Another Brain Teaser

[tlwaring]

Here's another one...
Bridges are often supported by arches in the shape of a parabola. A model for a specific parabola arch that supports a bridge is the function f(x)=-2/49x2sqr + 10/7x, where x is the horizontal distance from the base of the arch in feet and f(x) is the vertical height of the bridge.

Will this arch be tall enough for a road crew to build a county road under the bridge?

a) Complete the square to find the maximum height of the arch.
b) Can the average vehicle (no more than 6ft tall) be driven under the arch? Verify algebraically and then explain your reasoning.
c) The county decides to create a two lane road 20 feet wide under the arch. Will this road fit between the bases of the arch?
d) Can the average vehicle drive in either land under the arch and not scrape the paint off the roof? Or scrape the roof off the car?

Your help is greatly appreciated.

 

[dagr8est]

Here's another one...
Bridges are often supported by arches in the shape of a parabola. A model for a specific parabola arch that supports a bridge is the function f(x)=-2/49x2sqr + 10/7x, where x is the horizontal distance from the base of the arch in feet and f(x) is the vertical height of the bridge.

Will this arch be tall enough for a road crew to build a county road under the bridge?

a) Complete the square to find the maximum height of the arch.
b) Can the average vehicle (no more than 6ft tall) be driven under the arch? Verify algebraically and then explain your reasoning.
c) The county decides to create a two lane road 20 feet wide under the arch. Will this road fit between the bases of the arch?
d) Can the average vehicle drive in either land under the arch and not scrape the paint off the roof? Or scrape the roof off the car?

Your help is greatly appreciated.

Please use some parenthesis to make the function more clear.

 

[tlwaring]

f(x) = -(2/49)(xsquared)+(10/7)(x)

Is this what you mean?

I do appreciate your help!

 

[tkhunny]

f(x) = -(2/49)(xsquared)+(10/7)(x)

Is this what you mean?

I do appreciate your help!

So, Complete the Square.

f(x) = -(2/49)*[x² - 35*x]
f(x) = -(2/49)*[x² - 35*x + ___________] + (2/49)*____________

(35/2)²

f(x) = -(2/49)*[x² - 35*x + (35/2)²] + (2/49)*(35/2)²
f(x) = -(2/49)*[x - 35/2]² + (25/2)

This gives tons of information. The middle of the arch is x = 35/2. The highest point of the arch is at x= 35/2. f(35/2) = 25/2 = 12.5 ft, more than sufficient for your average vehicle.

Now, solve -(2/49)*[x - 35/2]² + (25/2) = 6. This will tell you where the arch is tall enough to accomodate traffic. Outside those solutions, you will have it too short. Subtracting the values of these solutions will tell you how much room is between them. Will it be enough for the road?

You had better be very clear about running big trucks through here.