The size to which a pumpkin grows while on the vine is probably problematic, as our current visiting POLY pumpkins demonstrate.
What if we were watching a pumpkin on the vine grow over time and that its shape was totally spherical.
Now for your five (5) tasks:
1. Initially we see a pumpkin when its radius equals 2”. Calculate its volume and surface area at this point.
2. Determine what the volume and surface area of the pumpkin will be when its radius equals 4”.
3. Determine what the radius of the pumpkin would have been when its volume was 2 times its initial volume.
4. Determine what the radius would have been when the pumpkin had a surface area 4 times the surface area it had when its radius was 4”.
5. Consider the volume of our pumpkin when its radius was 3”. What if there were a cylindrically-shaped pumpkin having that exact same volume. Determine 3 different radius and height combinations for the cylinder pumpkin that would generate that same volume. By the way, all 6 numbers must be different and must be integers.
v = 4/3 pi r^3
a = 4 pi r^2
plug in r=2
plug in r=4
or, just multiply the volume by 8 and the area by 4
multiply r by ∛2
if area is 4x, radius is 2x
v(3) = 4/3 pi * 27 = 36pi
just come up with values for r and h such that r^2 and h are factors of 36 since cylinder volume
v = pi r^2 h
To solve these tasks, we will use formulas for the volume and surface area of a sphere:
1. Initially, when the radius equals 2 inches:
- Volume of a sphere = (4/3) * π * r^3
V = (4/3) * 3.1416 * 2^3 = 33.5103 cubic inches
- Surface area of a sphere = 4 * π * r^2
A = 4 * 3.1416 * 2^2 = 50.2656 square inches
2. When the radius equals 4 inches:
- Volume = (4/3) * π * r^3
V = (4/3) * 3.1416 * 4^3 = 268.0825 cubic inches
- Surface area = 4 * π * r^2
A = 4 * 3.1416 * 4^2 = 201.0619 square inches
3. Finding the radius when the volume is 2 times the initial volume:
- The initial volume is 33.5103 cubic inches, so 2 times that is 67.0206 cubic inches.
- Solving for the radius using the volume formula:
67.0206 = (4/3) * 3.1416 * r^3
r^3 = 67.0206 * (3/4) / 3.1416
r ≈ 3.0661 inches (rounded to 4 decimal places)
4. Finding the radius when the surface area is 4 times the surface area at a radius of 4 inches:
- The surface area at a radius of 4 inches is 201.0619 square inches, so 4 times that is 804.2476 square inches.
- Solving for the radius using the surface area formula:
804.2476 = 4 * 3.1416 * r^2
r^2 = 804.2476 / (4 * 3.1416)
r ≈ 6.4422 inches (rounded to 4 decimal places)
5. Considering a cylindrical pumpkin with the same volume as when the radius was 3 inches:
- Recall that the volume of a cylinder is given by V = π * r^2 * h.
- First, we need to find the volume when the radius is 3 inches:
V = (4/3) * 3.1416 * 3^3 = 113.0972 cubic inches.
- Now, let's find three different combinations of radius and height for the cylindrical pumpkin with the same volume:
Combination 1: r = 1 inch, h = 113.0972 inches
Combination 2: r = 2 inches, h = 28.2743 inches
Combination 3: r = 3 inches, h = 10.6991 inches
Note: The numbers provided for the radius and height combinations are just examples. There can be infinitely many combinations that satisfy the given volume.