Answer to Question #249733 in Analytic Geometry for phindi

Question

A highway underpass is parabolic in shape as shown on the diagram below. The curve of the underpass can be modelled by the quadratic function

h(x)=−110x2+95x,

where x and h(x) are in metres. Point D is the highest point of the underpass. Two beams, AB and CD, are inserted to reinforce the curve of the underpass at the top. The length of beam AB is 10,00 m. Determine the length of beam CD to two decimal places.

Answer 1

$h(x)=−110x^2+95x=-110(x^2-95x/110+(95/220)^2)+110\cdot (95/220)^2=$

$=-110(x-19/44)^2+1805/88$

So, the vertex of parabola is (19/44, 1805/88)

$AB=10\implies A(19/44-5,y_A),\ B(19/44+5,y_B)$

$CD=y_0-y_A=y_0-y_A$

where y₀ is y-coordinate of vertex

Then:

$y_A=y_B=-110\cdot5^2+1805/88=-2729.50$

$CD=1805/88+2729.5=2750.01$ m