Important Questions (Preparing For Final) Need Help

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May 13th 2006, 08:21 PM
OverclockerR520

Important Questions (Preparing For Final) Need Help

Hello,

My math final is monday and I've been up studying this weekend. I am pretty sure I've got most of my material down but I do have 3 questions I need the answers too. Out of 30 questions on a practice test I'm havign trouble with these three so I don't think I'm doing ot bad. Here they are with linked pictures for the diagrams I typed them out as best I could. Any help is much appreciated, thanks!

Question 1. An arch of a bridge is semi-elliptical, with major axis horizontal. The base of the arch is 30 feet across, and the highest part of the arch is 10 feet above the horizontal roadway.

a) Give the equation of the ellipse in standard form b) Find the height of the arch ten feet from the center of the horizontal roadway.

Here is link to pic of diagram for that problem...
http://putfile.com/pic.php?pic=5/13300103238.jpg&s=f5

Question 2. A satellite antenna dish has the shape of a paraboloid that is 10 feet across at the open end and is 3 feet deep. At what distance from the center of the dish should the reciever be placed to recieve the greatest intensity of sound waves?

Question 3. Shown in the figure is a system of 4 one way streets leading into the center of a city. The numbers in the figure denote the average number of vehicles per hour that travel in the directions shown. A toal of 300 vehicles enter the area and 300 vehicles leave the area every hour.

Signals at intersections A,B,C and D are to be timed in order to avoid congestion, and this timing will determine flow rates x1, x2, x3, and x4

a) If the number of vehicles entering an intersection per hour must equal the number of vehicles leaving the intersection per four, describe the traffic flow rates at each intersection with a system of equations.

b) If the signal at intersection C is timed so that x3 = 100, find x1, x2, and x4

Here is a link to the diagram for this problem...
http://putfile.com/pic.php?pic=5/13300110940.jpg&s=f5

Like I said the final is monday so any help is much appreciated, thanks! :)
May 13th 2006, 11:20 PM
CaptainBlack

Quote:

Originally Posted by OverclockerR520

Question 1. An arch of a bridge is semi-elliptical, with major axis horizontal. The base of the arch is 30 feet across, and the highest part of the arch is 10 feet above the horizontal roadway.

a) Give the equation of the ellipse in standard form
b) Find the height of the arch ten feet from the center of
the horizontal roadway.

Here is link to pic of diagram for that problem...
http://putfile.com/pic.php?pic=5/13300103238.jpg&s=f5

The equation of an ellipse with its major and minor axes oriented along
the co-ordinate axes, and with its centre at the origin is:

$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $

In the case of this arch $a=15$ feet and is the semi-major
axis and $b=10$ feet and is the semi-minor axis. So the
equation of the ellipse is:

$ \frac{x^2}{15^2}+\frac{y^2}{10^2}=1 $ .

Now if we substitute $x=10$ the resulting equation for $y$ will give us the required height of the arch 10 feet from the centre.

$ \frac{10^2}{15^2}+\frac{y^2}{10^2}=1 $ ,

rearranging:

$ y^2=10^2\left(1-\frac{10^2}{15^2}\right) $

or:

$ y\approx\pm 7.45 $

the positive root is the one which we require so the height is $7.45$ feet.

RonL
May 13th 2006, 11:34 PM
CaptainBlack

Quote:

Originally Posted by OverclockerR520

Question 2. A satellite antenna dish has the shape of a paraboloid that is 10 feet across at the open end and is 3 feet deep. At what distance from the center of the dish should the reciever be placed to recieve the greatest intensity of sound waves?

The standard equation of a parabola opening to the right, with vertex at
the origin is:

$ y^2=4\ a\ x $

where the focus is at $(a,0)$ .

You are told the dish is 10ft wide and 3ft deep, which means that the point
$(3,\pm 5)$ are on the parabola. Therefore:

$ 5^2=4\times a \times3 $

which may be rearranged to give:

$ a=25/12 $ .

The property of the parabola that leads to it being used for dish antennae is
that signals coming from a great distance are brought to a focus at the
focus of the parabola. So the distance from the centre of the dish at which
the receiver should be placed to receive the greatest signal is $25/12 \approx 2.1$ ft.

RonL
May 14th 2006, 05:29 AM
earboth

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Quote:

Originally Posted by OverclockerR520

...

Question 3. Shown in the figure is a system of 4 one way streets leading into the center of a city. The numbers in the figure denote the average number of vehicles per hour that travel in the directions shown. A toal of 300 vehicles enter the area and 300 vehicles leave the area every hour.

Signals at intersections A,B,C and D are to be timed in order to avoid congestion, and this timing will determine flow rates x1, x2, x3, and x4

a) If the number of vehicles entering an intersection per hour must equal the number of vehicles leaving the intersection per four, describe the traffic flow rates at each intersection with a system of equations.

b) If the signal at intersection C is timed so that x3 = 100, find x1, x2, and x4

Here is a link to the diagram for this problem...
http://putfile.com/pic.php?pic=5/13300110940.jpg&s=f5

Like I said the final is monday so any help is much appreciated, thanks! :)

Hello,

your problem is an example of Kirchhoff's junction rule.

to a)
You've got 2 entering points: A, C and 2 leaving points: B, D

If I recognized the numbers in your foto correctly (it's hardly something to detect on the foto! I've attached a diagram more or less for affirmation) you get one equation with each point:

$A: x_1+x_4=75+50$
$B: x_1+x_2=50+100$
$C: x_2+x_3=25+150$
$D: x_3+x_4=50+100$

This is a system of 4 linear equations, which hasn't exactly 1 solution.

to b).

Plug in the value and solve the system of linear equations. You'll get
$x_1=75\wedge x_2=75\wedge x_3=100\wedge x_4=50$

I hope this was of some help even if I misread the numbers on your foto.

Greetings

EB