Answer:
The level of the root beer is dropping at a rate of 0.08603 cm/s.
Explanation:
The volume of the cone is :
[tex]V=\frac {1}{3}\times \pi\times r^2\times h[/tex]
Where, V is the volume of the cone
r is the radius of the cone
h is the height of the cone
The ratio of the radius and the height remains constant in overall the cone.
Thus, given that, r = d / 2 = 10 / 2 cm = 5 cm
h = 13 cm
r / h = 5 / 13
r = {5 / 13} h
[tex]V=\frac {1}{3}\times \frac {22}{7}\times ({{{\frac {5}{13}\times h}}})^2\times h[/tex]
[tex]V=\frac {550}{3549}\times h^3[/tex]
Also differentiating the expression of volume w.r.t. time as:
[tex]\frac {dV}{dt}=\frac {550}{3549}\times 3\times h^2\times \frac {dh}{dt}[/tex]
Given: [tex]\frac {dV}{dt}[/tex] = -4 cm³/sec (negative sign to show leaving)
h = 10 cm
So,
[tex]-4=\frac{550}{3549}\times 3\times {10}^2\times \frac {dh}{dt}[/tex]
[tex]\frac{55000}{1183}\times \frac {dh}{dt}=-4[/tex]
[tex]\frac {dh}{dt}=-0.08603\ cm/s[/tex]
The level of the root beer is dropping at a rate of 0.08603 cm/s.
