SOLUTION: The line x+2y = 5 divides the circle x2 +y2 = 25 into two arcs. Calculate their lengths. The interior of the circle is divided into two regions by the line. Calculate their areas.
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Question 1158596: The line x+2y = 5 divides the circle x2 +y2 = 25 into two arcs. Calculate their lengths. The interior of the circle is divided into two regions by the line. Calculate their areas. Give three significant digits for your answers.
Found 2 solutions by Alan3354, solver91311:
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
x2 +y2 = 25
-------------------------------
Use ^ (Shift 6) for exponents.
eg, x^2 for x squared.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
The first thing that you must accomplish is to find the coordinates of the two points of intersection between the given circle and the given secant line.
There are several approaches, but I chose this one:
And
So
^2\ =\ -y^2\ +\ 25)
And I leave it as an exercise for the student to verify that this leads to the conclusion that the points of intersection are )
and )
The formula for the measure of an arc of a circle is 
where 
is the angle subtended measured in radians. Our next challenge is to find the value of for the arc 
. See figure:
By inspection, 
and 
, therefore 
But since the range of the 
function is 
, )
yields a negative angle in QIV. So the measure of the desired angle is given by:
\ +\ \pi)
And then
Since the major arc, namely 
, is simply the rest of the circle,
The arithmetic, as is my habit, is left to you.
Given the above solution to the arc length part of the problem, you should be able to solve the area part by use of the formula for the area of a circle, which I expect that you already know, and the formula for the area of a sector which is:
Don't forget to set your calculator to radian mode, don't round anything off until you get a final answer to a calculation, and never carry a rounded answer into a subsequent calculation.
John
My calculator said it, I believe it, that settles it
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