SOLUTION: An arch is in the shape of a parabola. It has a span of 256 meters and a maximum height of 32 meters. a. Find the equation of the parabola. b. Determine the distance from the c

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: An arch is in the shape of a parabola. It has a span of 256 meters and a maximum height of 32 meters. a. Find the equation of the parabola. b. Determine the distance from the c      Log On


   

Question 1204076: An arch is in the shape of a parabola. It has a span of 256 meters and a maximum height of 32 meters.
a. Find the equation of the parabola.
b. Determine the distance from the center at which the height is 12 meters.
Found 2 solutions by greenestamps, MathLover1:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let the origin be at the center of the base of the arch. The vertex is then (0,32) and the equation is of the form

y=ax%5E2%2B32

Determine the coefficient a using the fact that the feet of the arch are at (-128,0) and (128,0).

0=x%28128%5E2%29%2B32
x=%28-32%29%2F%28128%5E2%29=-1%2F512

ANSWER (a): The equation of the arch is y=%28-1%2F512%29x%5E2%2B32

To find the distance from the center where the height is 12, set y=12 in the equation and solve for x.

12=%28-1%2F512%29%28x%5E2%29%2B32
%281%2F512%29%28x%5E2%29=20
x%5E2=20%2A512=10%2A1024=10%2832%5E2%29
x=32sqrt%2810%29

That is 101.2 meters, to the nearest tenth.

ANSWER (b): approximately 101.2 meters

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Note the second question answered by tutor @MathLover1 is not the question that was asked....

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
a. Find the equation of the parabola.

a span of 256 meters , x coordinate of the vertex is 256%2F2=128 meters and a maximum height of 32
Vertex at ( 128 , 32 )

vertex form of the equation is
y=a%28x-h%29%5E2%2Bk , ( +h , +k ) is the vertex
y=a%28x-128%29%5E2%2B32
the parabola starts at ( 0 , 0 ) that is a point ( +x , +y )
0=a%280-128%29%5E2%2B32
a%28-128%29%5E2+=+-32
16384a=-32
a=-32%2F16384
a=-1%2F512
so, equation is
y=%28-1%2F512%29%28x-128%29%5E2%2B32

b. Determine the distance from the center at which the height is 12 meters.
the height of the arch 12 feet from the center
center is at 128, +12 meters from the center so x=128%2B12=140

y=%28-1%2F512%29%28140-128%29%5E2%2B32
y=1015%2F32
y=31.7 approximately