L22 Measurement in Three Dimensions. 22a Three Dimensions Warmup

Transcription

1 22a Three Dimensions Warmup Let s take a look at two-dimensional and three-dimensional objects below. A vertex (plural: vertices) (#VOC) in a 2 or 3-dimensional object is a point where two or more straight lines meet. {Mark with a dot the vertices in the two figures above} *A square has vertices. A square prism (at right above) has vertices. An edge (#VOC) in a 2 or 3-dimensional object is a line segment that joins two vertices. {Mark with a highlighter the edges of the two figures above} *A square has edges. A square prism has edges. A face (#VOC) of a 3-dimensional object is one of the 2-dimensional (flat) objects that form the boundary (surface) of the 3-dimensional (solid) object. {Shade one face of the square prism above} *A rectangular prism has faces. Why does the square above NOT have any faces?

2 Shape Guessing Game Cut up the pieces below for both the vocabulary words and the shapes, and place the shapes face down in a pile. One partner will randomly choose a shape from the pile, keeping their choice a secret from the other partner. They will then describe their chosen shape to their partner, using the previous vocabulary terms only: vertices, edges, and faces. Their partner will then choose from the vocabularly word cutouts the correct 3-dimensional shape described. Once the vocabulary term is determined, name a real-world object that has that particular shape. Glue both cutouts in the space below and take turns choosing from the shape pile until you ve exhausted all shapes. Once you are finished, your partner should blue their matching cutouts onto their paper. *Be sure to fill in your vocabularty booklet with your new vocabulary below. Formal definitions will come later.

3 Rectangular Pyramid Triangular Pyramid Triangular Prism Cone Sphere Hexagonal Prism Cube Square Prism Cylinder

4 This page is intentionally left blank to allow for cutouts on the previous page.

5 Many of the shapes we just reviewed are called polyhedrons (#VOC). A polyhedron is a 3-dimensional object made up of a finite number of polygon faces that are joined in pairs only, along their edges; the points where the edges intersect are the vertices. Which of the shapes we just studied are polyhedrons? Why are the others NOT polyhedrons? Label the rectangular prism below to designate its length (L), width (W), and height (H). (Note: the labeling is not necessarily unique) Determine which expression in the table below represents the volume (V) of the rectangular prism, and which represents the surface area (SA) of the rectangular prism: length + width + height (length) (width) (height) (length) (width) + height 2 x [ (length)(width) + (length)(height) + (width)(height)] Explain to your partner how you determined which choice is the correct expression for volume. *Hint: You could think of stacking layers of small cubes, each with edge length 1 unit, in a large prism. Explain how you determined which choice is the correct expression for surface area.

6 Practice Find the volume of each solid. Leave your answers in terms of π or in simplest radical form, AND as a decimal rounded to the nearest thousandths Given the following information, calculate the missing value for a rectangular prism. Be sure to include the units; this is a critical aspect of this exercise. a. Length = 6 inches; Width = 4 inches; Height = 3 inches Volume = Surface Area = b. Surface Area = 102 in 2 ; Width = 9 inches; Height = 2 inches Length = Volume = c. Volume = 60 feet 3 ; Width = 4 feet; Height = 3 feet Length = Surface Area =

7 A cylinder is completely determined by the radius of its base, and its height. Label the radius (r) and height (h) on the cylinder figure below. Determine which expression represents the volume of the cylinder and which represents its surface area (including the top and bottom). Hint for Surface Area: the portion of a cylinder not including the top and bottom can be formed by bending a rectangular piece of paper until the right edge meets the left edge. Equivalently, if you cut the cylinder along a line perpendicular to its base and unbend it to lay it flat, the resulting figure will be a rectangle. Thus, the surface area of that portion is the same as the area of a rectangle. 2π (radius)(height) 2π (radius) 2 + 2π (radius)(height) π (radius) 2 (height) 2π + radius 2 + height Explain how you determined which choice is the correct expression for volume. Explain how you determined which choice is the correct expression for surface area. In particular, how did you determine the length of the rectangle formed from the cylinder? Why does your volume answer seem to be reasonable, given the formula for the volume of a rectangular prism?

8 Practice Find the volume of each solid. Leave your answers in terms of π or in simplest radical form, AND as a decimal rounded to the nearest thousandths Given the following information about the cylinder, find the missing value. a. Diameter = 10 inches; Height = 6 inches; Volume = b. Volume = 14π feet 3 ; Height = 2 feet; Radius = 7. Grandma needs to purchase an attic fan for her attic. The attic is in the shape of a triangular prism with height 6 ft., base 24 ft., and length 40 ft., as shown in the diagram below. The attic has vents on each end and an attic fan will be placed at one end that is designed to suck outside air from the opposite end (the front-left triangle) and send it out of the attic at the other end. If the fan is rated to move 32,000 in 3 of air every second, how many minutes does it take the fan to remove the same amount of air that the attic contains? In other words, how long does it take to replenished the attic air with fresh air? Fan 24 ft. 6 ft. 40 ft.

9 Exit Pass 1. Grandma is building an outdoor shed in the shape of a rectangular prism. She would like the shed to be 36 cubic meters in volume. If the measurements must be in whole number units, list 4 different shed sizes (length, width, and height) that yield the desired volume (not including duplicate combinations; e.g. 2 x 3 x 4 is the same as 4 x 2 x 3). Be sure to include units for your measurement choices. Which of your 4 choices do you think is the most reasonable for the shed dimensions, and why? (Answers may vary.) 2. Grandma is making a pot of beef stew for dinner to thank her family for helping to build the shed. If she would like to make 8000π ml of stew (1 ml = 1 cm 3 ), the only 3 whole-number cylindrical pot sizes she could use have radius r = 5cm, 10cm, and 20 cm. What would the heights need to be for each? Be sure to include units. r = 5cm h = r = 10cm h = r = 20cm h = Which, if any, of your choices is a reasonable size for a stew pot? Why or why not?

10 Homework Find the volume of each solid. Leave your answers in terms of π or in simplest radical form, AND as a decimal rounded to the nearest thousandths cm 4 cm 3. A rectangular prism is formed by appropriately folding the faces of the shape to the right. Identify the dimensions (length, width, and height), then calculate the volume and surface area. If necessary, round to nearest tenths. 11 cm L22 Measurement in Three Dimensions Homework Find the volume of each solid. Leave your answers in terms of π or in simplest radical form, AND as a decimal rounded to the nearest thousandths cm 4 cm 3. A rectangular prism is formed by appropriately folding the faces of the shape to the right. Identify the dimensions (length, width, and height), then calculate the volume and surface area. If necessary, round to nearest tenths. 11 cm

11 4 yd 7 yd 4. A right cylinder is formed by bending the rectangle to the left until the right edge touches the left edge. The length of the top/bottom of the rectangle will be the circumference of the base of the cylinder. Recall that the formula for the circumference of a circle is C = 2πr, and the area of a circle is A = πr 2. Find the radius and circumference of the circular base and the height of the cylinder, then calculate the volume. Round to nearest tenths. 5. Susan has an exotic fish tank in the shape of a cylinder that is 26 inches tall. The diameter of the tank is 12 inches. If there are 2 inches of rocks in the bottom, how much water, in cubic inches, is needed to fill the tank? You can assume the water in between the rocks is negligible. 6. Campbell s manufactures a cylindrical soup can that has a diameter of 3 inches and a volume of 30 in 3. If the height stays the same and the diameter is doubled, by what factor will the can s volume increase (i.e. does it double, triple, etc.)? 4 yd 7 yd 4. A right cylinder is formed by bending the rectangle to the left until the right edge touches the left edge. The length of the top/bottom of the rectangle will be the circumference of the base of the cylinder. Recall that the formula for the circumference of a circle is C = 2πr, and the area of a circle is A = πr 2. Find the radius and circumference of the circular base and the height of the cylinder, then calculate the volume. Round to nearest tenths. 5. Susan has an exotic fish tank in the shape of a cylinder that is 26 inches tall. The diameter of the tank is 12 inches. If there are 2 inches of rocks in the bottom, how much water, in cubic inches, is needed to fill the tank? You can assume the water in between the rocks is negligible. 6. Campbell s manufactures a cylindrical soup can that has a diameter of 3 inches and a volume of 30 in 3. If the height stays the same and the diameter is doubled, by what factor will the can s volume increase (i.e. does it double, triple, etc.)?