A conical tank has a circular base with radius 5 ft and height 12 ft. If water is flowing out of the tank at a rate of 3 ft^3/min, how fast is the height of the water changing when the height is 7 ft?

Question

A conical tank has a circular base with radius 5 ft and height 12 ft. If water is flowing out of the tank at a rate of 3 ft^3/min, how fast is the height of the water changing when the height is 7 ft?

1 Answer

Impact of this question

2047 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-03-08T01:15:09

$0.112253$ $0.112253$ ft /sec. [to $6$ $6$ decimal places]

Explanation:

We are given that $d V$ $dV$ /dt $= 3$ $=3$ ........... $[1]$ $[1]$ [ the rate of change of volume with respect to time, t.

Assuming the cone is uniform with straight sides, then although $r and h$ $r and h$ are variables their ratio will remain constant,i.e.

$\frac{r}{5} = \frac{h}{12}$ $r/5=h/12$ , so $r = \frac{5 h}{12}$ $r=[5h]/12$ ............. $[2]$ $[2]$

Volume of such a cone is $V = \frac{1}{3} π r^{2} h$ $V=1/3pir^2h$ ......... $[3]$ $[3]$ , Substituting $r$ $r$ from ...... $[2],$ $[2],$ in......... $[3]$ $[3]$ we obtain,

Volume of cone $= \frac{π}{3} {[\frac{5 h}{12}]}^{2} h,$ $=pi/3[[5h]/12]^2h,$ = $\frac{25 π h^{3}}{3 [144]}$ $[25pih^3]/[3[144]]$ ....... $[4]$ $[4]$

differentiating $[4]$ $[4]$ with respect to time $[t]$ $[t]$ implicitly,

$d \frac{V}{d t} = \frac{25 π h^{3}}{3 [144]}$ $dV/dt=[25pih^3]/[3[144]$ $[d \frac{h}{d t}] [h^{3}]$ $[dh/dt][h^3]$ = $[\frac{75 π h^{2}}{3 [144]}] d \frac{h}{d t}$ $[[75pih^2]/[3[144]]]dh/dt$ , since $\frac{d}{d t} [h^{3}] = 3 h^{2}$ $d/dt[h^3]=3h^2$ .

We know $d \frac{V}{d t} = 3$ $dV/dt=3$ , So $3 = \frac{75 π h^{2}}{3 [144]}$ $3=[75pih^2]/[3[144]$ $d \frac{h}{d t}$ $dh/dt$ , and when $h = 7$ $h=7$

$3 = [[75] 49 \frac{π}{3 [144]} d \frac{h}{d t}$ $3=[[75]49pi/[3[144]]dh/dt$ and so $d \frac{h}{d t} = 9 \frac{144}{[75] [49 π]}$ $dh/dt=9[144]/[[75][49pi]$

$d \frac{h}{d t} = 0.112253$ $dh/dt=0.112253$ ft /sec#.

Science

Math

Humanities

... and beyond

A conical tank has a circular base with radius 5 ft and height 12 ft. If water is flowing out of the tank at a rate of 3 ft^3/min, how fast is the height of the water changing when the height is 7 ft?

1 Answer

Explanation:

Related questions

Impact of this question