You will mathematically and experimentally show how 4 figures made from the SAME size paper will produce different volumes.
Take 4 sheets of paper, you will use 8x10 in and create:
-A: tall rectangular prism
-B: short rectangular prism
-C: tall cylinder
-D: short cylinder
Mathematically:
Let the dimensions of the paper be L = 10 inches and W = 8 inches. We will find the volume of each figure by dividing the paper and folding it into the desired shapes.
A. Tall Rectangular Prism:
Let one edge of the paper be the height (H), and the other side be the base (B) of the prism. Therefore, the height H = 10 inches, and the base B = 8 inches. To make the prism, fold the paper in half so that the width and height remain the same.
Volume (Va) = B * W * H = 8 * 4 * 10 = 320 cubic inches
B. Short Rectangular Prism:
Let one edge of the paper be the height (H) and the other side to be twice the base (2B) of the prism. The height H = 8 inches, and the total base length is 10 inches. To make the prism, fold the paper in half so that the base length is doubled, and the width will become half.
Volume (Vb) = B * W * H = 5 * 8 * 8 = 320 cubic inches
C. Tall Cylinder:
To create a tall cylinder, roll the paper along its longer side. The height will be equal to the longer side of the paper, and the circumference will be equal to the shorter side.
Height (H) = 10 inches
Circumference (C) = 8 inches
Radius (r) = C / (2 * π) ≈ 1.27 inches
Volume (Vc) = π * r^2 * H ≈ 3.14 * 1.27^2 * 10 ≈ 161.29 cubic inches
D. Short Cylinder:
To form a short cylinder, roll the paper along the shorter side. The height will be equal to the shorter side of the paper, and the circumference will be equal to the longer side.
Height (H) = 8 inches
Circumference (C) = 10 inches
Radius (r) = C / (2 * π) ≈ 1.59 inches
Volume (Vd) = π * r^2 * H ≈ 3.14 * 1.59^2 * 8 ≈ 201.06 cubic inches
Experimentally:
1. Take four 8x10 inch sheets of paper.
2. Create the tall and short rectangular prisms (A and B) by folding the paper as described above and securing it with tape.
3. Create tall and short cylinders (C and D) by rolling the paper along the longer and shorter side as described above and securing it with tape.
4. Fill each shape with a measured amount of dry material (e.g., rice or beans). Record the volumetric measurements for each figure.
5. Compare the volumes found experimentally with the mathematical calculations.
The mathematical and experimental results should demonstrate that the four figures created have different volumes even though they were made from the same size paper.