Can you guys help me with this simple math problem ?

A 500 gallon tank initially contains 200 gallons of water with 5 lbs of salt dissolved in it. Water enters the tank at a rate of

Lapatulllka [165]

Until the concerns I raised in the comments are resolved, you can still set up the differential equation that gives the amount of salt within the tank over time. Call it $A(t)$ .

Then the ODE representing the change in the amount of salt over time is

$\dfrac{\mathrm dA}{\mathrm dt}=\text{rate in}-\text{rate out}$
$\dfrac{\mathrm dA}{\mathrm dt}=\dfrac{2\text{ gal}}{1\text{ hr}}\times\dfrac{\frac15(1+\cos t)\text{ lbs}}{1\text{ gal}}-\dfrac{2\text{ gal}}{1\text{ hr}}\times\dfrac{A(t)\text{ lbs}}{500+(2-2)t}$
$\dfrac{\mathrm dA}{\mathrm dt}=\dfrac25(1+\cos t)-\dfrac1{250}A(t)$

and this with the initial condition $A(0)=5$

You have

$\dfrac{\mathrm dA}{\mathrm dt}+\dfrac1{250}A(t)=\dfrac25(1+\cos t)$
$e^{t/250}\dfrac{\mathrm dA}{\mathrm dt}+\dfrac1{250}e^{t/250}A(t)=\dfrac25e^{t/250}(1+\cos t)$
$\dfrac{\mathrm d}{\mathrm dt}\left[e^{t/250}A(t)\right]=\dfrac25e^{t/250}(1+\cos t)$

Integrating both sides gives

$e^{t/250}A(t)=100e^{t/250}\left(1+\dfrac1{62501}\cos t+\dfrac{250}{62501}\sin t\right)+C$
$A(t)=100\left(1+\dfrac1{62501}\cos t+\dfrac{250}{62501}\sin t\right)+Ce^{-t/250}$

Since $A(0)=5$ , you get

$5=100\left(1+\dfrac1{62501}\right)+C\implies C=-\dfrac{5937695}{62501}$

so the amount of salt at any given time in the tank is

$A(t)=100\left(1+\dfrac1{62501}\cos t+\dfrac{250}{62501}\sin t\right)-\dfrac{5937695}{62501}e^{-t/250}$

The tank will never overflow, since the same amount of solution flows into the tank as it does out of the tank, so with the given conditions it's not possible to answer the question.

However, you can make some observations about end behavior. As $t\to\infty$ , the exponential term vanishes and the amount of salt in the tank will oscillate between a maximum of about 100.4 lbs and a minimum of 99.6 lbs.

5 0

4 years ago

Bob won a $200,000 lottery. He invests part of the money in real estate with an annual return of 3% and the rest in a money mark

SVETLANKA909090 [29]

Answer:

Bob invested $50,000 in real state and $150,000 in a money market.

Step-by-step explanation:

Assuming that x is the investment in real state:

(200,000-x)*0.045 + x * 0.03 = 8,250

Apply distributive property:

(200,000*0,045) - 0.045x +0.03x = 8,250

9,000 - 0.045x +0.03x = 8,250

We must clear x:

- 0.045x +0.03x = 8,250 - 9,000

-0.015x = -750

x = -750/-0.015

x = 50,000

So, the invest in real state is $50,000.

$200,000 - $50,000 = $150,000 is the invest in a money market.

4 0

3 years ago

Two increased by three equals the quotient of ten and two.

Marrrta [24]

Answer: 5 = 5 or true its an equal equation.

Step-by-step explanation:

The equation would look like this: 2 + 3 = 10 divided by 2. This equation is extremely easy it turns out to be 5 = 5 because 2 + 3 = 5 and 10 divided by 2 is 5 so your final answer is 5 = 5 or if its a true or false question, the answer is true it is an equal equation. Please mark me brainliest. -Cyrusskywalker

5 0

3 years ago

The length of a rectangle is 5 centimeters less than twice its width. The perimeter of the rectangle is 80 cm. What are the dime

Sonbull [250]

Hello,

Answer A
Let's the width of the rectangle

2a-5 its length
Teh half perimeter is 80/2=40
a+2a-5=40 ==>3a=45==>a=15 and 2a-5=30-5=25

5 0

3 years ago

Read 2 more answers

Volume is always measured in square units. True False

bogdanovich [222]

Hello,

The answer is option B "false".

Reason:

The answer is false because the volume can be measure in anyway for example: feet, yards.

If you need anymore help feel free to ask me!

Hope this helps!

~Nonportrit

4 0

3 years ago

Read 2 more answers