Answer
$a.\:\:\:\:C(x)=180\sqrt{x^2+640000}+1056000-100x$
$b.\:\:\:\:$ The least expensive point is less than $\ 2000 \mathrm{ft}\ $ from the point $\ P.$
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Work Step by Step
$a.\:\:\:$ First of all we need to find the distance from the power plant to the point $\:Q.\:$ Distance from $\:Q\:$ to $\:\mathrm{City}\:$ is $\:2(5280)-x=10560-x\:\mathrm{ft}.$ We have converted the miles to feet.
Now apply the pythagorean theorem to calculate the distance across the river, let us say represented by $\:d.$
$d^2=800^2+x^2$
$d=\sqrt{x^2+640000}$
So, the total cost for laying the cable would be:
$C(x)=180\cdot (\sqrt{x^2+640000})+100\cdot (10560-x)$
$C(x)=180\sqrt{x^2+640000}+1056000-100x$
$b.\:\:\:$ Pick a few numbers for $\:x\:$ and calculate the cost $\:C(x).$
$\:\:\:\:\:\:\:\mathrm{See\:the\:table\:above.}$
Based on the table, it seems that the cost is a minimum for some value of $\:x\:$ that is below $\:2000\:\mathrm{ft}.$
