If a circle has a radius of 1 and a sector defined by a 37 degree arc, what is the area, in cm^2, of the sector?

The complete circle has 360 degrees
 We look for the relationship that represents 37 degrees.
 We have then:
 37/360 = 0.103
 Then, the area of the circle is:
 A = (0.103) * (pi) * (1) ^ 2
 A = 0.324 cm ^ 2
 
 the area, in cm ^ 2, of the sector is:
 A = 0.324 cm ^ 2

a) Scientific notation is written as a unit digit, that can have other digits after the decimal place, multiplied by 10 to the power of n, where n can be any integer number.

Then 25x10^0 is not scientific notation, because there are two digits at the left of the decimal point.

The scientific notation would be:

2.5x10^1

b) Why is this useful?

Because we divide all our measures in powers of 10.

For example, you know that 1 meter  = 100c

But we can also write this as:

1m = 1.0*10^3 cm.

The use of scientific notation comes useful because we can have a fast read of the dimensions of our measure, and in another hand, is way easier work with the 10 as a base when we solve equations.