If the radius of the right circular cylinder is 4 cm and the height is 16 cm and the measure of the central angle of the shaded sector is 45 degrees, what is the volume of the slice of the cylinder that has the 45 degree sector as it top surface

well, 45° is 1/8 of a circle.

So, the volume is 1/8 of the whole.

To find the volume of the slice of the cylinder, we need to determine the height of the slice and the area of the sector.

First, let's calculate the height of the slice. Since the central angle of the shaded sector is 45 degrees, it is 1/8th of the total circumference of the circular top of the cylinder. The circumference of a circle can be calculated using the formula:

Circumference = 2 * π * radius

In this case, the radius is 4 cm. Plugging this value into the formula:

Circumference = 2 * 3.14 * 4 = 25.12 cm

Since we know the central angle of the sector is 45 degrees, the arc length (which is equivalent to the length of the curved top edge of the slice) can be calculated using the formula:

Arc Length = (Central angle / 360 degrees) * Circumference

Plugging in the values, we get:

Arc Length = (45 / 360) * 25.12 = 3.14 cm

Now, we can use the arc length as the height of the slice since the curved top edge of the slice corresponds to the arc. Therefore, the height of the slice is 3.14 cm.

Next, let's calculate the area of the sector. The formula for the area of a sector is:

Area of Sector = (Central angle / 360 degrees) * π * radius^2

Plugging in the values, we get:

Area of Sector = (45 / 360) * 3.14 * 4^2 = 3.14 cm^2

Finally, to calculate the volume of the slice, we multiply the area of the sector by the height of the slice:

Volume of Slice = Area of Sector * Height
Volume of Slice = 3.14 cm^2 * 3.14 cm
Volume of Slice = 9.86 cm^3

Therefore, the volume of the slice of the cylinder that has the 45-degree sector as its top surface is 9.86 cubic centimeters.