Tower 1 is 60 ft high and tower 2 is 20 ft high. The towers are 140 ft apart. A guy wire is to run from point A to the top of each tower. [See a picture of this situation on page 274 of the textbook.]

How many feet from tower 1 should point A be so that the total length of guy wire is minimal? Do not include units in your answer.
Distance from tower 1 to point A is???

without a diagram, I assume point A is an anchor in between the towers. If its distance from T1 is x, then the cable length is

d = √(60^2+x^2) + √(20^2+(140-x)^2)
to find the minimum, we want dd/dx = 0
That happens when x=105

To find the optimal distance from tower 1 to point A, we can use the concept of minimizing the total length of the guy wire.

Let's assume that the distance from tower 1 to point A is x feet. Since the towers are 140 feet apart, the distance from tower 2 to point A would be 140 - x feet.

To calculate the total length of the guy wire, we need to consider the length of each individual segment. We have the vertical segment from tower 1 to point A, which is 60 feet. We also have the diagonal segment from point A to the top of tower 1. Finally, we have the diagonal segment from point A to the top of tower 2.

Using the Pythagorean theorem, we can find the length of the diagonal segments. The diagonal segment from point A to the top of tower 1 can be calculated as √(x^2 + 60^2) feet, and the diagonal segment from point A to the top of tower 2 can be calculated as √((140 - x)^2 + 20^2) feet.

Therefore, the total length of the guy wire can be expressed as:

Total length = 60 + √(x^2 + 60^2) + √((140 - x)^2 + 20^2)

To find the distance from tower 1 where the total length of the guy wire is minimal, we need to minimize the above expression.