Question 751408: A parabola shaped tunnel is 10m high at the centre and spans 18m wide at the base. A 7.5 m high vehicle needs to use the tunnel;, what is the maximum width of the vehicle that can fit through the tunnel?
I need to draw a diagram to represent the tunnel on a coordinate number plane , and find the equation of the parabola. using algebra and coordinate geometry to determinate the maximum width of the truck
I try to solve this problem many times but my results seems to be wrong, please help me I'm so stuck
Ally
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! The parabola is upside down and has the vertex at (0,10). Notice how this is the highest point.
The two roots are (-9,0) and (9,0). The horizontal distance between these two points is exactly 18 m. Basically how I found these two points is that I divided the distance of 18 m in half to get 9 m, then you walk 9 m in each direction to land on (-9,0) and (9,0)
So we have the three points: (0,10), (-9,0) and (9,0)
The first point is the vertex. The first point is also the y-intercept.
The next two points are the x-intercepts or roots.
Let's use the roots to find the equation
x = -9 or x = 9
x+9 = 0 or x-9 = 0
k(x+9)(x-9) = 0 ... for some fixed number k (we don't know it yet, but we'll find it soon)
k(x^2 - 81) = 0
The equation so far is y = k(x^2 - 81)
Since the y-intercept is (0,10), we can plug in x = 0 and y = 10 to find k
y = k(x^2 - 81)
10 = k(0^2 - 81)
10 = k(0 - 81)
10 = k(-81)
10 = -81k
-81k = 10
k = 10/(-81)
k = -10/81
So the equation that models the tunnel is  which distributes to  and that simplifies to 
So the equation in standard form is 
Now we're told that the truck is 7.5 m, so this means that the height y is 7.5, so we can plug in y = 7.5 and solve for x to find the clearance
 or 
 or 
So when or , the height is exactly 7.5 m, which means that the truck's width must be less than 2*4.5 = 9 feet across so it can fit in the tunnel.
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