Question 1014571: A child has a toy chest with wheels. When the wheels of the toy chest turn through an angle of 2 radians, the toy chest moves forward by 3.75 inches.
What is the length, in inches, of the radius of each wheel?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the length of an arc is equal to (the central angle of the arc divided by the central angle of the whole circle * the circumference of the circle.
if you let l = the length of the arc and a = the central angle of the arc and w = the central angle of the whole circle and c = the circumference of the circle, then the formula becomes:
l = (a/w) * c
since the circumference of a circle is equal to 2 * pi * r, this formula becomes:
l = (a/w) * 2 * pi * r
r is the radius of the circle.
since the central angle of the whole circle is equal to 2 * pi radians, this formula becomes:
l = (a / (2 * pi)) * (2 * pi * r)
since a = 2 radians and l = 3.75 inches, this formula becomes:
3.75 = (2 / (2 * pi)) * (2 * pi * r)
simplify to get:
3.75 = (1/pi) * (2 * pi * r)
this formula can also be written as:
3.75 = (1 * 2 * pi * r) / pi
the pi in the numerator and denominator cancel out and you are left with:
3.75 = (1 * 2 * r) which becomes 3.75 = 2 * r
solve for r to get r = 3.75 / 2 = 1.875 inches.
the radius of the wheels of the toy chest are 1.875 inches in length.
with a radius of 1.875 inches, the wheels, turning through an angle of 2 radians, will move 3.75 inches on the ground.
the formula for the length of the arc is, once again, ...
l = (a/w) * (2 * pi * r)
r is the radius which is equal to 1.875 inches.
w is central angle of the whole circle which is equal to 2 * pi radians.
a is equal to the central angle of the arc which is equal to 2 radians.
the formula becomes:
l = (2 / (2 * pi)) * (2 * pi * 1.875)
this can also be written as:
l = (2 * 2 * pi * 1.875) / (2 * pi)
the (2 * pi) in the numerator and denominator cancel out and you are left with:
l = 2 * 1.875) which results in l = 3.75 inches.
your solution, as far as i can tell, is:
the radius of the wheels is 1.875 inches in length.
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