# Cups A and B are cone shaped and have heights of #12 cm# and #15 cm# and openings with radii of #6 cm# and #4 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?

Cup A will not overflow and will fill up to a height of

As volume of Cup B is less than that of A, when cup B is full and its contents are poured into cup A, cup A will not overflow.

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Since the volume of a cone is directly proportional to the cube of its radius and inversely proportional to its height, we can compare the volumes of cups A and B to determine if cup A will overflow when the contents of cup B are poured into it.

The volume of cup A is given by (V_A = \frac{1}{3} \pi (6^2)(12) = 144\pi) cubic cm.

The volume of cup B is given by (V_B = \frac{1}{3} \pi (4^2)(15) = 80\pi) cubic cm.

When cup B is poured into cup A, the total volume in cup A will be (V_A + V_B = 144\pi + 80\pi = 224\pi) cubic cm.

This volume is less than the total volume cup A can hold. Therefore, cup A will not overflow. To find how high cup A will be filled, we need to calculate the height of the liquid in cup A. Using the formula for the volume of a cone, we can solve for the height:

[V_A = \frac{1}{3} \pi r_A^2 h_A]

Solving for (h_A):

[h_A = \frac{3V_A}{\pi r_A^2}]

Substituting the known values:

[h_A = \frac{3(224\pi)}{\pi(6^2)} = \frac{3(224)}{36} = \frac{672}{36} = 18.67] cm.

So, cup A will be filled to a height of approximately 18.67 cm.

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