Cups A and B are cone shaped and have heights of #16 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?

Question

Cups A and B are cone shaped and have heights of #16 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?

Theodore Abad · Accepted Answer

#11.95# cm high of cup A will be filled.

Height and radius of cup A is #h_a=16# cm , #r_a=6# cm Volume of cup is #1/3*pi*r^2*h :.V_a=1/3*pi*6^2*16# or #V_a ~~ 603.19# cubic cm. Height and radius of cup B is #h_b=15# cm , #r_b=4# cm # :.V_b=1/3*pi*4^2*15 or V_b ~~ 251.33# cubic cm. Since #V_a > V_b # , the content will not overflow. The ratio of radius and hight of cup A is #r/h=12/32 =3/8 #. The ratio of radius and hight of cup A filled with water is #r_w/h_w=6/16=3/8 or r_w=(3*h_w)/8# The volume of water cone is #V_w=251.33# cubic cm. #:.1/3*pi*r_w^2*h_w =251.33# or # 1/3*pi*((3*h_w)/8)^2*h_w =251.33# or #h_w^3= (251.33*64)/(3*pi)~~1706.684 # or #h_w= root(3)1706.684 ~~ 11.95(2dp)# cm #11.95# cm high of cup A will be filled [Ans]

Allison Allen · Answer

undefined 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. Given that cup A has a height of 16 cm and an opening radius of 6 cm, and cup B has a height of 15 cm and an opening radius of 4 cm, we can calculate their volumes using the formula for the volume of a cone:

$$ V = \frac{1}{3}πr^2h $$

For cup A:
$$ V_A = \frac{1}{3}π(6^2)(16) = 192π $$

For cup B:
$$ V_B = \frac{1}{3}π(4^2)(15) = 80π $$

Since the volume of cup B (80π) is less than the volume of cup A (192π), pouring the contents of cup B into cup A will not cause cup A to overflow.

To find out how high cup A will be filled after pouring the contents of cup B into it, we need to calculate the volume of the liquid transferred from cup B to cup A. This volume is the difference between the volume of cup A and the volume of cup B:

$$ V_{\text{transferred}} = V_A - V_B = 192π - 80π = 112π $$

Now, we can calculate the height to which cup A will be filled using the volume of the transferred liquid and the formula for the volume of a cone:

$$ V_{\text{transferred}} = \frac{1}{3}πr^2h_{\text{filled}} $$

Substituting the known values:
$$ 112π = \frac{1}{3}π(6^2)h_{\text{filled}} $$
$$ 112 = \frac{1}{3}(36)h_{\text{filled}} $$
$$ h_{\text{filled}} = \frac{112 \times 3}{36} = \frac{336}{36} $$
$$ h_{\text{filled}} ≈ 9.33 \text{ cm} $$

Therefore, after pouring the contents of cup B into cup A, cup A will be filled to a height of approximately $ 9.33 \text{ cm} $.