a fountain in a pond has a pump that recirculates 75 gallons of water in 1/4 of an hour.express this rate as a unit rate in gall
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Answer:
and
Step-by-step explanation:
Let's first isolate the trig function.
Add 1 one on both sides:
Divide both sides by :
Now recall .
or
The first ratio I have can be found using in the first rotation of the unit circle.
The second ratio I have can be found using you can see this is on the same line as the
so you could write
as
.
So this means the following:
is true when
where is integer.
Integers are the set containing {..,-3,-2,-1,0,1,2,3,...}.
So now we have a linear equation to solve:
Add on both sides:
Find common denominator between the first two terms on the right.
That is 24.
(So this is for all the solutions.)
Now I just notice that it said find all the solutions in the interval .
So if and we let
, then solving for
gives us:
( I just added
on both sides.)
So recall .
Then .
Subtract on both sides:
Simplify:
So we want to find solutions to:
with the condition:
That's just at and
So now adding to both gives us the solutions to:
in the interval:
.
The solutions we are looking for are:
and
Let's simplifying:
and
and
and