On a movie set, an archway is modeled by the equation
y = -0.5x2 + 3x, where y is the height in feet and x is the horizontal distance in feet. A laser is directed at the archway at an angle modeled by the equation -0.5x + 2.42y = 7.65 such that the beam crosses the archway at points A and B. At what height from the ground are the points A and B?
A. 1.5 feet and 3.5 feet
B. 1.5 feet and 4 feet
C. 3.5 feet and 4 feet
D. 4 feet and 4 feet

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- Pandemonium

- katieb

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- mathmate

Have you started thinking about It?

- Pandemonium

ive tried solving it but i dont understand on how to start this

- mathmate

To solve ANY math word problem, you go with the first step
1. read the question, understand everything that is supplied. Read it three times if necessary to understand the question perfectly. Then write a summary of the given information. Can you do that now?

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- Pandemonium

so look at the equation given and then read the question and what i need to find?

- mathmate

After having fully understood the question, then
1. write a summary of the given information

- Pandemonium

ok

- Pandemonium

So i understand the question and its asking for me to find the height of a cross beam that crosses through points A and B

- mathmate

I would put the summary like this:
Given:
Equation of arch : y=-0.5x^2+3x
Equation of beam: -0.5x + 2.42y = 7.65, or \(y=\frac{0.5x+7.65}{2.42}
Need to find intersection points of arch and beam

- mathmate

* \( \Large y=\frac{0.5x+7.65}{2.42}\)

- mathmate

This way, you have a clear mind of what is mathematically needed, and don't have to worry about wordings. But do check that you have copied all the numbers correctly.
From now on until the answer line, you are working with mathematics.
The answer line is the \(last\) sentence of your solution that interprets the results of math calculations in the terms of the question.

- Pandemonium

alright i have written down all the numbers correctly

- Pandemonium

So from here on until we get to the answer line, will we only work with mathematics?

- mathmate

Then we can proceed to the second step
2. Graph the curves/lines.
Use a graphics calculator if your school allows it. If not, even better, use a sheet of paper and pencil. Can you draw that for me (use the "draw" button below). This step will help you find approximately the required answer, or check the final answer for gross mistakes.

- Pandemonium

alright so what exactly am i suppose to draw?

- mathmate

Draw the arch and the beam, approximately.

- Pandemonium

so is it a triangle type or square?

- Pandemonium

|dw:1434978937794:dw|

- Pandemonium

Like that?

- mathmate

An arch is a curve, and the beam is a line.
Are you familiar with quadratic functions and linear functions?

- Pandemonium

yea im familiar with it but i could never really understand it

- Pandemonium

|dw:1434979035876:dw|

- mathmate

That's the idea, but we have to put them in the context of the question.

- Pandemonium

and would i do that?

- mathmate

First, examine the equations.
The arch is a quadratic function
y=-0.5x^2+3x
the leading coefficient, or the coefficient of x^2, is -0.5.
The negative sign means that the function is concave down, the way you drew it.
The constant term (with no variable) is zero, means the function passes through the origin.
If you complete the square, meaning writing it as
y=-0.5(x-3)^2+4.5
It means that the maximum of the function is at x=3.
So we end up with a graph like this:
|dw:1434979494256:dw|

- mathmate

Is that clear so far?

- Pandemonium

yea im learning this now :D

- mathmate

Good! Now we'll work on the beam.
\(\Large y=\frac{0.5x+7.65}{2.42}\)

- Pandemonium

so we plug in something to that x and solve it?

- mathmate

Kind of, but we will first write this in slope intercept form and get rid of the fractions!

- Pandemonium

ok

- mathmate

So that makes the equation \(approximately\)
y=0.207x+3.161

- Pandemonium

ok then whats next?

- mathmate

Can you sketch the line on the graph?

- mathmate

I'll give you something for the vertical scale:
|dw:1434979999842:dw|

- mathmate

yes!

- mathmate

Yes, these are the points we're looking for.

- mathmate

Recall that in the equation
y=0.207x+3.161
3.161 is the y-intercept
0.207 is the slope.

- Pandemonium

so i draw i line from the 3 up and somewhere in between the 4.5 and 0 across?

- mathmate

What you can do is to find the point where x=6, and joint the point with the y-intercept.
Point at y-intercept (0,3.161), and the point at x=6 has y=0.207*6+3.161=4.4
So you just need to join the points (0,3.161) and (6,4.4) to locate the beam.

- Pandemonium

|dw:1434980631856:dw|

- Pandemonium

like that?

- mathmate

Not quite.
The slope is positive 0.2, so the line cannot be decreasing. Also, the point on the right is (6,4.4), you have drawn (6,0) :(
|dw:1434980741403:dw|
I have drawn the dots, now you do "connect the dots". :)

- Pandemonium

|dw:1434980772402:dw|

- Pandemonium

i saw where i made the mistake
Sorry bout that

- mathmate

That looks good!
Can you now estimate the values of x at the intersection points?

- Pandemonium

|dw:1434980899618:dw|
Are those the intersection points?

- mathmate

Exactly!
|dw:1434980969153:dw|
Can you now read off the values of x at the intersection points, approximately, of course.

- Pandemonium

ok so my estimate would be 1.4, 3.3 and then 3.9,4.1

- mathmate

Good, so your estimates for would be 1.4 and 3.9, right?

- Pandemonium

yea

- Pandemonium

and thats close to the answer 1.5 and 4 feet

- mathmate

You can always continue to solve algebraically the solution (which turn out to be close to 1.5765 and 4.0102) but the graphical solution is good enough for you to choose the multiple choice answers, right?

- Pandemonium

yea

- Pandemonium

so the answer should be B if im correct

- mathmate

The graphical solution method is very powerful. Once you master it, you can solve non-linear problems like this in one, maybe 2 minutes!

- Pandemonium

hahaha this one took a long time :D

- mathmate

Can you please repeat the answer with the values instead of the letters?

- Pandemonium

B. 1.5 feet and 4 feet

- mathmate

Yes, that is correct, since this one is the closest to your estimate of 1.4 and 3.9, right?

- Pandemonium

yea

- mathmate

Hope you practise the graphics method more when you have a chance!

- Pandemonium

i will thanks so much @mathmate

- mathmate

You're welcome, Pande! :)

- Pandemonium

you helped me alot to learn that and get me a better understanding of it :D

- mathmate

Yes, understanding is the key to learning math! Well done!

- Pandemonium

ok im closing this down bc i have one question to do and this one involves linear-quadratic system

- mathmate

You can try it a similar way!

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