Question

A homeowner installs a solar heating system, which generates 20 savings at a rate of 300e t dollars per year, t years after installation. Find a formula for the total savings after t years. If the system cost 30000 dollars to install, when will the savings match the installation cost

Answer 1

Answer:

"35.84 years" is the appropriate solution.

Step-by-step explanation:

The given values are:

Rate of savings generation,

$s'(t)=300e^{\frac{t}{20} }$

System cost to install,

= $30000

Now,

The total savings will be:

⇒  $s(t)=\int\limits s'(t) \ dt$

On substituting the values, we get

⇒         $= \int\limits 300e^{\frac{t}{20} } \ dt$

⇒  $s(t)=\frac{300e^{\frac{t}{20} }}{\frac{1}{20} }+c$

When,

Savings = 0

t = 0, then

⇒  $0=6000e^o+c$

⇒  $c=-6000$

The formula will be:

⇒  $s(t)=6000[e^{\frac{t}{20} }-1]$

On putting the values in the above formula, we get

⇒  $30000=6000(e^{\frac{t}{20} }-1)$

           $5=e^{\frac{t}{20}}-1$

        $e^{\frac{t}{20} }=6$

On taking log, we get

⇒  $\frac{t}{20} =log \ 6$

      $t=20 log 6$

        $=35.84 \ years$