I know when you decrease the height of a cone by half that the volume decreases by half from plugging in the numbers in the formula. However, I don't how to explain in words why a decrease in height by half results in a decrease in the volume by half.
Thanks..
The left part equals to the right part from the formula. If you increase or decrease left(right) part the same happens to the other part.
original area: a = 1/3 pi r^2 h
new area, using half the height is
1/3 pi r^2 (h/2) = (1/3 pi r^2 h)/2 = a/2
To explain why a decrease in the height of a cone by half results in a decrease in the volume by half, we need to understand the relationship between the height and volume of a cone.
The volume of a cone is given by the formula V = (1/3)πr^2h, where V is the volume, π is a constant (approximately 3.14159), r is the radius of the base, and h is the height.
Let's consider two identical cones, one with a height h and another with a height h/2. Both cones have the same radius, as it was not specified that it changes.
When we reduce the height of a cone to h/2, we are effectively reducing the vertical distance from the apex to the base by half. Since volume is a measure of how much space an object occupies, it intuitively makes sense that decreasing the height of the cone would result in a decrease in volume.
Now, let's put this into the formula. If we substitute h/2 for h in the volume formula, we get V' = (1/3)πr^2(h/2).
To simplify, we can factor out the (1/3)πr^2 from the equation to get V' = (1/3)πr^2 * (h/2).
We can further simplify the equation by canceling out the common factors. (1/3) * (1/2) is equal to 1/6. So, V' = (1/6)πr^2h.
Comparing the original volume formula V = (1/3)πr^2h with the new formula V' = (1/6)πr^2h, we can see that V' is half of V.
Therefore, decreasing the height of a cone by half results in a decrease in the volume by half because the volume is directly proportional to the height in a cone, as expressed by the formula.