Question

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

Answer 1

Have you started thinking about It?

Answer 2

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

Answer 3

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?

Answer 4

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

Answer 5

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

Answer 6

ok

Answer 7

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

Answer 8

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

Answer 9

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

Answer 10

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.

Answer 11

alright i have written down all the numbers correctly

Answer 12

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

Answer 13

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.

Answer 14

alright so what exactly am i suppose to draw?

Answer 15

Draw the arch and the beam, approximately.

Answer 16

so is it a triangle type or square?

Answer 17

Like that?

Answer 18

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

Answer 19

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

Answer 20

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

Answer 21

and would i do that?

Answer 22

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: