Bell work: Factor: -4x – 28x – 48x 3 2

Answer: -4x(x + 3)(x + 4)

Lesson 72: Factors that are Sums, Pyramids and Cones

Sometimes a trinomial has a common factor that is a sum, as we in the following examples.

Example: Factor (a + b)x – (a + b)x – 6(a + b) 2

Answer: (a + b)(x – x – 6) (a + b)(x – 3)(x + 2) 2

Example: Factor (x + y)x + 9x(x + y) + 20(x + y) 2

Answer: (x + y)(x + 4)(x + 5)

Practice: Factor m(x – 1)x + 7mx(x – 1) + 10m(x – 1) 2

Answer: m(x – 1)(x + 2)(x + 5)

Pyramids: a geometric solid with one face a polygon (base) and the other faces triangles (lateral faces) with a common vertex. The altitude of a pyramid is the height.

A right pyramid is a pyramid whose axis is a right angle to the base. In a right pyramid, the axis is also its latitude. In your book we will be working with right pyramids.

There is a certain type of right pyramid that are important. These right pyramids are called regular pyramids. A regular pyramid is a right pyramid whose base is a regular polygon. The lateral races are identical isosceles triangles.

The height of a lateral face is called the slant height of the regular pyramid. Slant height is defined only for regular pyramids. In a regular pyramid, the altitude and a slant height determine a right triangle.

A pyramid is classified and named according to the shape of its base.

A cone is like a pyramid except that its base a closed curve instead of a polygon. The curved surface between the vertex and the base is called the lateral surface.

The segment joining the vertex to the center of the base is called the axis of the cone. The altitude of a cone is the perpendicular segment from the vertex to the plane of the base.

A right cone is a cone whose axis is at right angles to the base. The axis is also the altitude. The distance form the vertex to any point of the circle of the base is the slant height.

Volume of pyramids and cones: The volume of a pyramid or a cone is equal to one third the area of the base times the height. 1/3bh= Volume

Example: Find the volume of the right rectangular pyramid. Height = 8 inches 9 inches 7 inches

Answer: Area of base = (9in)(7in) = 63in Volume = 1/3(63in )(8in) = 168in 2 2 3

Example: A right circular cone has a base of radius 6 feet and a height of 8 feet. Find the volume of the right circular cone.

Answer: Area of base = π(6ft) = 36πft Volume = 1/3(36πft )(8ft) = ft

We define the lateral surface area of a pyramid to be the sum of the areas of all the lateral faces. To find the surface area of a pyramid, we add the area of the base to the lateral surface area.

Example: Find the surface area of a regular square pyramid with a slant height of 5 and a base length of 6. Dimensions are in centimeters.

Answer: Area of base = 36cm Area of one face = ½(5)(6) = 15cm Surface area = 4(15cm ) + 36cm = 96cm

Lateral surface area of a right circular cone = π(radius)(slant height)

Example: A right circular cone has a base of radius 8m and a slant height of 10m. Find the surface area of the right circular cone.

Answer: Surface area = area of base + lateral surface area = πr + πrl = π(8m) + π(8m)(10m) = m 2 2 2

HW: Lesson 72 #1-30