.
The span of the parabolic arch in Figure B is 8 m.
At a distance of 2 m from the center, the vertical clearance is 4.5 m.
Find the height of the arch.
Hint: Choose a convenient coordinate system in which the equation of the parabola
will have the form x2 4p(y k). Figure B: https://ctrl.vi/i/z9urPS17M
~~~~~~~~~~~~~~~~~~~~~~~
I will solve the problem, but will not follow the hint,
because I have better way to solve.
Place the origin of the coordinate system at the ground, where the left branch
of the parabola meets the ground.
Then the parabola has left x-intercept at x= 0 and right x-intercept at x= 8.
So, I can write the parabola equation in the form
y(x) = -a*(x-0)*(x-8), or y = -ax*(x-8).
Here "a" is some positive real number. With the sign "-" before "a", the parabola is opened downward.
The parameter "a" is the only unknown now, and now my task is to find its value.
About this parabola, I know that the value y(x) at x = 6 is 4.5; so I write this equation
y(6) = 4.5.
It is the same as
-a*6*(6-8) = 4.5, or -6a*(-2) = 4.5, or 12a = 4.5.
From this equation, I find a = = = = 0.375.
So, the parabola equation is y(x) = -0.375x*(x-8).
The maximum height is at x= 4: = y(4) = -0.375*4*(4-8) = -0.375*4(-4) = 0.375*16 = 6.
ANSWER. The maximum height of the arch is 6 meters.
Solved.
======================
When people from outside ask me to solve a problem, I don't like to use
somebodies' hints - because often I have my own, much better plan/idea/solution in my head.
///////////////////
If you want to see many other similar and different solved problems, look into the lesson
- Word problems on engineering constructions of parabolic shapes
in this site.
You will find there complete solutions with detailed explanations,
so you will be able to learn the entire subject from there.
Happy learning ( ! )