Cups A and B are cone shaped and have heights of #35 cm# and #21 cm# and openings with radii of #12 cm# and #18 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?

Find the volume of each cup using the formula #V=πr^2h/ 3#.

If Cup A is filled with water to the brim and carried to Cup B, then Cup B will not overflow, but the shape that the water will create will be a similar shape to the cup because the frustum of the cone at any point is similar to the cone itself. So find the volume scale factor by dividing the volume of Cup B by Cup A. To find the linear scale factor just cube root it. The divided the height of Cup B by the linear scale factor to find out how high the water goes.

To determine whether cup A will overflow when the contents of cup B are poured into it, we need to compare the volumes of the two cups.

The volume of a cone can be calculated using the formula: ( V = \frac{1}{3} \pi r^2 h ), where ( r ) is the radius of the base and ( h ) is the height.

For cup A:

• Radius (( r_A )) = 12 cm
• Height (( h_A )) = 35 cm

For cup B:

• Radius (( r_B )) = 18 cm
• Height (( h_B )) = 21 cm

Calculate the volumes of cup A and cup B using the given dimensions.

Next, pour the contents of cup B into cup A. Add the volume of cup B to the current volume of cup A.

If the total volume of cup A after pouring is less than its maximum volume (before overflow), then cup A will not overflow. Otherwise, it will overflow.

If cup A does not overflow, calculate the height to which cup A is filled by comparing the total volume with the maximum volume of cup A and using the formula for the volume of a cone to solve for the height.