A power plant it located on the bank of a river that is .5 miles wide. Wiring is to be laid across the river and then along the shore to a substation 8 miles downstream. It costs $12,000 per mile for underwater wiring adn $8,000 per mile for wiring on land. If $72,000 is to be spent on the project, how far from the substation should the wiring come to shore?
Draw the picture its a right triangle ΔABC A is where the power station is B is directly opposite the power station on the other bank so the right angle is at B C is where the wiring from the power station intersects the shoreline extend BC to D (where the substation is located) so AB 0.5 miles Mark BC x and mark CD 8 - x Use the Pythagorean Theorem to find an expression for AC (the length of underwater wiring): AC √(x? + 0.5?) AC √(x? + 0.25) Now the cost of underwater wiring 12000 √(x? + 0.25) the cost of the wiring along the shoreline 8000 (8 - x) the total cost allowed 72000 so 12000 √(x? + 0.25) + 8000 (8 - x) 72000 3 √(x? + 0.25) + 2 (8 - x) 18 3 √(x? + 0.25) + 16 - 2x 18 3 √(x? + 0.25) 2x + 2 Square both sides: 9 (x? + 0.25) 4x? + 8x + 4 9x? + 2.25 4x? + 8x + 4 5x? - 8x - 1.75 0 x [-b ± √(b? - 4ac)] / 2a x [8 ± √(64 + 35)] / 10 x [8 ± √99] / 10 x [8 + √99] / 10 [rejecting the negative possibility b/c the length of wiring is positive] so the distance from the substation to where the wiring comes to shore 8 - [8 + √99] / 10 6.2 miles (to 1 dec pl) btw If they want it left as an exact answer then do this: distance 8 - [8 + √99] / 10 [80 - 8 - √99] / 10 [taking a common denominator and be careful with the signs] [72 - √99] / 10 miles
This one is challenging because of the tricky wording. You have to do most of the problem before you can truly understand the last sentence. First, let's compute the least amount of money we can spend 1) Go straight across the river -- 0.5 miles * $12k / mile $6,000. 2) Go directly down the shore -- 8 miles * $8k / mile $64,000. 3) Add them together for a minimum cost of $70,000 ($6+$64). Now, you have $72k to spend, but you only need $70k to go straight over and down. The words how far from the substation should the wiring come to shore? must mean the company would like to run the wiring UNDERWATER for part of the 8 miles. So, how much can you run wiring underwater for the extra $2k we have to spend? If it costs $12k per mile $2k * 1mile/12k 1/6 mile extra So, $2k will get you an extra 1/6 mile downstream underwater. Or, stated another way, the wiring should come to shore 7 and 5/6 miles from the substation. 8 - 1/6 7 5/6.
