Cups A and B are cone shaped and have heights of #35 cm# and #21 cm# and openings with radii of #7 cm# and #11 cm#, respectively. If cup B is full and its contents are poured into cup A, will cup A overflow? If not how high will cup A be filled?

To find the quantity inside a cup, we need to find the volume.

We can observe that volume of cup A is smaller than that of cup B.

So, cup A will overflow when contents of cup B are poured into cup A.

No, cup A will not overflow. To find out how high cup A will be filled, we need to calculate the volume of cup B and compare it to the remaining volume in cup A after filling. The volume of a cone is given by the formula ( V = \frac{1}{3}\pi r^2 h ), where ( r ) is the radius of the base and ( h ) is the height of the cone.

First, we calculate the volume of cup B, which is full. Substituting the given values, we get ( V_B = \frac{1}{3}\pi (11)^2 \times 21 ).

Next, we calculate the volume of cup A, which is initially empty, but will be partially filled after pouring the contents of cup B into it. Substituting the given values, we get ( V_A = \frac{1}{3}\pi (7)^2 \times 35 ).

Then, we subtract the volume of cup B from the volume of cup A to find out how much space is left in cup A after pouring. This gives us the volume of liquid in cup A after pouring.

( V_{\text{remaining in A}} = V_A - V_B ).

Finally, we divide the remaining volume by the base area of cup A (( \pi (7)^2 )) to find out how high cup A will be filled.

( \text{Height filled in A} = \frac{V_{\text{remaining in A}}}{\pi (7)^2} ).

Calculating these values will give us the height to which cup A will be filled.