Question 184046
I would plot the parabola and put one x-intercept at the origin
(0,0) and the other x-intercept at (20,0). That would put the 
y-intercept at the origin.
The general equation is
{{{y = ax^2 + bx + c}}}
I have to find a, b, and c
since one intercept is at (0,0), I can say
{{{y = ax^2 + bx + c}}}
{{{0 = a*0^2 + b*0 + c}}}
{{{c = 0}}}
And the other x-intercept is at (20,0)
{{{y = ax^2 + bx }}}
{{{0 = a*20^2 + b*20}}}
{{{400a = -20b}}}
{{{b = -20a}}}
The arch is 14 ft high at the center
This is the vertex. The x-value of the vertex is
{{{(-b)/(2a) = (-(-20a))/(2a)}}}
{{{ (-b)/(2a) = 10}}}
This is the x-coordinate of the vertex
The problem says {{{y = 14}}} at the vertex
so, (10,14) is a point on the curve
{{{y = ax^2 + bx }}}
{{{14 = a*10^2 + b*10 }}}
From above, {{{b = -20a}}}
{{{14 = 100a - 200a}}}
{{{-100a = 14}}}
{{{a = - .14}}}
and
{{{b = -20a}}}
{{{b = -20*(-.14)}}}
{{{b = 2.8}}}
The equation is
{{{y = -.14x^2 + 2.8x}}}
I'll plot it
{{{ graph( 500, 500, -2, 25, -2, 16, -.14x^2 + 2.8x) }}}
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a.) What is the tallest 8ft wide truck that can pass through the arch?
If it goes straight down the middle, it will cover 4 ft on either side
The center of the arch is at {{{x=10}}}. 4 on either side would
be {{{x = 6}}} and {{{x = 14}}}. I should get the same y-value
for both of these
{{{y = -.14x^2 + 2.8x}}}
{{{y = -.14*6^2 + 2.8*6}}}
{{{y = -5.04 + 16.8}}}
{{{y = 11.76}}}
The tallest truck is 11.76 ft tall
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b.) what is the widest 12ft high truck that can pass through the arch?
It is 6 ft on either side of {{{x=10}}}
{{{x = 4}}}
{{{x = 16}}}
I can use either one to find {{{y}}}
{{{y = -.14x^2 + 2.8x}}}
{{{y = -.14*16^2 + 2.8*16}}}
{{{y = -35.84 + 44.8}}}
{{{y = 8.96}}}
The tallest truck is 8.96 ft tall