Things Needed Today (TNT) Math CC7/8 – Feb 7 Things Needed Today (TNT) Pencil/Math Notebook/Calculator Filling & Wrapping 4.4-4.5 Math Notebook: Topic: Filling Cones & Spheres & Comparison HW: Worksheet Packet

What’s Happening Today? HW ? Warm Up Lesson 4.4-4.5 Vol. of Cones & Spheres & Comparing Spheres, cylinder and cones Begin HW?

Warm Up (no calculator!) In the equation what is the value of y, if x = - 18 ? Write an equation for a line that is parallel to the line 2 3 y = x - 4 2 3 y = x - 4 Same slope, different y-intercept! y = - 16 Solve the equations. -12(-y - 5) = 13y + 2 q + 14 = 8(q + 7) 12y + 60 = 13y + 2 -13y -13y -y + 60 = 2 -60 -60 -y = -58 -1 -1 y= 58 q + 14 = 8q + 56 -8q -8q -7q + 14 = 56 -14 -14 -7q = 42 -7 -7 q= - 6

Cylinders are not the only important solid figures with curved edges and faces. Balls in the shape of spheres play an important role in many of our most popular sports. Cones are used to served ice cream and direct traffic around construction zones.

You can describe any sphere using this single dimension. The size & shape of any Sphere DEPENDS on the size of the RADIUS, which is the distance from any point on the surface to its center . You can describe any sphere using this single dimension. Take notes Sphere

The size and shape of a cone DEPENDS on the size of two dimensions, the HEIGHT and the RADIUS of its circular base. r h Take notes cone

In the following problem, you will develop strategies for calculating volumes of spheres and cones and see how changes in their dimensions cause changes in their volumes. It is hard to measure the volume of a sphere or a cone by filling it with an exact number of unit cubes. Fortunately, there is a helpful relationship among the volumes of cylinders, spheres, and cones. A simple experiment will reveal the relationship and the formula for the volumes of spheres and cones. In the problem, you will make a sphere, a cone and a cylinder each with the same diameter and the same height.

Question If a sphere and a cone have the same dimensions as a cylinder, how do the volumes compare? What formulas for the volume of a sphere and the volume of a cone can you write using these relationships?

Interactive Tools CMP3 – Pouring & Filling PBS Learning – Cylinders, Spheres, & Cones CMP3 – Pouring & Filling

A sphere is 2/3 the volume of a cylinder … Do the results of the experiment suggest a relationship among the volumes of a cylinder, a sphere, and a cone? If so , describe the relationship A sphere is 2/3 the volume of a cylinder … with the same diameter and height. A cone is 1/3 the volume of a cylinder… with the same diameter and vertical height.

Sphere Cylinder V = π r h V = V = (3.14) (r) (r) (h) V = 50.3 in cubed Take notes h = 4 in Diameter of 4 in. A sphere is 2/3 the volume of a cylinder with the same diameter and height. Cylinder Sphere 2πr h 3 2 2 V = π r h V = V = (3.14) (r) (r) (h) V = 50.3 in cubed 33.51 in. cubed V = (volume of cylinder) Skip #3 - 4

h V = πr 1 3 A cone is 1/3 the volume of a cylinder… B 1) What is the relationship between the volume of the cone and the volume of the cylinder? Take notes A cone is 1/3 the volume of a cylinder… with the same diameter and vertical height. 2) How do this relationship and the formula for the volume of a cylinder suggest a formula for the volume of a cone? 1 3 V = h πr 2

Use your formula for the volume of a sphere to complete the table. What patterns do you see in how the volume grows as the radius increases? The volume increases very rapidly as the radius increases. When the radius is doubled, how does the volume increase? When the radius is doubled, the volume increases by a factor of 8! How is the effect of scaling up the radius similar to the patterns you noticed when you scaled up rectangular prisms? When the radius is tripled, the volume increase by a factor of 27!

(64) V = V = V = 64 cubic in. V = 2 3 V = 42.67 in or D Suppose a cylinder, cone, and sphere all have the same height and that the volume of the cylinder is 64 cubic inches. Use only the relationships from Questions A & B. Describe how to find the volumes of the other shapes. 1) The sphere 2) The cone V = V = V = 64 cubic in. V = 2 V = 42.67 in or 42 and 2/3 cubic in. (64) V = 1 (64) 3 = 21.33 in or 21 and 1/3 cubic in. 3 3 3

Esther and Jasmine buy ice cream from Chilly’s Ice Cream Parlor Esther and Jasmine buy ice cream from Chilly’s Ice Cream Parlor. Esther gets a scoop of ice cream in a cone, and Jasmine gets a scoop in a cylindrical cup. Each container has a height of 8 centimeters and a radius of 4 centimeters. Each scoop of ice cream is a sphere with a radius of 4 centimeters.

Question What are some relationships you can use involving a cone, a sphere, & a cylinder with the same dimensions?

No, if it completely melts it will fill 2/3 of the cup. Suppose Esther and Jasmine both allow their ice cream to melt. A Will the melted ice cream fill Esther’s cone exactly? No, it will fill the cone and overflow!! The sphere of ice cream has twice the volume of the cone. Will the melted ice cream fill Jasmine’s cup exactly? B No, if it completely melts it will fill 2/3 of the cup. C How many scoops of ice cream can be packed into each container? The cone will hold ½ a scoop of melted ice cream. The cup will hold 1.5 or 3/2 scoops of ice cream.

Formulas to find Volume for a Sphere & Cone Note: Formulas for Sphere and cone when you do not have a cylinder with the same dimension to compare to their volume 4 3 V = 1πr h πr 2 V = 3 3 Cone Sphere

Cone V = 1πr h 2 3 Cone: has 1 circular base and a lateral surface Formula for volume of a cone: V= 1/3Bh b: base h: height

Practice V = πr h 2 V= π5*5 7.6 V ≈ 199 cubic meters

Practice V ≈ 32.7 cubic in

Practice V ≈ 201.1cubic ft

Sphere Sphere: has only 1 surface, which is curved, and has no base Formula for volume of a sphere: π: 3.14 r: radius

Let’s practice (round to the nearest tenths) 2 cm V ≈ 33.5 cubic cm

Let’s practice 10 cm V ≈ 523.6 cubic cm

Let’s practice 8 cm V ≈ 268.1 cubic cm

Homework: F&W 4.4 Worksheet packet