Cups A and B are cone shaped and have heights of #24 cm# and #26 cm# and openings with radii of #10 cm# and #9 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 #24 cm# and #26 cm# and openings with radii of #10 cm# and #9 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?

Aaliyah Anderson · Accepted Answer

no overflow

Formula for volume of a cone #V=1/3*pi*r^2*h# volume of 1st cone #V_A=1/3*pi*10^2*24# volume of 2nd cone #V_B=1/3*pi*9^2*26# volume of 2nd cone #V_A/V_B=(1/3*pi*10^2*24)/(1/3*pi*9^2*26)=400/351>1# #:.V_A>V_B# So the cup A will not over flow when contents of full of B is poured in A

Wyatt Anderson · Answer

undefined 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 of the cone.

For cup A:
Radius $ r_A = 10 $ cm
Height $ h_A = 24 $ cm

For cup B:
Radius $ r_B = 9 $ cm
Height $ h_B = 26 $ cm

Now, we calculate the volumes of both cups:

Volume of cup A:
$$ V_A = \frac{1}{3}\pi \times (10^2) \times 24 $$

Volume of cup B:
$$ V_B = \frac{1}{3}\pi \times (9^2) \times 26 $$

Next, we compare the volumes. If $ V_B $ is less than or equal to $ V_A $, then pouring the contents of cup B into cup A will not cause overflow.

If $ V_B > V_A $, then cup A will overflow. However, if $ V_B $ is only slightly greater than $ V_A $, cup A may not overflow completely.

Once we determine whether cup A will overflow, we can calculate how high it will be filled by comparing the volumes. If cup A does not overflow, the remaining space in cup A after pouring in the contents of cup B will determine how high cup A is filled.