Calculus optimization: you are building a new house on a cartesian plane whose units are measured in miles. your house is to be located at the point (2,0). unfortunately, the existing gas line follows the curve y=sqrt (16x^2+5x+16). it costs 400? **ive seen people solve it but im not sure where they got the 34

The cost of the pipeline would be 4.47049 × 400 = 1788.196 units of currency. Check the attached file on how to get the 4.47049 miles.

Using the Pythagorean theorem, 12^2 + 12^2 = 288 (equal lengths of a right-angle triangle). the square root of 288 is 16.9 (could be rounded to 17).

Rounding up to 17, the perimeter (P) would be 41. Without rounding, it would be 40.9. 

Using the 17 measurement as the Base, the height would come out to 4.82. This can be used to find the Area, which comes out to being 41. 

If you would like to double check, the measurements based on the picture are: 
Sides: 12, 12, 17 
Angles: 45, 45, 90

I hope this helps!