SOLUTION: By how much does the arc intercepted by a central angle of 38 degrees exceed the chord intercepted by the same angle on a circle of radius 43 ft.? Please help me :(

Question 1096676: By how much does the arc intercepted by a central angle of 38 degrees exceed the chord intercepted by the same angle on a circle of radius 43 ft.?

Please help me :(
Found 2 solutions by KMST, MathTherapy:
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
NOTE:
I assume this is math homework,
and not physics homework.
I also assume that the answer is expected to be a length in feet,
because if the expected answer was 1.9%,
There would be no need to know the radius.
In that case, you would compare the length of
a arc on a unit circle with
(the measure of the arc in radians)
to the sine of ,
to find that

A PICTURE AND THEN AN ANSWER IN 1,000 WORDS:

LENGTH OF THE ARC:
Using radians:
The angle measure is in radians.
The length of an arc of that measure is
angle in radians times radius.
For a circle of radius, it is .
Without radians:
The whole circumference is .
The arc is a fraction of that. It is .
So, the arc length is .
That is approximately .

LENGTH OF THE CHORD:
Connecting the ends of the chord to the center of the circle,
you form an isosceles triangle.
It has two legs measuring forming an angle measuring .
It's base is the chord, whose length we need to find
If Law of cosines was taught in class, you may be expected to use it.
.
So,
That is approximately .

Otherwise, you could split that triangle into two right triangles,
and use trigonometry to find the length of half the chord as
,
so the length of the chord is twice that,
or approximately .

By how much does the arc exceed the chord?
We calculate the difference as about ,
So, I would answer .
Answer by MathTherapy(10557) (Show Source): You can put this solution on YOUR website!
By how much does the arc intercepted by a central angle of 38 degrees exceed the chord intercepted by the same angle on a circle of radius 43 ft.?

Please help me :(

Let central angle be O, radii AO and BO, and intercepted chord & intercepted arc, AB
∡A = 38^o
Length of arc AB:
∡A and ∡B =
Draw an altitude from vertex O to AB, and name it C
cos ∡A =

--------- Cross-multiplying
AC = 14
Since the altitude from vertex O, or OC BISECTS chord AB, AC = . Therefore, the INTERCEPTED CHORD or
Arc AB exceeds chord AB by .
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