Bell Work: Practice set (a) (b) and (c) on page 739.

Answer: ½ √5/5 2√5/5

Lesson 114: Calculating Area as a Sweep

Imagine using a push broom to clean a dirty floor or sidewalk Imagine using a push broom to clean a dirty floor or sidewalk. As you push the broom it sweeps an area. The area the broom sweeps is equal to the width of the broom times the distance the broom is pushed.

Now imagine sweeping an area with one side of the broom held in place Now imagine sweeping an area with one side of the broom held in place. The broom sweeps a circular area. The area equals the width of the broom times the average (mean) distance the broom is pushed.

We will apply this concept of sweeping to calculate some areas and surface areas. We start with a circle, then we calculate the lateral surface area of a cylinder and a cone.

Look at the minute hand of a watch or clock Look at the minute hand of a watch or clock. In one hour the minute hand sweeps all the way around the circular face of the clock. Every point inside the circle the length of the minute hand is swept across by that hand. Notice that the minute hand represents the radius of the circular face of the clock.

As radius CA turns, it sweeps the area of a circle. Example: As radius CA turns, it sweeps the area of a circle. How wide is “broom CA” (radius)? How far does point a move in one full turn? How far does point C move in one full turn? What is the average distance the points from A to C move during one full turn? The area of the circle is the product of the length of “broom CA” and the average distance the points of the broom move. What is the area? 5 cm C A

Point A moves the distance of the circumference = 10πcm Answer: The length is 5cm Point A moves the distance of the circumference = 10πcm Point C moves 0 cm The average (mean) is the average of 10π and 0 = 5πcm The product of the radius and its average = 25πcm 2

Recall that the lateral surface area of a cylinder is represented by the label of a can. We can calculate the lateral surface area of a cylinder as a sweep. We show two ways:

Method 1: Sweep with the circumference Imagine the upper rim of a can is a circular broom. As it slides down the can it sweeps the entire lateral surface of the can. The length of the broom is the circumference of the can. The distance the broom moves is the height of the can.

Method 2: sweep with the height Imagine a broom segment sweeping around the can. The length of the “broom” is the height of the can. The distance it moves is the circumference of the circle.

With both methods we find the lateral surface area by multiplying the circumference of the can and the height of the can. lateral surface area of a cylinder = circumference x height S = 2πrh or S = πdh

We can use similar calculations to develop a formula for the lateral surface area of a cone. Lateral surface area = slant height x average of bases

Example: Calculate the lateral surface area of the cone Example: Calculate the lateral surface area of the cone. The diameter of the base is 2 inches and the slant height is 4 inches.

Answer: Surface area = slant height x average of bases = 4 x 0 + 2π 2 = 4π inches

Finally, imagine sweeping an area with a broom that becomes narrower and narrower. The broom sweeps an area the shape of a trapezoid of triangle. The area of the trapezoid or triangle equals the average (mean) width of the broom times the distance it is pushed.

Lesson 114: #1-25