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3 years ago

HELP ASAP!!! DUE SOON!!! Linear,exponential, or neither?

Mathematics

1 answer:

malfutka [58]3 years ago

3 0

Answer:

It is therefore a exponential function, as y increases with x.

Step-by-step explanation:

coordinates: ( 7 , 5 ) , ( 16 , 10 )

Slope: $\frac{y2-y1}{x2-x1}$

: $\frac{10-5}{16-7}$

: $\frac{5}{9}$

equation: y - y1 = m( x - x1 )

: $y - 5 = \frac{5}{9}(x - 7)$

: $y = \frac{5}{9}x+\frac{10}{9}$

check if third coordinates matches:

$y = \frac{5}{9}(25)+\frac{10}{9}$

$y = 15$ ......does not match, so its not linear.

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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

All trapezoids are parallelograms. A. True B. False

ale4655 [162]

Answer:

True

parallelogram is a quadrilateral with two pairs of parallel sides. A trapezoid has one pair of parallel sides .

6 0

3 years ago

which of the following plotted points on the graph represent the zeros of the function g(x) = (x^2 - 3x - 10)(x + 4)

Natalka [10]

Answer: The answer would be (-2,0),(-4,0), and (5,0)

Step-by-step explanation:

Guessing the answer choices are

(-2,0)

(4,0)

(0,10)

(-4,0)

(0,-2)

(0,-10)

(5,0)

3 0

2 years ago

What is the soulution set of the equation 3x^2=48

valentina_108 [34]

Answer:

x = ± 4

Step-by-step explanation:

given 3x² = 48 ( divide both sides by 3 )

x² = 16 ( take the square root of both sides )

x = ± $\sqrt{16}$ = ± 4

x ∈ {- 4, 4 }

8 0

3 years ago

Read 2 more answers

24 m de panza costa 168 de lei.Cat costa 56 m de panza de acelasi fel?Va rogg dau coroana!!!

masya89 [10]

Answer:

392 lei găsiți rata unitară de 24 și 168 împărțind 168 la 24 pentru a obține 7, apoi înmulțiți 56 cu 7 pentru a obține răspunsul dvs., care este 392

Step-by-step explanation:

Translated- Question: 24 m of canvas costs 168 lei. How much does 56 m of canvas of the same kind cost? Please give me the crown!!!

Answer: 392 lei you find the unit rate of 24 and 168 by dividing 168 by 24 to get 7 then you multiply 56 by 7 to get your answer which is 392.

Sper că acest lucru vă ajută

O zi bună!

Hope this helps

Have a great day!

4 0

3 years ago