A tunnels arch is parabola shaped. It spans 8 meters wide, and is 5 meters high at a distance of 1 meter from the tunnel's edge. What is the maximum height of the tunnel?

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Answer 1 · 2018-05-10T12:07:11

# 80/7 # meters is the maximum.

Let's place the vertex of the parabola on the y axis by making the form of the equation:

# f(x) = a x^2 + c#

When we do this, an #8# meter wide tunnel means our edges are at #x=pm 4.#

We're given

#f(4)=f(-4) = 0#

and

#f(4-1)=f(-4 + 1)=5 #

and asked for #f(0).# We expect #a<0# so that's a maximum.

#0 = f(4) = a(4^2) + c#

# c = -16 a#

# 5 = f(3) = a(3^2) + c #

#9a + c = 5 #

#9a + -16 a = 5#

#-7a = 5#

#a = -5/7#

Correct sign.

#c = -16 a = 80/7 #

#f(0) = 80/7 # is the maximum

Check:

We'll pop #y=-5/7 x^2 + 80/7 # into the grapher:

graph{y=-5/7 x^2 + 80/7 [-15.02, 17.01, -4.45, 11.57]}

Looks correct at #(\pm 4,0) and (pm 3, 5). quad sqrt#