A sphere is dilated by a scale factor of 4 to create a new sphere. (That means the new shape will be 4 times bigger than the original shape. )
The volume of the new sphere compares with the volume of the original sphere: original sphere volume x = new sphere volume.
Word Bank:
1 4^3 4 4^2 8
16 64
Answer: 4^3
Well, well, well, it seems we have a dilated sphere on our hands! Let's do some math, shall we?
Now, the scale factor is 4, which means the new sphere will be 4 times bigger than the original one. So, we need to find the relationship between the volumes.
The formula for the volume of a sphere is V = (4/3)πr^3. Since the radius is multiplied by 4, we can say that the new sphere's volume is (4/3)π(4r)^3. Simplifying that, we get (4/3)π(64r^3).
Now, we can compare the volumes of the new and original spheres. Let's say the original sphere's volume is x. So, our equation becomes:
x = (4/3)π(64r^3)
With some rearranging, we can find that x = (4/3)(π)(4^3)r^3. So, using the word bank, the answer would be:
original sphere volume x = 4^3
Voila! We've solved the dilated sphere mystery!
To find the scale factor for the volume of a sphere, we need to consider that the scale factor for the volume of a three-dimensional object is the cube of the scale factor for its linear dimensions.
In this case, the scale factor is given as 4, which means the new sphere is 4 times bigger than the original sphere.
To find the scale factor for the volume, we need to raise the scale factor (4) to the power of 3 (since it is a three-dimensional object).
So, the scale factor for the volume is 4^3 = 64.
Therefore, the equation for the volume comparison is:
original sphere volume x 64 = new sphere volume.
To compare the volumes of the original and new spheres, we need to understand the relationship between the scale factor and the volume of a sphere.
The volume of a sphere is given by the formula V = (4/3)πr^3, where V is the volume and r is the radius of the sphere.
In this case, the sphere is being dilated by a scale factor of 4. This means that all lengths (including the radius) of the original sphere are multiplied by 4 to create the new sphere.
Since the scale factor is applied to the radius, the new radius will be 4 times the original radius. Therefore, if the original radius is denoted as r, the new radius will be 4r.
To find the new volume of the sphere, we substitute the new radius (4r) into the volume formula:
New sphere volume = (4/3)π(4r)^3 = (4/3)π(64r^3) = (256/3)πr^3
Comparing this to the original sphere volume (V), we can set up the equation:
Original sphere volume x = new sphere volume
V x = (256/3)πr^3
Therefore, the correct answer from the word bank is "256/3".