Cups A and B are cone shaped and have heights of #16 cm# and #12 cm# and openings with radii of #6 cm# and #8 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?

Question

Cups A and B are cone shaped and have heights of #16 cm# and #12 cm# and openings with radii of #6 cm# and #8 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?

Caroline Aucoin · Accepted Answer

Calculate and compare the volumes.

#V = 1/3 * pi * r^2 * h# Calculate the two volumes. If B is greater than A, it will overflow. If it is less, use the volume calculated to find the equivalent height of A.
#V_A = 1/3 * pi * r_A^2 * h_a#
#V_B = 1/3 * pi * r_B^2 * h_b#

#V_A = 1/3 * pi * 6^2 * 16# ; #V_A = 603.2#
#V_B = 1/3 * pi * 8^2 * 12# ; #V_B = 804.2#

Cup B has #1/3# more volume, so Cup A will overflow.

Evelyn Arboleda · Answer

undefined No, cup A will not overflow when the contents of cup B are poured into it. To determine how high cup A will be filled, we need to calculate the volume of cup B and see if it fits within the volume of cup A.

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.

For cup A:
Radius ($r$) = 6 cm
Height ($h$) = 16 cm

For cup B:
Radius ($r$) = 8 cm
Height ($h$) = 12 cm

First, calculate the volume of cup B. Then, determine if this volume can fit within the volume of cup A. If it does, then cup A will not overflow.

$$ V_B = \frac{1}{3}\pi \times (8 \text{ cm})^2 \times 12 \text{ cm} $$

$$ V_B = \frac{1}{3}\pi \times 64 \text{ cm}^2 \times 12 \text{ cm} $$

$$ V_B = \frac{1}{3}\pi \times 768 \text{ cm}^3 $$

Next, calculate the volume of cup A:

$$ V_A = \frac{1}{3}\pi \times (6 \text{ cm})^2 \times 16 \text{ cm} $$

$$ V_A = \frac{1}{3}\pi \times 36 \text{ cm}^2 \times 16 \text{ cm} $$

$$ V_A = \frac{1}{3}\pi \times 576 \text{ cm}^3 $$

Since $ V_B = \frac{1}{3}\pi \times 768 \text{ cm}^3 > \frac{1}{3}\pi \times 576 \text{ cm}^3 = V_A$, cup B's contents can fit entirely within cup A. Hence, cup A will not overflow. To find out how high cup A will be filled, you can use the formula for the volume of a cone and solve for height using the known volume.