A power station is on one side of a river that is 0.5 mile wide, and a factory is 5.00 miles downstream on the other side of the river. It costs $15 per foot to run overland power lines and $23 per foot to run underwater power lines. Estimate the value of x that minimizes the cost.

a. 1.00

b. 0.43

c. 0.00

d. 2.00

e. 1.44

The question did not define x, which is probably shown in a diagram.

Let us define x as the distance downstream of the power station across the river, to which point the under-river power line connects.

Then the cost function
C(x)=15*(5-x)+23*sqrt(0.5^2+x^2)
You need to minimize C(x) with respect to x.
Can you take it from here?

I will try but there is no diagram.

The solution for x does fit one of the answers. So the interpretation should be correct.

To estimate the value of "x" that minimizes the cost, we need to consider whether it is more cost-effective to run power lines overland or underwater.

Let's assume that "x" represents the distance downstream (in miles) from the power station to the point where the power lines cross the river. Consequently, the remaining distance from the crossing point to the factory will be (5 - x) miles.

The cost of running overland power lines would be $15 per foot multiplied by the total distance of (0.5 + x) miles. Similarly, the cost of running underwater power lines would be $23 per foot multiplied by the hypotenuse of the right triangle formed by the river and the distance along the river from the crossing point to the factory, which is given by the square root of [(0.5)^2 + (5 - x)^2] miles.

To find the value of x that minimizes the cost, we can set up the cost equations:

Cost_overland = 15 * (0.5 + x)
Cost_underwater = 23 * sqrt[(0.5)^2 + (5 - x)^2]

Now, we can substitute the given options for "x" and calculate the costs for each option. The option with the lowest cost will be the estimate for the value of "x" that minimizes the cost.

Let's perform the calculations for each option:

a. x = 1.00
Cost_overland = 15 * (0.5 + 1) = 22.5
Cost_underwater = 23 * sqrt[(0.5)^2 + (5 - 1)^2] = 23 * sqrt(18) ≈ 92.57

b. x = 0.43
Cost_overland = 15 * (0.5 + 0.43) = 19.95
Cost_underwater = 23 * sqrt[(0.5)^2 + (5 - 0.43)^2] ≈ 84.15

c. x = 0.00
Cost_overland = 15 * (0.5 + 0) = 7.5
Cost_underwater = 23 * sqrt[(0.5)^2 + (5 - 0)^2] ≈ 82.53

d. x = 2.00
Cost_overland = 15 * (0.5 + 2) = 52.5
Cost_underwater = 23 * sqrt[(0.5)^2 + (5 - 2)^2] ≈ 57.88

e. x = 1.44
Cost_overland = 15 * (0.5 + 1.44) ≈ 31.80
Cost_underwater = 23 * sqrt[(0.5)^2 + (5 - 1.44)^2] ≈ 75.43

Comparing the costs for each option, we can see that option (b) with x = 0.43 has the lowest cost of approximately $19.95.

Therefore, the estimate for the value of "x" that minimizes the cost is 0.43.

The correct answer is b. 0.43.