An island is 3 mi from the nearest point on a straight​ shoreline; that point is 6 mi from a power station​. A Utility company plans to lay electrical cable underwater from the island to the shore and then underground along the shore to the power station. Assume that is costs ​2800​/mi to lay cable underwater and ​$1400​/mi to lay underground cable. At what point should the underwater cable meet the shore to minimize the cost of the project. Write an equation for the total​ cost, C​, as a function of​ x, where x is the distance from the shoreline closest to the island and the point where the cable leaves the shore. Find interval of interest, the mi where cost is minimized, and minimum installation cost.

I figured out the answers.

To find the point on the shoreline where the underwater cable should meet in order to minimize the cost of the project, we need to analyze the cost function for the cable installation.

Let x represent the distance from the shoreline where the cable leaves the shore. Therefore, the distance from the power station to the point where the cable leaves the shore would be (6 - x).

The cost of laying cable underwater is given as $2800/mi and the cost of laying underground cable along the shore is given as $1400/mi.

The cost of laying the underwater cable would be 2800*x, as the cost is per mile and x represents the distance from the shoreline where the cable leaves the shore.

The cost of laying the underground cable would be 1400*(6 - x), as the distance from the power station to the shoreline is (6 - x).

Therefore, the total cost function, C, can be written as:

C(x) = 2800*x + 1400*(6 - x)

Simplifying the equation, we get:

C(x) = 2800x + 8400 - 1400x
C(x) = 1400x + 8400

To find the interval of interest, we need to identify the range of valid values for x. Since the island is 3 mi from the nearest point on the shoreline, x can take on values between 0 and 3.

To find the point where the cost is minimized, we need to find the derivative of the cost function, set it equal to zero, and solve for x.

C'(x) = 1400

Setting C'(x) = 0, we get:

1400 = 0

Since this is a constant and not equal to zero, there are no critical points. Therefore, the minimum cost occurs at the endpoints of the interval.

Considering the values of x in the interval of interest, the minimum installation cost can be found by evaluating the cost function at the endpoints.

C(0) = 1400(0) + 8400 = 8400
C(3) = 1400(3) + 8400 = 12600

Therefore, the minimum installation cost is $8400, which occurs when the underwater cable meets the shore at the point closest to the island.

To solve this problem, let's break it down step by step.

Step 1: Determine the total cost function as a function of x.
Let's denote the distance at which the underwater cable meets the shore as x. The distance from the power station to the meeting point is then (6 - x) miles. The total cost can be calculated as follows:

Cost of underwater cable = $2800/mi * 3 mi
Cost of underground cable = $1400/mi * (6 - x) mi

Therefore, the total cost function C(x) can be expressed as:
C(x) = 2800 * 3 + 1400 * (6 - x)

Step 2: Find the interval of interest.
In this case, the distance from the shoreline closest to the island to the point where the cable leaves the shore should be between 0 and 6 miles. So the interval of interest is 0 ≤ x ≤ 6.

Step 3: Find the minimum cost.
To find the minimum cost, we need to find the value of x that minimizes the cost function C(x). We can achieve this by taking the derivative of C(x) with respect to x and setting it equal to zero.

dC(x)/dx = -1400

Setting -1400 equal to zero, we find that x = 4. This is a critical point.

Step 4: Confirm if it is a minimum or maximum.
To confirm whether x = 4 is a minimum or maximum, we need to examine the second derivative of C(x). If the second derivative is positive, it confirms that x = 4 is a minimum.

Taking the second derivative of C(x):
d²C(x)/dx² = 0

Since the second derivative is zero, we'll need to go back to the first derivative and examine the behavior to determine if it's a minimum or maximum.

dC(x)/dx is a constant (-1400), independent of x. Therefore, x = 4 is a minimum.

Step 5: Calculate the minimum installation cost.
We can substitute x = 4 into the cost function C(x) to find the minimum installation cost.

C(4) = 2800 * 3 + 1400 * (6 - 4)
= 8400 + 2800
= $11,200

Therefore, the minimum installation cost is $11,200.

In summary, the underwater cable should meet the shore 4 miles from the shoreline closest to the island. The minimum installation cost for the project is $11,200.