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 paper (I used 8 ½ x 11 printer paper you will use 8x10 in) 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.
A: The dimensions of the paper are 8x10 in, which means the base of the prism will be 8x10 in. To create a tall rectangular prism, the height will be 8 in and the length will be 2.5 in (since 10 divided by 4 is 2.5).
Volume of A: 8 x 8 x 2.5 = 160 cubic inches
B: To create a short rectangular prism, the height will be 2 in (half of 4 paper divisions) and the length will be 5 in (half of 10 paper divisions).
Volume of B: 8 x 5 x 2 = 80 cubic inches
C: To create a tall cylinder, the height will be 8 in (same as paper height) and the diameter will be 2.2 in (circumference of 7 in divided by 3.14, then divided by 2 for the radius).
Volume of C: 1.1 x 1.1 x 3.14 x 8 = 24.78 cubic inches
D: To create a short cylinder, the height will be 2 in (half of 4 paper divisions) and the diameter will be 2.2 in (same as C).
Volume of D: 1.1 x 1.1 x 3.14 x 2 = 7.77 cubic inches
Therefore, the volumes with the proper units for each shape are:
A: 160 cubic inches
B: 80 cubic inches
C: 24.78 cubic inches
D: 7.77 cubic inches
To find the dimensions and volumes of each shape, we will use the provided paper size of 8x10 inches.
A: Tall Rectangular Prism
The dimensions of the paper used for shape A are 8x10 inches. To find the dimensions of the prism, we divide each side by 4.
Length = 8/4 = 2 inches
Width = 10/4 = 2.5 inches
Height = The height of the prism is not specified, so you can choose any value you want. For example, let's say the height is 6 inches.
To find the volume, we use the formula for the volume of a rectangular prism:
Volume = Length x Width x Height
Volume = 2 inches x 2.5 inches x 6 inches = 30 cubic inches
B: Short Rectangular Prism
Using the same method as above, we divide each side of the paper by 4.
Length = 8/4 = 2 inches
Width = 10/4 = 2.5 inches
Height = Let's say the height is 3 inches.
Volume = Length x Width x Height
Volume = 2 inches x 2.5 inches x 3 inches = 15 cubic inches
C: Tall Cylinder
To find the dimensions of the cylinder, we need to first find the radius of the circle top using the provided circumference of 11 inches.
Circumference = Diameter x 3.14
11 inches = Diameter x 3.14
Divide both sides by 3.14 to find the diameter:
11 inches / 3.14 = Diameter ≈ 3.508 inches
Since we need the radius for the volume formula, we divide the diameter by 2.
Radius = Diameter / 2 = 3.508 inches / 2 ≈ 1.754 inches
Height = The height of the cylinder can be any value you choose. Let's say the height is 5 inches.
To find the volume, we use the formula for the volume of a cylinder:
Volume = π x Radius^2 x Height
Volume = 3.14 x (1.754 inches)^2 x 5 inches ≈ 48.31 cubic inches
D: Short Cylinder
Using the same method as above, the calculations are the same except for the height. Let's say the height is 2 inches.
Volume = π x (1.754 inches)^2 x 2 inches ≈ 12.93 cubic inches
The volumes of the shapes are:
A: 30 cubic inches
B: 15 cubic inches
C: 48.31 cubic inches
D: 12.93 cubic inches
Please note that these calculations are based on the assumption that the given dimensions and formulas are correct.
To find the volume of each shape, we will use the given dimensions of the paper and the formulas provided.
First, let's find the dimensions for shape A (tall rectangular prism) using the paper dimensions of 8x10 inches.
A:
Length = 8 inches
Width = 10 inches
Height = (10 / 4) = 2.5 inches
Volume of Prism = Length x Width x Height
Volume of A = 8 inches x 10 inches x 2.5 inches = 200 cubic inches
Next, let's find the dimensions for shape B (short rectangular prism) using the same paper dimensions.
B:
Length = 8 inches
Width = 10 inches
Height = (8 / 4) = 2 inches
Volume of B = 8 inches x 10 inches x 2 inches = 160 cubic inches
Now, let's find the dimensions for shape C (tall cylinder) using the circumference formula and the given paper dimensions.
C:
Circumference = Diameter x 3.14
11 inches = Diameter x 3.14
Solving for Diameter:
Diameter = 11 inches / 3.14 ≈ 3.5 inches
To find the radius (r) for the volume formula, divide the diameter by 2:
Radius (r) = Diameter / 2 = 3.5 inches / 2 = 1.75 inches
Height (h) of the cylinder does not change and remains the same as the height of the paper, which is 10 inches.
Volume of Cylinder = Radius x Radius x 3.14 x Height
Volume of C = (1.75 inches) x (1.75 inches) x 3.14 x 10 inches ≈ 96.77 cubic inches
Finally, let's find the dimensions for shape D (short cylinder) using the same paper dimensions.
D:
Circumference = Diameter x 3.14
8 inches = Diameter x 3.14
Solving for Diameter:
Diameter = 8 inches / 3.14 ≈ 2.55 inches
Radius (r) = Diameter / 2 = 2.55 inches / 2 = 1.275 inches
Height (h) remains the same as the height of the paper, which is 10 inches.
Volume of D = (1.275 inches) x (1.275 inches) x 3.14 x 10 inches ≈ 50.93 cubic inches
So the volumes of the four shapes using 8x10 in paper are as follows:
Volume of A = 200 cubic inches
Volume of B = 160 cubic inches
Volume of C ≈ 96.77 cubic inches
Volume of D ≈ 50.93 cubic inches