Cups A and B are cone shaped and have heights of #32 cm# and #14 cm# and openings with radii of #15 cm# and #12 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 #32 cm# and #14 cm# and openings with radii of #15 cm# and #12 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?

Sadie Adams · Accepted Answer

height = #20.93cm#

Volume cone = #(pi r^2 h)/3#

#V_A = (pi xxcolor(red)(15^2xx32))/3 and V_B = (pixxcolor(blue)(12^2 xx14))/3#

We only need to know which one is bigger, not the actual volumes. We can therefore ignore #pi and 3# as they are common to both.

By inspection we can see that :#color(red)(V_A) >color(blue)(V_B)#

So A will not overflow, but how high will the water reach?

The water in A and the whole cup A are similar figures. Therefore we can compare the ratio of the cubes of the heights with the ratio of their volumes.

#(h/H)^3 =V_B/V_A#

#(h/H)^3 = h^3/H^3 = h^3/32^3 = (color(blue)(12^2xx14))/(color(red)(15^2xx32)) = V_B/V_A#

#h^3 = (32^3xx12^2xx14)/(15^2xx32) = 9,1750.4#

#h = root3(9,1750.4) = 20.93cm#

Mason Adams · Answer

Cup A's height will be 7.68cm

Volume of cone#=1/3pir^2h# A#=1/3 pi^2*15^2*32# #=1/3*3.141592654*225*32# #=1/3*22619.467# #=22619.467/3# Vol. A#=7539.822cm^3# Vol.B#=1/3pi12^2*12# #=1/3*3.41592654*144*12# #=1/3*5428.672105# #=5428.672105/3# #Vol.B=1809.557cm^3# New vol cup A=#1/3*pi15^2*x=1809.557cm^3# #235.619x=1809.557# #x=1809.557/235.619# #x=7.68cm#

Aurora Ayala · Answer

a. not overflow
b. height of cup A = 8.96cm

#Volume of con = 1/3*pi*r^2*h#. where r = radius and h = height #Va=1/3*pi*15^2*32# #Va = 2400pi# #Vb=1/3*pi*12^2*14# #Vb=672pi# so, cup A wont overflow because it is bigger than cup B. let say T is the height of cup A when content in cup B poured to cup A. Va = Vb #cancel(1/3)*cancelpi*15^2*T = cancel(1/3)*cancelpi*12^2*14 =(672*pi)# #15^2*T = 12^2*14# #T= (12^2*14)/(15^2)# #T = 8.96 cm#

Caroline Aucoin · Answer

undefined No, cup A will not overflow when the contents of cup B are poured into it. Cup A will be filled to a height of 14.2 cm.