## IMPORTANT Model Parabolic Bridge as Quadratic Equation

parabolic bridge application a parabolic

bridge is 45 meter wide the height of

the bridge 5 meter from the outside edge

is 10 meters determine the quadratic

function that models the parabolic

bridge find the maximum height of the

bridge so let's make a sketch of this

and then try to solve it right so let's

say this is the parabolic bridge and

that's a level ground now in this case

we will assume that when we are trying

to make the equation of the bridge we

could have our axis at the origin here

or in the center either way we can solve

it right so I'll solve it using my

coordinate axes on this side on one edge

of the bridge and you do the same thing

with the vertex or the y-axis right in

the center and see what results do you

care right so here now if I place my

y-axis here in x-axis along the base of

the break in that case the width given

to us is 45 so this point will

correspond to 45 and we also know that

height of the bridge 5 meter from

outside edges 10 that means if I move 5

meters from here then the height is 10

similarly I can extend this from the

symmetry 5 meters from here that means

this would be a point at 40 the height

will be 10 right so that is a given

situation now if we use these as our

x-intercepts then we can write down the

equation of this bridge as y equals to a

times X times X minus 45 so that

represents the equation of this bridge

in factored form now since we know one

of the points we can find what a is and

that will give us the equation of the

parabola so let's substitute this X is 5

and Y is 10 so we get 10 equals to a

times

five times five minus 45 now five -

forty five is forty times five equals

two times a equals 210 so we can now

solve it so it ten equals to a times

let's write five times forty first right

and then from here a is equals to well

this is minus forty so a will be equals

to 10 over 5 times 4 is 20 so minus 200

right so that is what you get value of a

or which can be simplified and written

as minus 1 over 20 so minus 1 over 20 is

the value of a for us in this equation

so that becomes that gives us the

equation of the parabola and now let's

write down our equation we're just the

part a determine the quality function

that models the parabola brace quality

function will be y equals 2 is minus 1

over 20 x times X minus 45 so this is

the quality function which models the

given situation now Part B is find the

maximum height of the bridge now the

maximum height of the bridge will be

right in the center where x equals to 20

right so we can substitute 20 here and

find our equation so we get y equals 2

minus 1 over 20 times X is 20 set X is

20 okay not 20 but half of 45 sorry 22.5

so it will be it will be 45 divided by 2

right half of 45 so midway in between so

maximum will be at 0 plus 45 divided by

2 that is the axis of symmetry sorry Oh

H is 22 point five so it is twenty two

point five minus 45 I should write here

times twenty two point five times five

minus 45 so that gives us

the maximum for the bridge so we can

calculate this which gives us twenty two

point five minus forty five will be

minus twenty two point five and this

minus and that will become positive

twenty two point five so we get twenty

two point five times twenty two point

five divided by twenty as the maximum

height of the bridge so that gives us so

we can write down here as twenty two

point five square divided by 20 which is

equal to twenty five point three one two

five which is twenty five point three

one two five so we can round it to

approximately twenty five point three

meters so the height of this bridge is

going to be twenty five point three

meters so that is how we can calculate

the answer for this particular question

now as an exercise what you should do is

you should plot your y-axis right in the

center so then these will be twenty two

point five apart all right - twenty two

point five and plus twenty two point

five and find the equation of the

parabola right as an exercise what so

for this one we do get our answer and

that is the equation of the bridge is

here that is Part A and second the

maximum height is 25 point three meters

thank you