transform y=(x-2)^3+1 into a volume equation based off of v(r)=4/3pier^3
To transform the equation y = (x-2)^3 + 1 into a volume equation based on the formula V(r) = (4/3)πr^3, we need to identify the appropriate variables to substitute.
In the given equation y = (x-2)^3 + 1, y represents the dependent variable (the output) and x represents the independent variable (the input). In the volume equation V(r) = (4/3)πr^3, V represents the volume, and r represents the radius.
To make these two equations compatible, we need to determine how y, x, V, and r relate to each other. Notice that y and V represent the output or volume, while x and r represent the input or radius. Therefore, to make the transformation, we need to establish a connection between x and r.
In the equation y = (x-2)^3 + 1, the term (x-2)^3 represents the cube of a quantity. To create a connection to the volume equation, we can rewrite this term as (r-2)^3, where r is the radius. By doing this, we are essentially equating x-2 to r, since (r-2)^3 = (x-2)^3.
Thus, the transformed equation becomes:
y = (r-2)^3 + 1
This equation represents the volume equation based on the formula V(r) = (4/3)πr^3.
To transform the equation y = (x - 2)^3 + 1 into a volume equation based on V(r) = (4/3)πr^3, we need to replace the variables appropriately.
In the given equation, we have y = (x - 2)^3 + 1. Let's substitute y with V and x - 2 with r.
V = (r)^3 + 1.
However, this equation does not match the volume equation V(r) = (4/3)πr^3 exactly. To modify the equation to match, we can multiply both sides by (4/3)π:
(4/3)πV = (4/3)π[(r)^3 + 1].
Now, the equation is transformed into a volume equation based on V(r) = (4/3)πr^3:
V(r) = (4/3)πr^3 + (4/3)π.
To transform the equation y=(x-2)^3+1 into a volume equation based off of V(r) = (4/3)πr^3, we need to find a way to relate the variables x and y to the radius r.
The given equation y=(x-2)^3+1 represents a cubic function shifted horizontally by 2 units to the right and vertically by 1 unit upward. We can manipulate this equation to find the relationship between x and r.
Let's start by substituting y with V in the equation:
V = (x-2)^3 + 1
Next, we need to express x in terms of r. Since we are dealing with a volume equation, we want to express x as a function of the radius r.
To find x in terms of r, we need to reverse the process of shifting the cubic equation horizontally by 2 units:
x = r + 2
Now, substitute x back into the equation:
V = (r+2-2)^3 + 1
V = r^3 + 1
Finally, rewrite the equation based on the given volume equation V(r) = (4/3)πr^3:
V = (4/3)πr^3
Therefore, the transformed equation for the given cubic equation y=(x-2)^3+1, in terms of volume V(r), is V = (4/3)πr^3.