Question

a bridge is supported by three arches. The function that describes the arches is h(x) = 0.25x^2 + 2.375x where h(x) is the height, in meters, of the arch above the ground at any distance, x, in meters, from one end of the bridge. How far apart are the bases of each arch?

Answer 1

\[h(x) = 0.25x^2+2.375x\]The bases are going to be at the points where \(h(x) = 0\). Can you solve \(0 = 0.25x^2+2.375x\)?

Answer 2

That equation must not be correct. You need a negative term in there somewhere, or it just keeps getting higher....

Answer 3

Are you sure it isn't \(h(x) = -0.25x^2+2.375x\)?

Answer 4

yes omg you're right
so sorry!

Answer 5

what does that mean!?

Answer 6

Well, it's going to be an inverted parabola. at x = 0, that's the end of the bridge. at the other spot where y = 0, that's the first base.

Answer 7

so, you need to solve the equation to find where h(x) = 0. There are two solutions, although one of them is at x = 0. The other one is the spot where the base goes. Make sense?

Answer 8

This is also the path that a baseball would follow if you threw it up in the air (with no air resistance, that is)

Answer 9

okay...

Answer 10

whats next

Answer 11

Well, did you solve the equation?

Answer 12

ahh no, maybe im not getting it-

Answer 13

where is the second base

Answer 14

Okay, no problem, we'll fix that! :-)

So, we want to find the value of \(x\) such that \(h(x) = 0\) to find the other end of our arch. \[h(x) = 0 = -0.25x^2+2.375x\]A little experience tells me that those ugly numbers are all 1/8ths, but even easier is to just multiply through by 1000 and get rid of the decimals altogether.
\[0 = -0.25x^2+2.375x\]\[0*1000=-0.25*1000*x^2+1000*2.375x\]\[0=-250x^2+2375x\]We can factor that, or use the quadratic equation, which do you prefer?

Answer 15

im more familiar with factoring, but whichever is best to teach

Answer 16

Why don't you try factoring that and show me what you get...

Answer 17

would you mind doing it but showing exactly how you got there, its easier for me to see the proper steps, especially with crazy high numbers : $

Answer 18

Well, do you see any common factors in -250x^2 + 2375x?

Answer 19

5?

Answer 20

Yes. A whole mess of them, in fact :-) What else?

Answer 21

As a first step, can't you write x(-250x+2375) ?

Answer 22

25?, 10?

Answer 23

\[-250x^2+2375x = 0\]Factor out \(x\)
\[x(-250x + 2375) = 0\]For that to be true, either \(x=0\) or \(-250x + 2375 = 0\) I bet you can solve both of those equations :-)

Answer 24

ive never factored without a middle number

Answer 25

\(x=0\) is the case for the bank end of the arch. \(-250x + 2375 = 0\) is the case for the other end of the arch.

Answer 26

I'm sure you have, you just don't remember it as such. You maybe haven't solved an equation by factoring where that is true, but I guarantee that you must have factored something like \(a^2 + ab\) when you were first learning how to factor :-)

Answer 27

uhhh im so confused, what do i do next ?

Answer 28

Solve \[-250x + 2375 = 0\]

Answer 29

is it 9.5

Answer 30

i had to use square root factor

Answer 31

No. No square root needed here.
\[-250x + 2375 = 0\]Add 250x to both sides
\[-250x + 250x + 2375 = 250x\]\[2375=250x\]Now what do you think is the value of \(x\)?

Answer 32

9.5

Answer 33

Oh, so sorry...9.5 is correct. I wrote it down as 19/2 :-)

Answer 34

haha thank you soo much!

Answer 35

I'd say I was testing your confidence, but... :-)

Answer 36

Okay, so at x = 9.5, the height of the arch should be 0 again. Let's check:

-0.25(9.5)^2 +2.375(9.5) = 0
-0.25*90.25+2.375*9.5=0
-22.5625 + 22.5625 = 0
Success!

Answer 37

Here's a plot of h(x) from x = 0 to x = 10:

Answer 38

So, the bases are 9.5 meters apart, because that's how far it is from x = 0 to x = 9.5.

Answer 39

I'll quickly do it with the quadratic formula, just for illustration:

we had \[h(x) = -0.25x^2+2.375x\]so our quadratic would be\[-0.25x^2+2.375x = 0\]giving us \(a=-0.25, b = 2.375, c = 0\). Our solutions would be thus\[x = \frac{-2.375\pm\sqrt{2.375^2-4(0)(-0.25)}}{2*(-0.25)} = \frac{-2.375\pm2.375}{-0.5} = 0, 9.5\]