the savings in electricity minus the investment in the heat pump.
$\frac{\$1310}{year}(20 years) - \$15000 = \$11200$
