Answer:
A
The distance in revolutions is [tex]n = 7 \ revolutions[/tex]
The distance in degrees is [tex]\theta =2529 ^ o[/tex]
The distance in radian is [tex]\theta_{radian } = 44.15 \ rad[/tex]
B
[tex]\theta_a = 1.26 \ rad[/tex]
Explanation:
From the question we are told that
The diameter of the pie is [tex]d = 10.00 \ in[/tex]
The distance covered by the rim is [tex]D = 208 \ in[/tex]
The number which the pie is divided into is k = 5
Generally the radius of the pie is mathematically represented as
[tex]r = \frac{d}{2}[/tex]
=> [tex]r = \frac{10}{2}[/tex]
=> [tex]r = 5 \ in[/tex]
Generally the distance covered by the rim is mathematically represented as
[tex]D= 2 \pi r n[/tex]
=> [tex]208 = 2* 3.142 * 5 * n[/tex]
=> [tex]n = 7 \ revolutions[/tex]
Generally converting to degrees
[tex]\theta = n * 360[/tex]
=> [tex]\theta = 7 * 360[/tex]
=> [tex]\theta =2529 ^ o[/tex]
Generally converting to radian
[tex]\theta_{radian } = \frac{\theta * \pi}{180 }[/tex]
=>[tex]\theta_{radian } = \frac{2520 * 3.142 }{180}[/tex]
=>[tex]\theta_{radian } = 44.15 \ rad[/tex]
Generally the angular size of one piece of the pie is
[tex]\theta_a = \frac{2 * \pi }{ k}[/tex]
=> [tex]\theta_a = \frac{2 * 3.142}{ 5}[/tex]
=> [tex]\theta_a = 1.26 \ rad[/tex]
(a) The angular distance is 0.168 revolution, 1.056 radians and 60.5⁰.
(b) The angular size of one slice in radians is 2π radian.
The given parameters;
The angular distance traveled by the pie plate is calculated as follows;
2πr = 5.28
[tex]2\pi = \frac{5.28}{r} \\\\2 \pi = \frac{5.28}{5} \\\\2\pi = 1.056 \ rad\\\\\frac{1 \ rev}{2\pi \ rad } \times 1.056 \ rad = 0.168 \ rev[/tex]
[tex]1.056 \ rad \times \frac{360^0}{2\pi \ rad} = \ 60.5^0[/tex]
Thus, the angular distance is 0.168 revolution, 1.056 radians and 60.5⁰.
The circumference of the pie is calculated as follows;
C = 2πr
C = 2π x 5
C = 10π
Diving the pie into 5 equal slice;
[tex]Fraction \ of \ one \ part \ in \ radian= \frac{10\pi}{5} = 2\pi\ radian[/tex]
Thus, the angular size of one slice in radians is 2π radian.
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