Cost A company wishes to run a utility cable from point A of...

Cost A company wishes to run a utility cable from point $A$ of the shore (see the figure on the next page) to an installation at point $B$ on the island. The island is 6 miles from the shore. It costs $\$ 400$ per mile to run the cable on land and $\$ 500$ per mile underwater. Assume that the cable starts at $A$ and runs along the shoreline, then angles and runs underwater to the island. Find the point at which the line should begin to angle in order to yield the minimum total cost.

Cost A company wishes to run a utility cable from point $A$ of the shore (see the figure on the next page) to an installation at point $B$ on the island. The island is 6 miles from the shore. It
costs $\$ 400$ per mile to run the cable on land and $\$ 500$ per mile underwater. Assume that the cable starts at $A$ and runs along the shoreline, then angles and runs underwater to the island. Find the point at which the line should begin to angle in order to yield the minimum total cost.

Key Concepts

Optimization

Optimization is the process of finding the most effective solution—typically the minimum or maximum of a function—subject to constraints. In calculus, this often involves setting derivatives equal to zero to locate critical points that potentially yield the optimal outcome.

Cost Function Modeling

Cost function modeling involves defining a mathematical expression that accurately describes the total cost in a problem based on various input variables. This process is crucial in translating real-world scenarios into a form where calculus can be applied to find optimal solutions.

Differentiation

Differentiation is a fundamental tool in calculus that calculates the rate at which a function changes with respect to one of its variables. It is essential for optimization problems, as finding where the derivative equals zero can reveal points where the function achieves a minimum or maximum.

Pythagorean Theorem

The Pythagorean theorem is used to calculate distances in geometric problems, particularly in right-angled triangles. In optimization problems involving spatial relationships, it helps express one distance in terms of another, thereby linking geometric conditions with algebraic expressions.

Critical Points Analysis

Critical point analysis involves identifying where the derivative of a function is zero or undefined, as these points are potential candidates for local extrema. Evaluating these points helps determine where the function attains its minimum or maximum value, which is a key step in optimization.

Transcript

00:02 Alright, so we're running a utility cable.

00:06 I'm going to draw a diagram.

00:10 The diagram looks like this.

00:12 Here is the shoreline.

00:18 And this point is point a and this point right here is point c and the point on the island is point b and that vertical distance that we've drawn here from the point from point b to point c is six units long.

00:42 And when i say six units long, that's actually miles.

00:45 So six miles long.

00:47 Then we know that the distance from a to c is nine miles.

00:53 I'm going to call the distance from c to this point right here, somewhere in between, as x.

01:02 And so then that makes the distance from that point to a, nine minus x.

01:09 We're trying to find the location of this point.

01:17 That is what we're trying to find.

01:20 And we want to minimize the cost to run the cable along the shore there and then underwater to point b.

01:32 So there's where we're running the cable.

01:35 Minimize the cost to run the cable along that green.

01:41 So first off, do you see this right triangle here? we need to find an expression for the length along that diagonal.

01:56 Using pythagorean theorem, you would have that that hypotenuse is the square root of x squared plus six squared.

02:11 Now let's write a formula for our cost in terms of x.

02:17 We're told in the problem that it cost $500 per mile underwater, and it costs $400 per mile to run the cable on land.

02:32 So we're going to have 400, 400 times 9 minus 6.

02:46 That's the cost for running the cable on land.

02:51 Then we have 500 times the square root of x squared plus 36.

03:05 That's the cost for running the cable underwater.

03:11 Now we want to minimize that cost, so we'll take the derivative.

03:19 Notice that the derivative of 400 times 9 minus 6 will be 400 times negative...

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