QuestionHW12: Problem 5 Previous Problem Problem List Next Problem much money to lay pipe in the water as it does on land, how far down the shoreline from $P$ should the pipe from the island reach land in order to minimize the total construction costs?
Distance from $P=$ $\square$
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Studdy Solution
Verify that this critical point minimizes the cost by evaluating the second derivative or using a sign chart. Since the problem is constrained by the shoreline from $P$ to 10 miles down, check the endpoints and the critical point:
The critical point $x = 2\sqrt{5}$ is within the range $0 \leq x \leq 10$.
Therefore, the minimum cost occurs at $x = 2\sqrt{5}$.
The pipe should reach the land at a distance of:
$$\boxed{2\sqrt{5}}$$