The centers of two belt pulleys, with radii of 3 cm and 6 cm, respectively, are 13 cm apart. Find the total area K enclosed by the belt.

I assume you made a diagram

Because of the symmetry, let's just look at the top part of the diagram

Let the centre of the smaller circle be O and the centre of the larger circle be P
Let the point of contact of the belt to the smaller circle be A and to the larger circle B.
Draw a line parallel to AB from O to hit BP at C
You now have a rectangle AOCB and a right-angled triangle OPC with 90° at C
look at the right-angled triangle , you have OP = 13 and PC = 3
We can find angle OPC = Ø by
cosØ = 3/13
Ø = 76.658°
also by Pythagoras, OC = √160
thus AB=√160

We are now in a position to find the area of the rectangle and the area of the triangle.
Double that and you have the area of the stuff between the radii.

So all we need now is the area of the larger and smaller circle covered by the belts.
Would you agree with me that the angle covered by the belt in the larger circle is 206.685° ?
So now all you have to do is set up a ratio of
arclength1/(2π(6)) = 206.685/360
and for the smaller circle:
arclength2/(2π(3)) = 153.332/360

you do the arithmetic

dihoihihihih

<p>This is Mathematics</p>

To find the total area enclosed by the belt, we need to consider both the inner and outer areas of the belt.

First, let's calculate the circumference of each pulley:
Circumference of the smaller pulley = 2 * pi * radius of the smaller pulley
Circumference of the smaller pulley = 2 * 3.14 * 3 cm = 18.84 cm

Circumference of the larger pulley = 2 * pi * radius of the larger pulley
Circumference of the larger pulley = 2 * 3.14 * 6 cm = 37.68 cm

Next, we find the length of the belt that connects the two pulleys. This can be calculated as the sum of the distances around each pulley plus the distance between the two pulleys:
Length of the belt = Circumference of the smaller pulley + Circumference of the larger pulley + Distance between the pulleys
Length of the belt = 18.84 cm + 37.68 cm + 13 cm = 69.52 cm

To calculate the total area enclosed by the belt, we multiply the length of the belt by the width of the belt. Here, the width of the belt is the distance between the two pulleys:
Total area enclosed by the belt (K) = Length of the belt * Width of the belt
Total area enclosed by the belt (K) = 69.52 cm * 13 cm = 904.16 cm^2

Therefore, the total area enclosed by the belt is 904.16 cm^2.

I really have no idea how to find the theta of the angles, help please