Now use the formula for volume of a sphere and complete the same 3 transformations used in Part 2A on the volume equation to build the final equation.
Vr= 43r3
To complete the same 3 transformations used in Part 2A on the volume equation Vr = 4/3r^3, we can follow these steps:
1. Multiply both sides of the equation by 3 to eliminate the fraction:
3 * Vr = 3 * (4/3r^3)
This simplifies to:
3Vr = 4r^3
2. Divide both sides of the equation by 4 to isolate r^3:
(3Vr)/4 = (4r^3)/4
Simplifying further:
(3Vr)/4 = r^3
3. Take the cube root of both sides to solve for r:
∛((3Vr)/4) = ∛(r^3)
This simplifies to:
r = ∛((3Vr)/4)
So, the final equation for the volume of a sphere becomes:
r = ∛((3V)/4)
To build the final equation, let's go through the three transformations used in Part 2A on the volume equation.
1. Scaling transformation: Multiply the equation by a constant. Let's use the scaling factor of 4/3.
Vr' = (4/3) * Vr
= (4/3) * (4/3 * πr^3)
= (16/9) * πr^3
2. Translation transformation: Add or subtract a constant. Let's subtract a constant, the equation will remain the same.
Vr'' = Vr'
= (16/9) * πr^3
3. Reflection transformation: Replace the variable with its negative.
V(-r) = (16/9) * π(-r)^3
= (16/9) * π(-r^3)
= - (16/9) * πr^3
Therefore, the final equation after applying all three transformations is:
Vr_final = - (16/9) * πr^3
To apply the same transformations used in Part 2A on the volume equation, which is Vr = 4/3 * π * r^3, we can break it down step by step. Here's how you can complete the transformations:
1. Translation Transformation:
Let's replace V with Vr to indicate that the volume depends on the radius (r). So our new equation becomes Vr = 4/3 * π * r^3.
2. Scaling Transformation:
To scale the equation, we need to multiply both sides by a scaling factor. Let's multiply both sides by 1/3 to address the 4/3 term:
Vr * (1/3) = (4/3) * π * r^3 * (1/3).
Simplifying, we have:
Vr/3 = (4π/9) * r^3.
3. Vertical Stretch/Compression Transformation:
To apply vertical stretch or compression, we multiply both sides by a scaling factor. We'll multiply the equation by 3 to address the denominator on the left side:
3 * (Vr/3) = 3 * (4π/9) * r^3.
Simplifying, we get:
Vr = (4π/3) * r^3.
So, by combining these transformations, we have the final equation for the volume of a sphere:
Vr = (4π/3) * r^3.