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 the volume of the 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 are ALL YOU HAVE to provide. No pictures needed.
A: tall rectangular prism
Dimensions: 8 x 2.5 x 10 in
Volume = 8 x 2.5 x 10 = 200 cubic inches
B: short rectangular prism
Dimensions: 2 x 5 x 8 in
Volume = 2 x 5 x 8 = 80 cubic inches
C: tall cylinder
Dimensions: radius = 1.75 in, height = 10 in
Volume = 1.75 x 1.75 x 3.14 x 10 = 96.625 cubic inches
D: short cylinder
Dimensions: radius = 1.75 in, height = 5 in
Volume = 1.75 x 1.75 x 3.14 x 5 = 48.313 cubic inches
To find the volume of each shape A, B, C, and D using the provided 8x10 in paper, you will follow these steps:
1. For shapes A and B (the prisms):
a. The length, width, and height of the prism can be found by dividing the paper dimensions by 4.
Shape A: Length = 8/4 = 2 in, Width = 10/4 = 2.5 in, Height = 8/4 = 2 in
Shape B: Length = 8/4 = 2 in, Width = 10/4 = 2.5 in, Height = 10/4 = 2.5 in
b. Calculate the volume of each prism using the formula Volume = length x width x height.
Volume of shape A = 2 in x 2.5 in x 2 in = 10 in³
Volume of shape B = 2 in x 2.5 in x 2.5 in = 12.5 in³
2. For shapes C and D (the cylinders):
a. You need to find the radius of the circle top for each cylinder.
b. Start by finding the diameter using the given circumference formula: Circumference = diameter x 3.14.
The circumference is given as 11 in, so we can calculate the diameter:
11 in = diameter x 3.14
Divide both sides by 3.14:
diameter = 11 in / 3.14
diameter ≈ 3.50477 in (approximated to 5 decimal places)
c. To calculate the radius, divide the diameter by 2:
radius = diameter / 2
radius ≈ 3.50477 in / 2 ≈ 1.75238 in (approximated to 5 decimal places)
Round it to 1.75 in (to match the provided measurements)
d. Now that you have the radius and the height (given as 8 in) for both cylinders, calculate the volume using the formula Volume = radius² x 3.14 x height.
Volume of shape C = 1.75 in x 1.75 in x 3.14 x 8 in ≈ 96.636 in³ (approximated to 3 decimal places)
Volume of shape D = 1.75 in x 1.75 in x 3.14 x 10 in ≈ 120.795 in³ (approximated to 3 decimal places)
The four calculated volumes, with their proper units for volume, are as follows:
- Volume of shape A: 10 in³
- Volume of shape B: 12.5 in³
- Volume of shape C: 96.636 in³
- Volume of shape D: 120.795 in³
To find the volumes of the four shapes using 8x10 in paper, we will follow the given instructions.
A: Tall Rectangular Prism
The dimensions of the paper are 8x10 in. To create a tall rectangular prism, we divide each dimension by 4.
Length (l) = 8 in / 4 = 2 in
Width (w) = 10 in / 4 = 2.5 in
Height (h) = 8 in (Original paper height)
Volume of Prism (V) = l x w x h
V = 2 in x 2.5 in x 8 in
V = 40 in³
B: Short Rectangular Prism
We use the same dimensions of the paper, but this time we divide by 4 differently.
Length (l) = 10 in / 4 = 2.5 in
Width (w) = 8 in / 4 = 2 in
Height (h) = 8 in (Original paper height)
Volume of Prism (V) = l x w x h
V = 2.5 in x 2 in x 8 in
V = 40 in³
C: Tall Cylinder
To find the dimensions of the cylinder, we use the diameter we calculated using the circumference formula.
Diameter (d) = 3.5 in
Radius (r) = d / 2 = 3.5 in / 2 = 1.75 in
Height (h) = 8 in (Original paper height)
Volume of Cylinder (V) = r x r x 3.14 x h
V = 1.75 in x 1.75 in x 3.14 x 8 in
V = 77 in³
D: Short Cylinder
We use the same dimensions as the tall cylinder.
Diameter (d) = 3.5 in
Radius (r) = d / 2 = 3.5 in / 2 = 1.75 in
Height (h) = 8 in (Original paper height)
Volume of Cylinder (V) = r x r x 3.14 x h
V = 1.75 in x 1.75 in x 3.14 x 8 in
V = 77 in³
Therefore, the volume of shape A is 40 in³, shape B is 40 in³, shape C is 77 in³, and shape D is 77 in³.