7B Math: Efficiency Expert Portfolio
Directions:
You will mathematically and experimentally show how 4 figures made from the SAME size paper will produce different volumes.
Take 4 sheets of 8x10 in paper and create:
-A: tall rectangular prism
-B: short rectangular prism
-C: tall cylinder
-D: short cylinder
Mathematically you will figure the dimensions of your shapes just using your paper dimensions versus a ruler.
A: B: C: D:
Volume of Prism: l x w x h
Volume of Cylinder: r x r x 3.14 x h
So we will use our paper dimensions to find the shape measurements needed to find the volume of each one.
A and B….the prisms are easy. We just take the dimension and divide it by 4 to find our measurement.
C and D ….the cylinders are a bit more tricky. We know the circumference of our circle tops and the heights of our cylinders but we need to find the radius of our circle top.
Circumference = diameter x 3.14
11 = d x 3.14
Divide both sides by 3.14 to find the diameter
11/3.14 = d
3.5 = d but in finding volume of cylinder we need the radius.
d/2 = r 3.5/2 = 1.75 in
You will provide the volume of each shape A,B,C and D using 8x10 in paper. USE PROPER UNITS FOR VOLUME. The 4 volumes with units is ALL YOU HAVE to provide. No pictures needed.
OPTIONAL: Now you can test your volumes by actually making these open ended paper shapes and filling them with rice or beans or some material to figure out which one holds the most. It is best to make your shapes out of a thicker paper but I understand if you don’t have thicker paper.
Here is what I need for full credit!
Paper dimensions ( length and width, you may use 8x10 inches) You may NOT use the 8 ½ x 11 inches I used.
Volumes (use correct unit) of tall prism A, short prism B, tall cylinder C and short cylinder D.
hint: in the example my paper is 8 ½ x 11 in our volume will be in³
The experiment portion (filling your open ended shapes with rice or beans to see which holds the most) is OPTIONAL but is a fun, hands on way to support your math.
Using 8x10 inch paper, we can create the four shapes as follows:
A: Tall Rectangular Prism:
Divide the 10 inches into 4 equal parts, so each side will be 2.5 inches wide. The height will be 8 inches.
Dimensions: 8x2.5x2.5 inches
B: Short Rectangular Prism:
Divide the 8 inches into 4 equal parts, so each side will be 2 inches wide. The height will be 10 inches.
Dimensions: 10x2x2 inches
C: Tall Cylinder:
To find the circumference, we'll use the 10-inch width of the paper for the long side of the cylinder.
Circumference = diameter x 3.14
10 = d x 3.14
Divide both sides by 3.14 to find the diameter
10/3.14 = d
3.18 ≈ d
To find the radius, divide the diameter by 2
r = 1.59 inches
Height of the cylinder will be 8 inches.
D: Short Cylinder:
To find the circumference, we'll use the 8-inch width of the paper for the long side of the cylinder.
Circumference = diameter x 3.14
8 = d x 3.14
Divide both sides by 3.14 to find the diameter
8/3.14 = d
2.54 ≈ d
To find the radius, divide the diameter by 2
r = 1.27 inches
Height of the cylinder will be 10 inches.
Now let's find the volumes:
Volume of Tall Prism A (V = l x w x h):
V = 8 x 2.5 x 2.5
V = 50 in³
Volume of Short Prism B (V = l x w x h):
V = 10 x 2 x 2
V = 40 in³
Volume of Tall Cylinder C (V = r x r x 3.14 x h):
V = 1.59 x 1.59 x 3.14 x 8
V ≈ 63.64 in³
Volume of Short Cylinder D (V= r x r x 3.14 x h):
V = 1.27 x 1.27 x 3.14 x 10
V ≈ 50.77 in³
In conclusion, the four shapes made from the 8x10 inch paper would have the following volumes:
A: 50 in³
B: 40 in³
C: ≈ 63.64 in³
D: ≈ 50.77 in³
From the calculations, Tall Cylinder C appears to have the largest volume, which can be verified through the optional experiment.