Example 3-6-5 - Maple Help

All Products Maple MapleSim

Home : Support : Online Help : Education : Study Guides : Calculus : Chapter 3 - Applications of Differentiation : Examples : Section 3-6 : Example 3-6-5

Chapter 3: Applications of Differentiation

Section 3.6: Related Rates

Example 3.6.5

A right-circular conical tank, whose cross-section through its axis is shown in Figure 3.6.4, is being filled with water at the constant rate $λ$ .

At time $\overset{&Hat;}{t}$ , find $\overset{&period;}{h} (\overset{&Hat;}{t})$ , the rate of change of the height of the water, where $\overset{&Hat;}{t}$ is the moment when the volume is $k$ times the volume of the tank, $0 < k \leq 1$ .

The dimensions of the tank and the rate of fill are all in consistent units. The height of the tank is $H$ , while the radius of the opening is $R$ . The varying radius of the circle at the level of the water is $r (t)$ (green dotted line in Figure 3.6.5(a)), and the varying height of the water is $h (t)$ .

Hint: The volume of the tank is $\frac{π}{3} R^{2} H$

p1:=plot([[0,0],[2,5],[-2,5],[0,0]],style=line,color=black):
p2:=plot([[0,0],[0,5]],style=line,linestyle=dot,color=red):
p3:=plot([[0,3.5],[7/5,3.5]],style=line,linestyle=dot,color=green):
p4:=plots:-textplot({[1,5.2,typeset(R)],[-1,5.2,typeset(R)],[-.3,4.3,typeset(H)],[.7,3.3,typeset(r(t))],[.3,2.4,typeset(h(t))]},font=[Times,12]):
p5:=plots:-textplot({[-.2,3.5,A],[1.6,3.5,B],[.2,0,O],[0,5.2,C],[2,5.2,E]},font=[Times,BoldRoman,14]):
plots:-display(p||(1..5),scaling=constrained, axes=none);

Figure 3.6.5(a) Conical tank

Solution

Mathematical Solution

•	In Figure 3.6.5(a), triangles OAB and OCE are similar, from which the proportion $\frac{r (t)}{h (t)} = \frac{R}{H}$ and the equation $r (t) = \frac{R}{H} h (t)$ follows.

•	The time-varying volume of water in the tank is then $V (t) = \frac{π}{3} {(\frac{R}{H} h (t))}^{2} h (t) = \frac{π}{3} {(\frac{R}{H})}^{2} h^{3} (t)$ .

•	The height of the water at time $\overset{&Hat;}{t}$ , that is, $h (\overset{&Hat;}{t}) = H k^{1 / 3}$ , is found from the equation

$V (\overset{&Hat;}{t}) = \frac{π}{3} {(\frac{R}{H})}^{2} h^{3} (\overset{&Hat;}{t}) = k (\frac{π}{3} R^{2} H)$

•	The rate of change of $h (t)$ is obtained by setting $\overset{&period;}{V}$ equal to $λ$ , and evaluating at $t = \overset{&Hat;}{t}$ .

$V (t)$	$= \frac{π}{3} {(\frac{R}{H})}^{2} h^{3} (t)$
$\overset{&period;}{V} (t)$	$= π {(\frac{R}{H})}^{2} h^{2} (t) \overset{&period;}{h} (t)$
$λ$	$= π {(\frac{R}{H})}^{2} {(H k^{1 / 3})}^{2} \overset{&period;}{h} (\overset{&Hat;}{t})$
$λ$	$= π R^{2} k^{2 / 3} \overset{&period;}{h} (\overset{&Hat;}{t})$

•	The result is the surprising $\overset{&period;}{h} (\overset{&Hat;}{t}) = \frac{λ}{π R^{2} k^{2 / 3}}$ , a quantity independent of the height of the tank, but not the radius.

Maple Solution

The following interactive calculations are an implementation of those in the Mathematical Solution, with the one exception that the symbol $\overset{&Hat;}{t}$ is not used. (It's use is problematic for the Context Panel.)

Solve the equation $V (t) = k V_{\max}$ for $h (\overset{&Hat;}{t})$

•	Control-drag or type the equation $V (t) = k V_{\max}$ . Press the Enter key.

•	Context Panel: Solve≻Isolate Expression for≻ $h (t)$

•	Context Panel: All Values

•	Context Panel: Select Element≻1

•	Context Panel: Simplify≻Assuming Positive

$\frac{π}{3} {(\frac{R}{H})}^{2} h^{3} (t) = k (\frac{π}{3} R^{2} H)$

$\frac{1}{3} \frac{π R^{2} {h (t)}^{3}}{H^{2}} = \frac{1}{3} k π R^{2} H$

$\overset{isolate for h(t)}{\to}$

$h (t) = RootOf (- H^{3} k + {_Z}^{3})$

$\overset{all values}{\to}$

$h (t) = {(k H^{3})}^{1 / 3}, h (t) = {(k H^{3})}^{1 / 3} {(- 1)}^{2 / 3}, h (t) = - {(k H^{3})}^{1 / 3} {(- 1)}^{1 / 3}$

$\overset{select entry 1}{\to}$

$h (t) = {(k H^{3})}^{1 / 3}$

$\overset{assuming positive}{\to}$

$h (t) = H k^{1 / 3}$

(1)

Obtain $\overset{&period;}{V} (t)$

•	Control-drag (or type) the expression for $V (t)$ . Press the Enter key.

•	Context Panel: Differentiate≻With Respect To≻ $t$

$\frac{π}{3} {(\frac{R}{H})}^{2} h^{3} (t)$

$\frac{1}{3} \frac{π R^{2} {h (t)}^{3}}{H^{2}}$

$\overset{differentiate w.r.t. t}{\to}$

$\frac{π R^{2} {h (t)}^{2} (\frac{&DifferentialD;}{&DifferentialD; t} h (t))}{H^{2}}$

(2)

Solve the equation $\overset{&period;}{V} (t) = λ$ for $\overset{&period;}{h} (\overset{&Hat;}{t})$

•	Use the equation label to write the equation $\overset{&period;}{V} (t) = λ$ . Press the Enter key.

•	Context Panel: Solve≻Isolate Expression for≻diff(h(t),t)

•	Context Panel: Right-hand Side

$= λ$

$\frac{π R^{2} {h (t)}^{2} (\frac{&DifferentialD;}{&DifferentialD; t} h (t))}{H^{2}} = λ$

$\overset{isolate for diff(h(t),t)}{\to}$

$\frac{&DifferentialD;}{&DifferentialD; t} h (t) = \frac{λ H^{2}}{π R^{2} {h (t)}^{2}}$

$\overset{right hand side}{\to}$

$\frac{λ H^{2}}{π R^{2} {h (t)}^{2}}$

(3)

In the expression for $\overset{&period;}{h} (\overset{&Hat;}{t})$ , replace $h (\overset{&Hat;}{t})$

•	Expression palette: Evaluation template Reference quantities via equation labels

•	Context Panel: Evaluate and Display Inline

$\binom{}{} | \binom{}{}$ = $\frac{λ}{π R^{2} k^{2 / 3}}$

<< Previous Example Section 3.6 Next Section >>

© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.

For more information on Maplesoft products and services, visit www.maplesoft.com

Download Help Document

Maple

Maple 附加模块

Student Success Platform

提高存留率

MapleSim

MapleSim 附加模块

系统工程

项目服务

Maple T.A. and Möbius

教育

行业

汽车与航空航天

机器人

机器设计和工业自动化

其他

应用领域

产品价格

购买

院系/全校正版授权

Maplesoft精英维护升级计划

支持

产品培训

产品在线帮助

研讨会与活动

出版

资源中心

示例和应用

社区

关于 Maplesoft

媒体中心

用户社区

联系我们

Online Help

All Products Maple MapleSim

Maple

数学软件

Maple 附加模块

Student Success Platform

提高存留率

MapleSim

多学科系统级建模仿真

MapleSim 附加模块

系统工程

项目服务

Maple T.A. and Möbius

教育

行业

汽车与航空航天

机器人

机器设计和工业自动化

其他

应用领域

产品价格

购买

院系/全校正版授权

Maplesoft精英维护升级计划

支持

产品培训

产品在线帮助

研讨会与活动

出版

资源中心

示例和应用

社区

关于 Maplesoft

媒体中心

用户社区

联系我们

Online Help

All Products Maple MapleSim