Final answer:
The distance between the two towers of the Tacoma Narrows Bridge is approximately 28000 feet.
Step-by-step explanation:
The distance between the two towers of the Tacoma Narrows Bridge can be found by determining the x-values where the two suspension cables intersect. To do this, we need to set the two equations for the cables equal to each other and solve for x.
Let's set up the equation:
- Equation for the first cable: f(x) = \frac{1}{7000}(x-14000)^2
- Equation for the second cable: f(x) = 27
By setting these two equations equal to each other, we get:
\frac{1}{7000}(x-14000)^2 = 27
To solve this equation, we can multiply both sides by 7000 to get rid of the fraction:
(x-14000)^2 = 27 \times 7000
Simplifying further, we have:
(x-14000)^2 = 189000
Now, we can take the square root of both sides to isolate x:
x - 14000 = \pm \sqrt{189000}
Finally, we can add 14000 to both sides to find the values of x:
x = 14000 \pm \sqrt{189000}
Calculating this, we get two values for x, which represent the x-coordinates where the cables intersect. The distance between the two towers is the difference between these two x-values. Therefore, the distance between the towers is approximately 28000 feet.