Question 1198183
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Rewrite y = -0.2x^2+ 27 as y = -0.2(x-0)^2+ 27


The second equation is of the form
y = a(x-h)^2 + k
where,
a = -0.2
h = 0
k = 27


The vertex is (h,k) = (0,27) representing the highest point on this parabola.
We know the parabola opens downward because a < 0.


Therefore, the highest point on the tunnel is at 27 feet.
Put another way: the largest h(x) can get is h(x) = 27, which occurs when x = 0.



We'll plug in y = 0 to solve for x, so we can find the roots of this parabola. Doing this will then help us determine how wide the tunnel is at the base.
y = -0.2x^2+ 27
0 = -0.2x^2+ 27
0.2x^2 = 27
x^2 = 27/0.2
x^2 = 135
x = sqrt(135) or x = -sqrt(135)
x = 11.61895 or x = -11.61895
Those decimal values are approximate.


Use a graphing tool like Desmos to confirm the answers.


The distance between the roots is
|11.61895-(-11.61895)| = |11.61895+11.61895| = 23.2379 feet approximately.


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Answers:
Max height of tunnel = 27 feet 
width at the base = about 23.2379 feet 
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