Question

A train tunnel is modeled by the quadratic function h ( x ) = − 0.18 ^ 2 + 25 , where x is the distance, in feet, from the center of the tracks and h ( x ) is the height of the tunnel, also in feet. Assume that the high point of the tunnel is directly in line with the center of the train tracks.

Round your answers to the nearest tenth as needed.
a) What is the maximum height of the tunnel? feet.
b) How wide is the base of the tunnel? feet.

Answer 1

a) 25 feet

b) Base width 23.57 feet

Step-by-step explanation: The expression:

h(x) = -0,18*x² + 25

is a quadratic function ( a parable). as a < 1 open down

The vertex of the parable is V(x,y)

a) V(x) = - b/2a = 0/2a V(x) = 0 to find V(y) we make use of the original equation and plugging x = 0

y = - 0.18*x² + 25 ⇒ y = 0 + 25 ⇒ y = 25

The Vertex is V ( 0 , 25 )

Now vertex in this case is the maximum height.

h(max) = 25 feet

b) To find how wide is the base of the tunnel. We have to consider that for h = 0 we are at ground level therefore the two roots of the quadratic equation will give the wide of the base of the tunnel

Then

h (x) = -018*x² +25 ⇒ 0 = -018*x² +25 ⇒ x² = 25/0.18

x² = 138.89

x = ± 11.79 ft

So we found interception with x axis and wide of the base is

2 * 11.79 = 23.57 feet