Calculate the distance the point R (5, 8) and (5, -3)
Answers
Step-by-step explanation:
The time is
t
=
s
5
cos
α
and the distance from point
A
to point
B
is
d
=
2
s
5
cos
α
where
s
is the distance to shore and
α
is the angle which the boat points.
Explanation:
Sometimes, when we get a question like this, we don't have all of the information that is needed. In that case, we can put a place-holder (variable) in for that information and proceed with a solution. In this case, a diagram helps us decide what we have and what we need:
We are looking for the distance between points
A
and
B
, let's call that
d
. We know the speed of the boat,
5
m
/
s
, and we know that it starts by pointing itself at point
A
. Let's call the angle that the boat is pointed in
α
with respect to the direction to shore,
x
.
We also know that the river is moving at
2
m
/
s
which adds to the velocity of the boat giving the resultant velocity,
v
r
, of the boat shown in the green arrow. This resultant velocity is what causes the boat to arrive at point
B
.
Finally, we need to know the distance to the shore, let's call this
s
. The remainder is just geometric constructions.
The x-velocity of the boat is simply:
v
x
=
5
cos
α
We can calculate the time to get to shore from this and the distance to shore as:
t
=
x
v
x
=
s
5
cos
α
The y-velocity of the boat is given by:
v
y
=
5
sin
α
−
2
From this and the time we can get the distance traveled upstream as
c
=
v
y
⋅
t
=
(
5
sin
α
−
2
)
s
5
cos
α
If the river wasn't moving, the boat would have reached point
A
in the same amount of time. So we can use the same approach to find the distance it would travel in this case as
c
+
d
=
v
y
o
⋅
t
=
(
5
sin
α
)
s
5
cos
α
We can now subtract
c
from this total to get
d
d
=
(
5
sin
α
)
s
5
cos
α
−
(
5
sin
α
−
2
)
s
5
cos
α
which simplifies to
d
=
2
s
5
cos
α