To plot points that show the volume of a sphere for different values of the radius, we can use the formula for the volume of a sphere:
V = (4/3)πr^3
For different values of the radius, we can substitute those values into the formula and calculate the corresponding volume.
A. Plotting points for r = 1, 2, 3, and 4:
When r = 1, V = (4/3)π(1)^3 = (4/3)π ≈ 4.19
When r = 2, V = (4/3)π(2)^3 = (4/3)π(8) ≈ 33.51
When r = 3, V = (4/3)π(3)^3 = (4/3)π(27) ≈ 113.1
When r = 4, V = (4/3)π(4)^3 = (4/3)π(64) ≈ 268.1
We can plot these points on a graph with radius on the x-axis and volume on the y-axis:
B. Reasoning:
As we can see from the formula for the volume of a sphere, the volume increases rapidly as the radius increases. In fact, the volume increases with the cube of the radius, which means that a small increase in radius can result in a significant increase in volume. This is why we see such a rapid increase in volume as we increase the radius from 1 to 4.
The graph of the volume of a sphere versus its radius is a curve that starts at zero when the radius is zero (i.e., a point) and increases rapidly as the radius increases. It is a smooth, continuous curve that does not have any sharp turns or corners. This is because the formula for the volume of a sphere is a smooth, continuous function of the radius, with no sudden jumps or changes.