4.5
#34:
To get the best view of the Statue of Liberty in Figure 4.76, you should be at the
position where theta is a maximum.
If the statue stands 92 meters high, including
the pedestal, which is 46 meters high, how far from the base should you be? [Hint:
Find a formula for theta in terms of your distance from the base.
Use this function
to maximize theta, noting that 0<
theta<
pi/2.]
From Figure 4.76, we can create a free body diagram which includes only the triangle
with the sides and angles in question.
Since we are looking for the distance from the base
using the distance from the top of the pedestal, we will actually have two triangles.
A
diagram can be seen below in
Figure 1.
x
Figure 1: Diagram of the angles and distances from the Statue of Liberty
Looking at the entire triangle (the two smaller triangles put together), we have right angle
and can use tangent to express theta.
We will call the angle of the other smaller triangle,
y.
Looking at the smaller triangle, we can see that tan(y)=46/x.
Then we can look at the
triangle as a whole again and see that the entire angle is (
+y). Therefore, tan(
+y)=92/x.
ө
ө
We know that +yy= .
Using this stupid statement we can make a mathematical
ө
ө
marvel by adding parentheses. Thus, (
+y)y= . Genius!
We know that +y=arctan(92/
ө
ө
ө
x). We also know that y=arctan(46/x).
So, we can put these two equations together
forming: (
+y)y= arctan(92/x)arctan(46/x)=
.
Since we want to maximize theta, we
ө
ө
can take the derivative of this equation and solve for x (the distance from the base) [not
so much fun so please see
Appendix A
for the rest of the problem].
ө
46
92
46
{
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4.6
#
24:
A train is traveling at .8 km/min along a long, straight track, moving in the
direction shown in the figure.
A movie camera, .5 km away from the track, is
focused on the train.
a)
Express
z
, the distance between the camera and the train, as a function of
x.
We can express z(x) using the Pythagorean Theorem since the camera and the track
form a right angle.
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 Fall '08
 BLAKELOCK
 Calculus, Pythagorean Theorem, Hypotenuse, Angela, triangle, Idaho Potato Company

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