The volume of a sphere is 2,240 cubic yards. what will be the new volume if the dimensions are multiplied by 1/4?

the answer is Courtney -  

Some of these problems have a similar approach, so I'll start you on some and leave others for you

1 - For a cube, we know all sides are equal, so if we let s = length of a side, then the volume V=s3. If we double each side, then the new cube would have sides of length 2s, and we cube this to get the volume:

V2 = (2s)3 = 8s3

So the volume of the new cube would be 8 times the volume of the original cube.  

2 - Similar approach as #1. We know the volume of a rectangular prism is V = l*w*h. So now, if the length and width are doubled, the volume  

V2 = (2l)*(2w)*h = 4lwh

So the volume of the new prism is 4 times the volume of the original.  

3 - Similar to #1, left for the student.  

4 - Volume of a Cone V = 1/3 ∏ r2h. If the radius is doubled and the height tripled, we plug these new values into our formula:

V2 = 1/3 ∏ (2r)2(3h) = 1/3 ∏ 4r2*3h.  

Rearranging our factors by pulling the 4 and 3 out front, we get  

V2 = (4*3)(1/3 ∏ r2h), =  12 (1/3 ∏ r2h)

So the new volume is 12 times the original volume. Since they ask for the new volume, and the original volume is 4∏, I will leave that final calculation to the student.  

5 - Volume of a cone is Vcone=1/3 ∏ r2h.   Vcylinder =  ∏ r2h. So for a cone with the same base and height of a cylinder, the cone's Volume is 1/3 the Cylinder's volume.  

6 - Use the relationship in #5 to solve this one.  

7 - Use the formula for volume of a cone for this one.  

8 - Vcylinder = ∏ r2h . Plugging numbers in, we get 100∏ = ∏ (2)2h. I'll leave the calculations to the student.  

9 - This is similar to #2. Vcylinder = ∏ r2h . If we double the radius, our new volume is  

Vcylinder = ∏ (2r)2h . The rest is left to the student.  

10 - Similar to #2, only instead of multiplying 2 sides by 2, we multiply 3 sides by 1/4:

V = (1/4)l * (1/4)w * (1/4)h.  

11 - Straight application of the formula for volume of a cone. Plug in the numbers where appropriate and solve for h.

12 - Volume of a Sphere is Vsphere = 4/3 ∏r3 . Volume of a cone is Vcone=1/3 ∏ r2h.

If we want the two volumes to be equal, let's set them to each other and solve for h:

4/3 ∏r3 = 1/3 ∏ r2h  

Multiply each side by 3 to remove denominators:  

4 ∏r3 =  ∏ r2h

Divide by pi to remove it:

4 r3 =  r2h

Now divide both sides by r2:

4r = h

So the height of the cone must be 4 times the radius for the volume of the cone and cylinder to be equal.

Step-by-step explanation:

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