Assignment 6:

Serkan Hekimoglu and Jamie Parker

EMAT 6680

University of Georgia (Dr.Wilson)

Investigating maximum angle in order

to view the picture & determining most suitable agle for penalty kick

In this assignment we decided to investigate Problem 4 & Problem 5. Because they have same logic and they are interconnected.

A 4 by 4 hangs on a wall such that its bottom edge is 2 feet above your eye level. How far back from the picture should you stand, directly in front of the picture , or at some distance back x ? Find the maximum viewing angle?

In order to maximize angle C condider the following equations in terms of z

The resulting equation is A=arctan(3 tanB)

Since C=A-B, substituting A=arctan(3 tanB) we have C= arctan(3 tanB)-B.

To maximize C, we simply take the derivative of C= arctan(3 tanB)-B and set it equal to zero.

We can solve for B:

Keeping the relevant domain of B, (0,p/2), in mind, we find that when C is maximum B=p/6. Hence, to find the maximum viewing angle C= arctan(3 tanB)-B, simply substitute B=p/6 and evaluate .

So the maximum viewing angle is C=p/6

Determining the p/6 angle distance from G point

Since AGO is a 90 degree, angle we have

The angle we are seeking for is C, but we can see C=A-B, so we have :

We can look at the graph of C and z:

The tangent function:

we can find z that gives us a maximum C by setting the slope of tangent line of

then we get:

Simplifying the equation:

z=3.464

GENERAL RULES

Labels: Picture height is PH and total height isTH

then we can write:

C=angle A-angle B

tan C=tan(A-B)=(tan A-tan B)/(1+tan B*tan A) then

we find the general formula

and then we get

In this problem

We would like to investigate the different ratios

If

If

If

We investigated most suitable angle for penalty kick

In Animation

We saw that somewhere between the 5 yard line and 2.5 yard line, the angle improves as the penalty is accepted.

As you can see below:the closer to the goal line, the greater the angle improvement.

 Name of the Angle Degree of Agle 20.9 20.9 17.36 17.36 14.60 14.60 9.48 9.48 7.47 7.47

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