Cups A and B are cone shaped and have heights of #27 cm# and #23 cm# and openings with radii of #7 cm# and #10 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 #27 cm# and #23 cm# and openings with radii of #7 cm# and #10 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?

Wyatt Anderson · Accepted Answer

cup A will overflow.

The first step is to calculate the volumes ( V) of both cups. #color(orange)"Reminder" color(red)(|bar(ul(color(white)(a/a)color(black)(V_("cone")=1/3pir^2h)color(white)(a/a)|)))# #rArrV_(A)=1/3xxpixx7^2xx27=1156.327/3≈1385.44" cubic cm"# #rArrV_(B)=1/3xxpixx10^2xx23=7225.663/3≈2408.55" cubic cm"# Since #V_(B)>V_(A)" then cup A will overflow"#

Theodore Abad · Answer

undefined No, cup A will not overflow. When the contents of cup B are poured into cup A, the volume of cup A will increase but will not exceed its capacity. To find out how high cup A will be filled, we need to calculate the increase in volume when the contents of cup B are poured into it. Then, we can use the formula for the volume of a cone to find the corresponding height.

The volume of a cone is given by the formula:
$$ V = \frac{1}{3} \pi r^2 h $$

We can calculate the volumes of cups A and B using their respective heights and radii. Then, we find the difference in volume when cup B is poured into cup A. Using this difference in volume, we can calculate the corresponding increase in height of cup A.