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3 years ago

THIS IS THE T H I R D TIME IVE POSTED THIS P L E A S E ANSWER

Mathematics

1 answer:

Goshia [24]3 years ago

4 0

Find common denominators. 10 is the common denominator. Multiply 2 to numerator and denominator of -3/5

(-3/5)(2/2) = -6/10

x + 1/10 = -6/10

Isolate the x. Subtract 1/10 from both sides

x + 1/10 (-1/10) = -6/10 (-1/10)

x = -6/10 - 1/10

x = -7/10

-7/10 is your answer for x

hope this helps

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A tank with a capacity of 500 gal originally contains 200 gal of water with 100 lb of salt in the solution. Water containing1 lb

devlian [24]

Answer:

(a) The amount of salt in the tank at any time prior to the instant when the solution begins to overflow is $\left(1-\frac{4000000}{\left(200+t\right)^3}\right)\left(200+t\right)$ .

(b) The concentration (in lbs per gallon) when it is at the point of overflowing is $\frac{121}{125}\:\frac{lb}{gal}$ .

(c) The theoretical limiting concentration if the tank has infinite capacity is $1\:\frac{lb}{gal}$ .

Step-by-step explanation:

This is a mixing problem. In these problems we will start with a substance that is dissolved in a liquid. Liquid will be entering and leaving a holding tank. The liquid entering the tank may or may not contain more of the substance dissolved in it. Liquid leaving the tank will of course contain the substance dissolved in it. If Q(t) gives the amount of the substance dissolved in the liquid in the tank at any time t we want to develop a differential equation that, when solved, will give us an expression for Q(t).

The main equation that we’ll be using to model this situation is:

Rate of change of Q(t) = Rate at which Q(t) enters the tank – Rate at which Q(t) exits the tank

where,

Rate at which Q(t) enters the tank = (flow rate of liquid entering) x (concentration of substance in liquid entering)

Rate at which Q(t) exits the tank = (flow rate of liquid exiting) x (concentration of substance in liquid exiting)

Let $C$ be the concentration of salt water solution in the tank (in $\frac{lb}{gal}$ ) and $t$ the time (in minutes).

Since the solution being pumped in has concentration $1 \:\frac{lb}{gal}$ and it is being pumped in at a rate of $3 \:\frac{gal}{min}$ , this tells us that the rate of the salt entering the tank is

$1 \:\frac{lb}{gal} \cdot 3 \:\frac{gal}{min}=3\:\frac{lb}{min}$

But this describes the amount of salt entering the system. We need the concentration. To get this, we need to divide the amount of salt entering the tank by the volume of water already in the tank.

$V(t)$ is the volume of brine in the tank at time t. To find it we know that at t = 0 there were 200 gallons, 3 gallons are added and 2 are drained, and the net increase is 1 gallons per second. So,

$V(t)=200+t$

Therefore,

The rate at which $C(t)$ enters the tank is

$\frac{3}{200+t}$

The rate of the amount of salt leaving the tank is

$C\:\frac{lb}{gal} \cdot 2 \:\frac{gal}{min}+C\:\frac{lb}{gal} \cdot 1\:\frac{gal}{min}=3C\:\frac{lb}{min}$

and the rate at which $C(t)$ exits the tank is

$\frac{3C}{200+t}$

Plugging this information in the main equation, our differential equation model is:

$\frac{dC}{dt} =\frac{3}{200+t}-\frac{3C}{200+t}$

Since we are told that the tank starts out with 200 gal of solution, containing 100 lb of salt, the initial concentration is

$\frac{100 \:lb}{200 \:gal} =0.5\frac{\:lb}{\:gal}$

Next, we solve the initial value problem

$\frac{dC}{dt} =\frac{3-3C}{200+t}, \quad C(0)=\frac{1}{2}$

$\frac{dC}{dt} =\frac{3-3C}{200+t}\\\\\frac{dC}{3-3C} =\frac{dt}{200+t} \\\\\int \frac{dC}{3-3C} =\int\frac{dt}{200+t} \\\\-\frac{1}{3}\ln \left|3-3C\right|=\ln \left|200+t\right|+D\\\\$

We solve for C(t)

$C(t)=1+D(200+t)^{-3}$

D is the constant of integration, to find it we use the initial condition $C(0)=\frac{1}{2}$

$C(0)=1+D(200+0)^{-3}\\\frac{1}{2} =1+D(200+0)^{-3}\\D=-4000000$

So the concentration of the solution in the tank at any time t (before the tank overflows) is

$C(t)=1-4000000(200+t)^{-3}$

(a) The amount of salt in the tank at any time prior to the instant when the solution begins to overflow is just the concentration of the solution times its volume

$(1-4000000(200+t)^{-3})(200+t)\\\left(1-\frac{4000000}{\left(200+t\right)^3}\right)\left(200+t\right)$

(b) Since the tank can hold 500 gallons, it will begin to overflow when the volume is exactly 500 gal. We noticed before that the volume of the solution at time t is $V(t)=200+t$ . Solving the equation

$200+t=500\\t=300$

tells us that the tank will begin to overflow at 300 minutes. Thus the concentration at that time is

$C(300)=1-4000000(200+300)^{-3}\\\\C(300)= \frac{121}{125}\:\frac{lb}{gal}$

(c) If the tank had infinite capacity the concentration would then converge to,

$\lim_{t \to \infty} C(t)= \lim_{t \to \infty} 1-4000000\left(200+t\right)^{-3}\\\\\lim _{t\to \infty \:}\left(1\right)-\lim _{t\to \infty \:}\left(4000000\left(200+t\right)^{-3}\right)\\\\1-0\\\\1$

The theoretical limiting concentration if the tank has infinite capacity is $1\:\frac{lb}{gal}$

4 0

3 years ago

V 1. (-6) -(-2) is equal to: (-8 (+4 (-4) None of these

Marina CMI [18]

Answer:

I think the answer is -4

3 0

3 years ago

Read 2 more answers

2 times -1/4 times 3

NemiM [27]

Answer:

-1.5

Step-by-step explanation:

2 times -1/4 is -0.5. -0.5 times 3 is -1.5

5 0

3 years ago

The perimeter of a rectangular baking sheet is 58 inches and its area is 201.25 in.2. What are the length and width of the bakin

anzhelika [568]

Answer:

$Length=17.5\ in\\\\Width=11.5\ in$

Step-by-step explanation:

The formula for calculate the Area of a rectangle is:

$A=lw$

Where "l" is the lenght and "w" is the width.

And the formula for calculate the peimeter of a rectangle is:

$P=2l+2w$

Where "l" is the lenght and "w" is the width.

We know that the perimeter of the rectangular baking sheet is 58 inches and its area is 201.25 in². Then:

$A=201.25\ in^2\\\\P=58\ in$

The steps are:

1. Solve for the "l" from the formula $A=lw$ :

$A=lw\\\\l=\frac{A}{w}\\\\\l=\frac{201.25}{w}$

2. Substitute $l=\frac{201.25}{w}$ into the formula $P=2l+2w$ and solve for "w":

$58=2(\frac{201.25}{w})+2w\\\\58=\frac{402.5}{w}+2w\\\\58=\frac{402.5+2w^2}{w}\\\\58w=402.5+2w^2\\\\2w^2-58w+402.5=0$

Applying the Quadratic formula $x=\frac{-b\±\sqrt{b^2-4ac}}{2a}$ , we get:

$x=w=\frac{-(-58)\±\sqrt{(-58)^2-4(2)(402.5)}}{2(2)}\\\\w_1=17.5\\\\w_2=11.5$

3. Substitute $w_1=17.5$ into $l=\frac{201.25}{w}$ :

$l=\frac{201.25}{17.5}=11.5$

4. Substitute $w_2=11.5$ into $l=\frac{201.25}{w}$ :

$l=\frac{201.25}{11.5}=17.5$

Therefore, since the value of the lenght of a rectangle must be greater that the value of the width, we can conclude that the lenght and the width of the rectangular baking sheet are:

$l=17.5\ in\\\\w=11.5\ in$

8 0

3 years ago

What is the lcm of 5,10

kap26 [50]

Answer:

Least common multiple (LCM) of 5 and 10 is 10.

6 0

3 years ago

Read 2 more answers