What is 17.445 - 6.76 rounded to the nearest tenth
1 answer:
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Answer:
7.28 miles
Step-by-step explanation:
Suppose the distance at closest approach is represented by x. Then the distance to the point of closest approach at the first sighting is ...
d1 = x·tan(62°)
At the second sighting, the distance to the point of closest approach is ...
d2 = x·tan(38°)
The difference of these distances is 8 miles, so we have ...
d1 -d2 = 8 = x(tan(62°) -tan(38°))
Dividing by the coefficient of x, we find ...
x = 8/(tan(62°) -tan(38°)) ≈ 7.2764 . . . . miles
The point of closest approach is about 7.28 miles from the landmark.
Answer:
The Fourier series of f(x) converges to 3 at the points x= π+2kπ, where k is an integer.
Step-by-step explanation:
First, recall that the function f(x) is extended 2π periodic to the whole real line, in order to obtain a valid Fourier expansion. Remember that a Fourier series is formed by a sines and cosines, which are 2π-periodic.
So, the 2π-periodic expansion of f(x) is discontinuous at the points π+2kπ, in particular π and -π. Check the attached figure to a better understanding.
Now, the Dirichlet theorem on the convergence of a Fourier series tells us that the series converges to the function at the points of continuity, and at points of discontinuity the sum of the series is
.
Here we understand the notation and
as
and
.
In this particular case
.
For the limit , with
recall that our function is 2π-periodic, so the values of f near π, with x>π are the same when x is near -π and x>-π. Again, check the attached figure. So,
.
Thus,
.
Note: In the attached figure we only have drawn three repetitions of the 2π-periodic extension of , recall that the extension is <em>ad infinitum</em>. Also, the points drawn in the dotted lines are the sum of the series at the points of discontinuity.
