Cups A and B are cone shaped and have heights of #32 cm# and #15 cm# and openings with radii of #9 cm# and #3 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 determine if 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 is calculated using the formula: V = (1/3) * π * r^2 * h, where r is the radius of the base and h is the height of the cone.

First, we calculate the volumes of cups A and B:

Volume of cup A: V_A = (1/3) * π * (9 cm)^2 * 32 cm

Volume of cup B: V_B = (1/3) * π * (3 cm)^2 * 15 cm

Next, we add the volume of cup B to the volume of cup A to determine the total volume when cup B's contents are poured into cup A:

Total volume = V_A + V_B

If the total volume is less than the maximum volume that cup A can hold, then cup A will not overflow. Otherwise, cup A will overflow.

Additionally, if cup A does not overflow, we can calculate how high cup A will be filled by dividing the total volume by the base area of cup A (π * (9 cm)^2) to find the height of the liquid.

Let's calculate:

V_A = (1/3) * π * (9 cm)^2 * 32 cm V_B = (1/3) * π * (3 cm)^2 * 15 cm

V_A ≈ 2419.69 cm³ V_B ≈ 141.37 cm³

Total volume ≈ 2419.69 cm³ + 141.37 cm³ ≈ 2561.06 cm³

The maximum volume that cup A can hold is its own volume, which is ≈ 2419.69 cm³.

Since the total volume is less than the maximum volume that cup A can hold, cup A will not overflow when the contents of cup B are poured into it.

To find out how high cup A will be filled, we divide the total volume by the base area of cup A:

Height of liquid in cup A = Total volume / (π * (9 cm)^2)

Height of liquid in cup A ≈ 2561.06 cm³ / (π * 81 cm²) ≈ 10.43 cm

Therefore, cup A will be filled to a height of approximately 10.43 cm.