dvWater is flowing into a dam at a rate given bydt5000(i)where V is the volumeVof water in the dam (in megalitres) at time t days.3t+4If the initial volume of water in the dam is 2.5x10 megalitres, calculate thevolume of water in the dam after 7 days.(a)The dam has a capacity of 3x10 megalitres. Calculate the time it will take forthe dam to reach capacity.(b)

Answered: dv 5000 (i) Water is flowing into a dam…

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter3: Polynomial And Rational Functions

Section3.1: Quadratic Functions And Models

Problem 56E: When a certain drug is taken orally, the concentration of the drug in the patient's bloodstream...

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dv
Water is flowing into a dam at a rate given by
dt
5000
(i)
where V is the volume
V
of water in the dam (in megalitres) at time t days.
3t+4
If the initial volume of water in the dam is 2.5x10 megalitres, calculate the
volume of water in the dam after 7 days.
(a)
The dam has a capacity of 3x10 megalitres. Calculate the time it will take for
the dam to reach capacity.
(b)

Expert Solution

Step 1

Given:

$\frac{d V}{d t} = \frac{5000}{\sqrt{3 t + 4}}$

V is volume of water in dam in megaliters, t is time in days

At t = 0, V = 2.5x10⁴megaliters

Step 2

Part (a)

$\frac{d V}{d t} = \frac{5000}{\sqrt{3 t + 4}} \Rightarrow V = \int \frac{5000}{\sqrt{3 t + 4}} d t$

Applying u substitution:

Let $u = 3 t + 4$

$\Rightarrow d u = 3 d t \Rightarrow d t = \frac{d u}{3}$

Step 3

$\Rightarrow V = \int \frac{5000}{\sqrt{3 t + 4}} d t \Rightarrow V = \int \frac{5000}{\sqrt{u}} \frac{d u}{3} \Rightarrow V = \frac{5000}{3} \int \frac{d u}{\sqrt{u}} \Rightarrow V = \frac{5000}{3} (\frac{u^{\frac{1}{2}}}{\frac{1}{2}}) + C \Rightarrow V = \frac{10000}{3} \sqrt{u} + C \Rightarrow V = \frac{10000}{3} \sqrt{3 t + 4} + C$

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