SOLUTION: To obtain maximum strength engineers often design tunnels as parabolic arches. In such a design if the highest point of the arch is 19 m above the road and the road is 20 m wide ,

Question 888333: To obtain maximum strength engineers often design tunnels as parabolic arches. In such a design if the highest point of the arch is 19 m above the road and the road is 20 m wide , determine the equation of the parabolic arch. You may find the equation using any method (vertex form, factored form etc) but you must,
a) set the bottom left corner of the tunnel as the origin
b) put your final answer into standard form
Found 2 solutions by stanbon, josgarithmetic:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
To obtain maximum strength engineers often design tunnels as parabolic arches. In such a design if the highest point of the arch is 19 m above the road and the road is 20 m wide , determine the equation of the parabolic arch. You may find the equation using any method (vertex form, factored form etc) but you must,
a) set the bottom left corner of the tunnel as the origin
b) put your final answer into standard form
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Draw the picture.
You have 3 points at:: (0,0),(20,0),(10,19)
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Form: y = ax^2 + bx + c
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Using 0,0 c = 0
Using 20,0 you get 400a + 20b = 0
Using 10,19 you get 100a + 10b = 19
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Modify:
20a + b = 0
10a + b = 1.9
----
10a = -1.9
a = -0.19
---
Solve for "b"::
10a + b = 1.9
-1.9 + b = 1.9
b = 3.8
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Equation:
y = -0.19x^2 + 3.8x
=======================
Cheers,
Stan H.
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Answer by josgarithmetic(39616) (Show Source): You can put this solution on YOUR website!
Working from standard form will be easier.

The road being 20 m wide means that half-way from the origin is 10 m, the "x" value for the vertex. The y value for the highest point is 19 m. The vertex is therefore (10,19).

Equation is . You also are given as instruced in part (a) that (0,0) is one of the points. This means

.

The equation fully in standard form is .
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