A helicopter is on a path directly overhead line A B when it is simultaneously observed from locations A and B separated by 900 ft . The angle of elevation from A is 42 ° 30 ' and the angle of elevation from B is 30 ° 12 ' . a. What is the distance from each location to the helicopter? Round to the nearest foot. b. How high is the helicopter from the ground at the moment of observation? Round to the nearest foot.
A helicopter is on a path directly overhead line A B when it is simultaneously observed from locations A and B separated by 900 ft . The angle of elevation from A is 42 ° 30 ' and the angle of elevation from B is 30 ° 12 ' . a. What is the distance from each location to the helicopter? Round to the nearest foot. b. How high is the helicopter from the ground at the moment of observation? Round to the nearest foot.
Solution Summary: The author calculates the distances from the locations A and B to the helicopter, using the unit conversion factor (1°60
A helicopter is on a path directly overhead line
A
B
when it is simultaneously observed from locations
A
and
B
separated by
900
ft
. The angle of elevation from
A
is
42
°
30
'
and the angle of elevation from
B
is
30
°
12
'
.
a. What is the distance from each location to the helicopter? Round to the nearest foot.
b. How high is the helicopter from the ground at the moment of observation? Round to the nearest foot.
3.
Consider the sequences of functions f₁: [-π, π] → R,
sin(n²x)
An(2)
n
f pointwise as
(i) Find a function ƒ : [-T,π] → R such that fn
n∞. Further, show that fn →f uniformly on [-π,π] as n → ∞.
[20 Marks]
(ii) Does the sequence of derivatives f(x) has a pointwise limit on [-7, 7]?
Justify your answer.
[10 Marks]
1. (i) Give the definition of a metric on a set X.
[5 Marks]
(ii) Let X = {a, b, c} and let a function d : XxX → [0, ∞) be defined
as d(a, a) = d(b,b) = d(c, c) 0, d(a, c) = d(c, a) 1, d(a, b) = d(b, a) = 4,
d(b, c) = d(c,b) = 2. Decide whether d is a metric on X. Justify your answer.
=
(iii) Consider a metric space (R, d.), where
=
[10 Marks]
0
if x = y,
d* (x, y)
5
if xy.
In the metric space (R, d*), describe:
(a) open ball B2(0) of radius 2 centred at 0;
(b) closed ball B5(0) of radius 5 centred at 0;
(c) sphere S10 (0) of radius 10 centred at 0.
[5 Marks]
[5 Marks]
[5 Marks]
(c) sphere S10 (0) of radius 10 centred at 0.
[5 Marks]
2. Let C([a, b]) be the metric space of continuous functions on the interval
[a, b] with the metric
doo (f,g)
=
max f(x)g(x)|.
xЄ[a,b]
= 1x. Find:
Let f(x) = 1 - x² and g(x):
(i) do(f, g) in C'([0, 1]);
(ii) do(f,g) in C([−1, 1]).
[20 Marks]
[20 Marks]
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