The reception for a radio station is given by the equation (x+1)^2/225 + (y+3)^2/144=1

A straight road passes through his area with an equation of 3x=y

Where x and y are measured in miles in both questions.
A car enters the the reception area at point a and exits at point b, as shown in the graph below, traveling at 30 mph for this entire distance.
For how much time can the car receive this station.? Show all work

Question

The reception for a radio station is given by the equation (x+1)^2/225 + (y+3)^2/144=1

A straight road passes through his area with an equation of 3x=y

Where x and y are measured in miles in both questions.
A car enters the the reception area at point a and exits at point b, as shown in the graph below, traveling at 30 mph for this entire distance.
For how much time can the car receive this station.? Show all work

anonymous · Answer

the graph is backwards

mathstudent55 · Answer

The line in the graph cannot have the equation you wrote, 3x = y.

anonymous · Answer

he told us to use substitution to fine the two (x,y) coordinates and then use the distance formula

baru · Answer

@mathstudent55 
the camera image is mirrored, i guess the left side is the positive x axis

anonymous · Answer

yea sorry about that

mathstudent55 · Answer

Your line in the graph is less than 45 up from the horizontal.
That means the slope is between 0 and 1.
Your equation 3x = y is the same as y = 3x.
With a slope of 3, the line would be much more than 45 degrees up fro the horizontal.
Perhaps you mean the line to be 3y = x.

baru · Answer

in any case, we do not need a figure

baru · Answer

the curve mentioned is an ellipse, not centred at the origin, so we can disregard the figure

mathstudent55 · Answer

I understand the picture is a mirror image, but unless you have the correct equation for the line, you will not be answering the question based on the line in the figure.

baru · Answer

@nadiainderjit disregard the diagram completely, you can just use the equations

mathstudent55 · Answer

@nadiainderjit 
Can you show a picture of the problem exactly as it was given to you?

baru · Answer

to find a and b
solve the two equations, substiute for y or x using the line eq into the curve eq

anonymous · Answer

attachment

baru · Answer

substitute y=3x in the eq
$\frac{(x+1)^2}{225}+.....=...$

mathstudent55 · Answer

Thanks.
You wrote ... (y + 3)^2, but it's ... (y - 3)^2, so be careful when solving the problem.
I see that the equation is really 3x = y as you wrote.

Then do what @baru suggested.
Substitute 3x in for y in the equation of the ellipse.
You will have a quadratic equation in x.
Solve for x. You will get two solutions for x.
Then insert each value of x in the equation 3x = y to get the corresponding y-coordinates.
Each set of x and y is a point. Those two points are points A and B.
Then use the distance formula to find the distance between A and B.
Then use the speed to find how much time will take to travel that distance.

anonymous · Answer

i know i had to do that it was actually doing the math that confused me @mathstudent55

mathstudent55 · Answer

If you need help, just show what you have and where you are stuck, and I'll try to help.

anonymous · Answer

i got as far as 
$$\frac{ x ^{2}+2x ^{2}+1}{ 225 }+\frac{ 9x ^{2}-18x+9} {144}=1$$

mathstudent55 · Answer

Good.
Now multiply both sides by the LCD, 144 * 225.

mathstudent55 · Answer

Or you can just use the LCD to add the fractions on the left.

anonymous · Answer

so multiply by 32400

mathstudent55 · Answer

You can just do this to get the LCD:

$\dfrac{ (x ^{2}+2x ^{2}+1)144}{ 225 \times 144 }+\dfrac{ (9x ^{2}-18x+9)225} {144 \times 225}=1$

anonymous · Answer

$$\frac{ 144x ^{2}+288x+ 144}{32400}+\frac{ 2025x ^{2}-4050x+2025 }{ 32400 }=1$$

mathstudent55 · Answer

Good.
Now add the fractions using the common denominator.

mathstudent55 · Answer

Do you get this?

$\dfrac{2169x^2 -3762x + 2169}{32400} = 1$

anonymous · Answer

yes. Now what. Do i multiply by 32400 on both sides to get
$$2169x ^{2}-3762x+2169=32400$$

anonymous · Answer

then subtract to get 
$$2169x ^{2}-3762x-30231$$

baru · Answer

the numbers are big, so lets try to simply as far as possible. try dividing throughout by 9

anonymous · Answer

$$\left( 241x ^{2}-418x-33359 \right)$$
@baru

baru · Answer

*3359  (u hav put an extra 3)

do you know the quadratic formula?

anonymous · Answer

yea

$$-b \pm \sqrt{ \frac{ b2-4ac }{ 2a }}$$

mathstudent55 · Answer

So far you are correct.
Now use the quadratic formula to find 2 solutions for x.

mathstudent55 · Answer

$ 241x ^{2}-418x-3359 = 0$

baru · Answer

*2a of the denominator is outside the square root, and it divides 'b' also

baru · Answer

attachment

mathstudent55 · Answer

$\large x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

anonymous · Answer

(4.7, )
(-2.96, )

anonymous · Answer

(4.7, 14.1 )
(-2.96, -8.88)

anonymous · Answer

using the distance formula would the final distance be 24.25 miles

mathstudent55 · Answer

Your points are correct.

mathstudent55 · Answer

Your distance is correct.

mathstudent55 · Answer

Now just find the time using the given speed.

anonymous · Answer

how?

baru · Answer

speed=$ \frac{distance}{time}$

baru · Answer

speed is given.
you just found the distance. substitute these

anonymous · Answer

24.5/30= TIME

baru · Answer

yep

anonymous · Answer

so about 49 minutes

mathstudent55 · Answer

Correct. Great job!