SOLUTION: 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

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.
RELATED QUESTIONS
A parabola shaped tunnel is 10m high at the centre and spans 18m wide at the base. A 7.5... (answered by josmiceli)

A parabola shaped tunnel is 10m high at the centre and spans 18m wide at the base. A 7.5... (answered by rothauserc)

A parabola shaped tunnel is 10m high at the centre and spans 18m wide at the base. A 7.5... (answered by josgarithmetic)

a road 12 m wide runs through a tunnel 18m high the cross section of the tunnel is a... (answered by CharlesG2)

A tunnel has the shape of a semi-ellipse that is 15 ft. high at the center and 36 ft.... (answered by josgarithmetic)

A tunnel has the shape of a semi ellipse that is 15 ft high at the center, and 36 ft... (answered by greenestamps)

A tunnel has the shape of semi ellipse that is 15 ft. high at the center, and 36 ft.... (answered by Alan3354)

a tunnel has a shape od semi ellipse that 15ft hight at the center,and 36ft across at the (answered by KMST)

A right pyramid, 18m high, has a square base measuring 10m by 10m. If the top section 3m... (answered by MathLover1,ikleyn)