sm-1 pre-check name circles and prisms · pre-check name _____ circles and prisms ... surface area...

Cylinder Savvy Student Materials of 22

SM-1 Pre-check Name _______________________________ Circles and Prisms 1. Label the following parts of the circle. Then give reasonable values for the radius, diameter and circumference in centimeters. Radius If the radius is 10 cm, then the Diameter diameter is _______ cm and the Circumference circumference is _______ cm. 2. Write the formula and/or describe in your own words the process you would use to find the… Perimeter of a rectangle ___________________________________________________ Area of a rectangle ___________________________________________________ Surface Area of a rectangular prism ___________________________________________________ Volume of a rectangular prism ___________________________________________________ Circumference of a circle ___________________________________________________ Area of a circle ___________________________________________________ Find the circumference and area for each circle below. 3. radius = 4 in. 4. diameter = 9 ft. C = _______ A = _______ C = _______ A = _______


5. Sketch and label a circle with a radius of 1 in. Find its circumference and area. Circumference = ________ Area = ________ Now double the radius to 2 in. Find the new circumference and area. Circumference = ________ Area = ________ How was the circumference affected by the change in radius? _______________ How was the area affected? ________________________________________________________________ What would happen to the circumference and area if the original radius is 3 inches? Why? ________________________________________________________________ Compare answers of students’ who used 3.14 and those who used the pi (π) key on their calculators. How do their answers vary? _________ Which is more accurate? _______________


6. In the space below, sketch and label a rectangular prism with dimensions 4cm, 5cm, and 8cm. Find the surface area and volume of your prism. Surface Area: _______ Volume: _______ Now, sketch the same prism, but double the side that is 4mm. Compute the new surface are and volume of your prism. Surface Area: _______ Volume: _______ How did the surface area change? ___________________________________________ How did the volume change? _______________________________________________ What do you predict will happen to the surface area and volume if we now double the 5 millimeters to 10 millimeter? ________________________________________________________________ Predict how surface area and volume will change in the 2nd figure. Find surface areas and volumes for both figures. 7. 2m 4m 7m 7m 2m 2m


SM-2 Name(s) ___________________ ___________________ Surface Area of a Cylinder (Form A) Goal: Become familiar with the formula for finding the surface area of cylinders.

To find the surface area of the cylinder, we will need to: • Find the area of all surfaces (3 surfaces total)

• circle on top • circle on bottom • lateral surface (wrapped around top and bottom)

• add the 3 areas together

1. Label the following parts of this cylinder: radius (r=4cm), height (h=6cm), base, and lateral surface. What shape is the base? ________________

2. Imagine you cut your cylinder into pieces and lay them out flat. Label each radius and the height of the lateral surface (rectangular when laid flat).

3. To find the area of a circle, use the formula A r= π 2 .

What is the area of each of the circles in our cylinder? _________________ Put this information in the chart that follows on number 6.

4. We still need to find the area for the rectangle that wraps around the circles.

Because it wraps around the circles, its length is equal to the circumference of each circle. Calculate the circumference of the circle using . C = 2πr

Circumference of the circle = ______________

5. The last step to finding the lateral surface area is to multiply its length by its width. Do this substituting in the Circumference for length and inserting the given height. The formula for this is 2πr h⋅ .

Area of lateral surface = ______________________


6. Fill in the table to be sure you have all the necessary values. Once you have the area for each of the 3 surfaces, you can add them to find the total surface area. Note the Total Surface Area formula above. Can you see where the areas of the 2 circles are listed? Can you see where the area of the lateral surface is listed? Now that you know where the formula comes from, use it to find the surface area of several cylinders. Rewrite the equation for each problem, substituting in the correct values for the radius and height. Then simplify. The first one is done for you. SA = 2 22⋅ + ⋅π πr r h

7. r = 2in. 8. r = 10ft. 9. d = 4m h = 10in. h = 8ft. h = 4m SA = 2 ⋅ π 2( )2 + 2π 2( )10( ) = 2 ⋅ π 4( )+ 2π 20( ) = 8π + 40π = 48π ≈ 150.72 in.2 10. r = 3.5cm 11. d = 9yd 12. r = 5in. h = 6cm h = 5yd h = 20in.

Surface Name Formula for Area Area Circle #1 πr 2 Circle #2 πr 2

Lateral Surface 2πr h⋅ Total Surface Area: 2 22⋅ + ⋅π πr r h


SM-3 Name(s) ___________________ ___________________ Surface Area of a Cylinder with Manipulatives (Form B) Goal: Become familiar with the formula for finding the surface area of cylinders. Materials: cylindrical object (oatmeal container), scissors, ruler with centimeters

To find the surface area of the cylinder, we will need to: • Find the area of all surfaces (3 surfaces total)

• circle on top • circle on bottom • lateral surface (wrapped around top and bottom) • add the 3 areas together

1. Sketch, measure (in centimeters), and label the following parts of your

cylindrical object: radius, height, base, and lateral surface. What shape is the base of your figure? ________

2. Cut your cylinder into three pieces and lay them out flat. Sketch and label all necessary parts.

3. To find the area of one of the circles, use the formula A r= π 2 .

What is the area? _________________ Record your answer in the table that follows. What do you notice about the other circle? How will its area compare? _____________________________


4. Think carefully about the lateral surface of your cylinder. What shape is it? ______________________ What do you notice about the length (or base) of the polygon? ________________________________________________________ You should be able to see how the lateral surface wrapped around each circle, so its length is equal to each circle’s circumference. Measure with a ruler and compare the following values.

Length of Lateral Surface using a ruler = _____________

Circumference of the circle using formula C = 2πr = ____________

How do they compare? __________________________________

Next, multiply the circumference by the height of the lateral surface to find the area.

Area of lateral surface = ______________________

5. Fill in the table to be sure you have all the necessary values. Now that you know where the formula comes from, use it to find the surface area of several cylinders. Rewrite the equation, substituting in your values for the radius and height. Then simplify. The first one is done for you. SA = 2 22⋅ + ⋅π πr r h

6. r = 2in. 7. r = 10 t. 8. d = 4 m h = 10in. h = 8 ft. h = 4 m SA = 2 ⋅ π 2( )2 + 2π 2( )10( ) = 2 ⋅ π 4( )+ 2π 20( ) = 8π + 40π = 48π 150.72 in.2 9. r = 3.5 cm 10. d = 9 yd 11. r = 5 in. h = 6 cm h = 5 yd h = 20 in.

Surface Name Formula for Area Area Circle #1 πr 2 Circle #2 πr 2

Lateral Surface 2πr h⋅ Total Surface Area: 2 22⋅ + ⋅π πr r h


SM-4 Name _______________ Volume of a Cylinder Finding the volume of a cylinder can be easier to remember if you use the same formula for all prisms. For all right prisms, volume is simply: Area of the Base x Height Try it with these cylinders. Remember, the base of a cylinder is always a circle, so you will always need to begin by finding the area of a circle. 1. r = 2 in. 2. r = 10 ft. 3. d = 4 m h = 10 in. h = 8 ft. h = 4 m V=Area of Base · Height = πr 2 · h = π 2( )2 · (10) = π4 · (10) = 40π ≈ 125.6 in.3 4. r = 3.5 cm 5. d = 9 yd 6. r = 5 in. h = 6 cm h = 5 d h = 20 in.


SM-5 Exploration: Paper & Rice Goal: Demonstrate how changes in radius can affect the volume of cylinders without affecting surface area. Your group will need: • 1 bag of rice • 1 sheet of paper • tape • a ruler • 1 box lid • 1 pair of scissors • 1 measuring cup

Follow these instructions:

1. Have one person in your group cut the piece of paper exactly in half (hamburger-style). Label the half sheets A and B.

2. Measure and find the surface area of rectangle A. 3. Measure and find the surface area of rectangle B. 4. Tape A into a hotdog-style cylinder (longways). 5. Tape B into a hamburger-style cylinder (wider). 6. Predict below which cylinder will have greater volume.

1. While one student holds the cylinder in place measure and fill A with rice.

What is its approximate volume? ___________

2. Slip B over and around A. Lift A up and out of B so the rice begins to fill the

wider cylinder. What occurred once A was completely removed?

________________________

Answer the following: 1. Surface Area of A: __________ 2. Surface Area of B: __________


3. Prediction- Which cylinder will have greater volume? ________________

Why? _______________________________________

4. Why do you think the result occurs as it does?

________________________________________________________________ 5. Can you sketch 2 cylinders for which the volume would be the same, but the surface

area would be different? ________________________________________________________________


Schematics Needed for Cylinder Savvy Lesson CD/DVD

Diameter = 12 cm, Thickness (height) = 0.5mm

Mini-CD

Diameter = 80 mm, Thickness (height) = 0.5mm


Disc Storage Websites: 1. www.sleevetown.com/dvd-case.shtml 2. http://www.caselogic.com/search/index.cfm?Ne=100&N=4011+20025939 3. http://www.mediastoragecenter.com/scripts/prodlist.asp?idcategory=237&sortField=price&idsft=757&ntype=DE&


SM-6 Cylinder Savvy: Design 1 Name ______________________ Student Instructions You are a project designer/engineer hired to make a detailed plan for a CD or DVD storage container. You will plan three versions of your design and make comparisons between each one regarding materials needed (surface area) and disc capacity (volume). You will need: • a metric ruler

• a DVD or CD • a mini disc (or metric measurements for one)

Design 1: 1. Sketch a design for holding 12 discs securely. It must be in the shape of a

rectangular prism, triangular prism, cylinder, or other three-dimensional figure you can calculate the surface area and volume for. You can arrange discs so that they are in sleeves, stack or pile, snap into place, etc. Be creative. Just keep in mind how your overall dimensions will be affected by the discs, any covers or stabilizing materials (sleeves or plastic framing), and open space. Label length, width, and height of the total structure, as it appears from the outside.

2. Describe how your disc-holder works. How are discs stored, how does it open and close, and are there other important characteristics about your design that makes it unique? ________________________________________________________________

____________________________________________________________


3. What do you predict will be your overall surface area and volume in centimeters?

Surface area prediction: ____________ square centimeters

Disc storage volume prediction: ____________ cubic centimeters

4. Now find the total surface area of Design 1. _______________

5. Find the volume of Design 1. Only calculate the volume for space that would be

used to store discs. _________________

6. Were your predictions close? _________ Why or why not?

___________________________________________________________

7. Do you have any unused space that is a part of your design, but does not contain discs? If so, find the volume of the extra, unused space.

What percent of your total volume is for actual disc storage? _____________

Why would this be important to consider?

________________________________________________________________

____________________________________________________________


SM-7 Cylinder Savvy: Design 2 Name _______________ The manufacturer wants you to prepare a prototype that is similar in design, but will hold more discs. Read and complete the following.

1. Sketch a taller version of Design 1, capable of holding 24 discs. Change only one dimension in your plan, so that you can hold twice as many discs as before. That is, the outer appearance of your design should only grow in one dimension – height, length, or width. Label all outer dimensions.

2. How do you predict the change in height will affect your overall surface area and

volume in centimeters? Why? ___________________________________________________________________

___________________________________________________

Surface area prediction: ____________ square centimeters

Disc storage volume prediction: ____________ cubic centimeters

3. Now calculate the total surface area of Design 2. _______________


4. Find the volume of the portion in Design 2 which stores discs.

_________________

What percent of your total volume is for actual disc storage? _______________

Are there any changes in your percentage? Why or why not?

______________________________________________________________

______________________________________________________________

5. Were your predictions correct about your change in surface area and volume?

____________ If not, why? __________________________

_________________________________________________________

6. What conclusions can you draw about the way surface area changed?

________________________________________________________________

______________________________________________________

7. What conclusions can you draw about the way volume changed?

________________________________________________________________

______________________________________________________


SM-8 Cylinder Savvy: Design 3 Name _______________ Now the manufacturing company has realized there is a market out there for smaller, mini discs. Find the necessary measurements for a mini disc and then complete the following.

1. Sketch a narrower version of Design 1, for storing mini discs. Make it the same height as Design 1. (In other words, take your first design and change only the measurements needed to affect capacity according to radius/diameter.) Label all dimensions.

2. Sketch each of the 3 surfaces of your original design below. Then shade the regions that will still be a part of the new, smaller design. Use this to help you make your predictions.

3. How do you predict the change in radius/diameter will affect your overall surface

area and volume? Why? ___________________________________________________________

Surface area prediction: ____________ square centimeters Disc storage volume prediction: ____________ cubic centimeters

4. Find the surface area of Design 3. _______________ 5. Find the volume of Design 3. _________________ 6. What percent of your total volume is for actual disc storage? ___________

Are there any changes in your percentage? Why or why not? ___________________________________________________________


7. What general conclusions can you make about the way changing height or diameter affects the surface area of a design?

___________________________________________________________

8. What general conclusions can you make about the way changing height or

diameter affects the volume used up by discs in your design?

___________________________________________________________


SM-9

Silo: Cylinder Change in Radius Name _____________________ Extra Practice (following Cylinder Savvy) With the following problem, you will note how a change in radius can affect both surface area and volume.

1. Find the surface area and volume of a silo that is 30 feet tall and has a 10 foot diameter.

Sketch Surface Area Volume

2 22⋅ + ⋅π πr r h πr 2 x h

2. The farmer wants to double the storage space (volume) in his new silo, so he increases the diameter to 20 feet. Will this work? Sketch the new silo and calculate the new SA and V. Sketch Surface Area Volume

2 22⋅ + ⋅π πr r h πr 2 x h 3. How did doubling the radius affect the surface area? Why? Sketch the three

surfaces of the new silo and shade the area that has been added to each surface. ________________________________________________________________________________________________________________________________

4. How did it affect the volume? Why?

________________________________________________________________________________________________________________________________

5. Consider this problem. A circle with a radius of 1 is placed into a larger circle

with radius 2. Compare the areas. How does this problem compare to our silo situation?


6. Is there a solution to doubling the volume of a silo? What might you be able to do? Discuss with a partner two ways you could double the volume of the original silo. Be sure to include a sketch of each and label all dimensions.


SM-10 Candles: Cylinder Change in Height & Radius Name _______________________ Cylinder Savvy Extension With the following problem, you will note how a change in height will affect the surface area and volume of a cylinder.

1. The Wax Candle Company is planning a new line of candles and figuring out how best to wrap them. One employee wants to sell candles in 2 sizes, both with a diameter of 3 inches:

small (3in. X 4 in. tall) large (3in. X 6 in. tall) Find the amount of wax needed to create each one (volume) and the

approximate amount of tissue to wrap them (surface area). Record your answers in the table on number 2. Use 3.14 for pi in your calculations.

2. Wax costs $.10 per cubic inch. Tissue paper costs $.05 per square inch. Find the cost of making both a small and a large candle in the sizes listed for number one. Round your answer to the nearest cent.

Volume

(cu. in.) Surface Area (sq. in.)

Wax Cost ($.10/cu. in.)

Tissue Cost ($.05/sq. in.)

Small (3 x 4)

Large (3 x 6)

3. Another employee wants to sell candles that have diameters of 4 inches because

she claims it will save the company money on wax and on tissue for wrapping. Her designs will be:

small (4in. X 2in. tall) large (4in. X 4in. tall)

Fill out the table below in order to make comparisons to number 2.

Volume (cu. in.)

Surface Area (sq. in.)

Wax Cost ($.10/cu. in.)

Tissue Cost ($.05/sq. in.)

Small (4 x 2)

Large (4 x 4)


4. Which size, the 3 or 4 inch diameter will save the Wax Candle Company more

money? Why? Be sure to mention both the effect on surface area and volume.

________________________________________________________________

______________________________________________________

5. In your work, you used 3.14 instead of pi to make your calculations. Using pi would have given a more accurate measurement for surface area and volume. Why are using 3.14 and rounding your answers to the nearest cent acceptable for solving this situation? Name a situation for which it would be better to use pi to find a more accurate answer. ________________________________________________________________

________________________________________________________________

sm-1 pre-check name circles and prisms · pre-check name _____ circles and prisms ... surface area...

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