Tower 1 is 90 ft high and tower 2 is 40 ft high. The towers are 200 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?
To find the position of point A that minimizes the length of the guy wire, we can use the concept of optimization.
Let's assume that point A is x feet from tower 1. Now, we need to determine the length of the guy wire in terms of x.
We can create a right triangle with tower 1, point A, and the ground:
(top of tower 1)
* |
* | x
- - - * - - |- - - - - - - - - - - - - - - - - - - -
* | 90 ft
* |
(ground)
This triangle is similar to another right triangle formed by tower 2, point A, and the ground:
(top of tower 2)
* |
* | 200-x
- - - * - - |- - - - - - - - - - - - - - - - - - - -
* | 40 ft
* |
(ground)
Using the properties of similar triangles, we can set up the following proportion:
x / 90 = (200 - x) / 40
Now, let's solve this proportion to find the value of x:
Cross-multiplying:
40x = 90(200 - x)
Expanding:
40x = 18,000 - 90x
Combining like terms:
40x + 90x = 18,000
130x = 18,000
Dividing by 130:
x = 18,000 / 130
x ≈ 138.46
So, point A should be approximately 138.46 feet from tower 1 in order to minimize the length of the guy wire.