Answer:
Arc length = [tex]=\frac{5}{4} \pi \,km=1.25\ \pi\,\,km[/tex]
[tex]Area=\frac{25}{8}\, \pi\,\,km^2= 3.125\, \pi\,\,km^2[/tex]
Step-by-step explanation:
For the length of an arc of circumference, you need to recall the formula:
[tex]arc = R * \theta[/tex]
which is the product of the radius R times the angle, but only valid if the angle [tex]\theta[/tex] is given in radians (not degrees). So we need to change the 45 degree angle into radians. we can do so by remembering that [tex]\pi[/tex] radians is the same as 180°, and then the following proportion can be written to find what is the value (x) of 45° in radians:
[tex]\frac{45^o}{180^o} = \frac{x}\pi} \\x=\frac{45^o}{180^o} \,\pi\\x=\frac{\pi}{4}[/tex]
Now we find the length of the arc using the formula given above:
[tex]arc = R * \theta\\arc=5\,km\,\frac{\pi}{4} \\arc=\frac{5}{4} \pi \,km[/tex]
Now for the area of the sector, recall the formula:
[tex]Area = \frac{1}{2} R^2\,\theta[/tex]
with the same condition for [tex]\theta[/tex] of being expressed in radians (and not degrees). So the formula becomes:
[tex]Area = \frac{1}{2} R^2\,\theta\\Area= \frac{1}{2} (5\,km)^2\,\frac{\pi}{4} \\Area=\frac{25}{8}\, \pi\,\,km^2[/tex]
