A large tank is filled to capacity with 400 gallons of pure water. Brine containing 3 pounds of salt per gallon is pumped into the tank at a rate of 4 gal/min. The well-mixed solution is pumped out at a rate of 8 gals/min. Find the number A(t) of pounds of salt in the tank at time t.
Answer:
The number A(t) of pounds of salt in the tank at time t = [tex]\frac{3t(100 - t)}{25}[/tex]
Step-by-step explanation:
Given - A large tank is filled to capacity with 400 gallons of pure water. Brine containing 3 pounds of salt per gallon is pumped into the tank at a rate of 4 gal/min. The well-mixed solution is pumped out at a rate of 8 gals/min.
To find - Find the number A(t) of pounds of salt in the tank at time t.
Proof -
Given that,
Capacity of water = 400 gallons
rate in flow = 4 gal/min
rate out flow = 8 gal/min
Concentration in = 3 lbs/gal
Concentration out = (Amount of salt at time t )/ (Solution in tank at time t)
Now,
Let
A(t) = Amount of salt in the tank at time t
Now,
Initially tank has 400 gallons of water, So
A(0) = 0 ................(1)
Now,
Rate of change in the amount of salt = [tex]\frac{d}{dt} A(t)[/tex]
= (Rate in flow )( Concentration In) - (Rate out flow )( Concentration out)
= (4) (3) - (8) ([tex]\frac{A(t)}{400 - 4t)}[/tex]
= 12 - [tex]\frac{2A(t)}{100 - t)}[/tex]
⇒[tex]\frac{d}{dt} A(t)[/tex] + [tex]\frac{2A(t)}{100 - t)}[/tex] = 12
Now,
Integrating Factor, I.F = [tex]e^{\int {\frac{2}{100 - t} } \, dt }[/tex]
= [tex]e^{-2ln(100 - t)}[/tex]
= (100 - t)⁻²
⇒I.F = (100 - t)⁻²
The solution becomes
A (I.F) = ∫ (I.F)(12) dt + C
⇒A (100 - t)⁻² = ∫12(100 - t)⁻² dt + C
⇒A (100 - t)⁻² = -12(100 - t)⁻¹ (-1) + C
⇒A(t) = [tex]12(100 - t)+ C(100 - t)^{2}[/tex]
Now,
We have A(0) = 0
⇒C = -12/100
∴ we get
A(t) = 12(100 - t) - [tex]\frac{12}{100}[/tex] (100 - t)²
= 12(100 - t) [ 1 - [tex]\frac{100 - t}{100}[/tex] ]
= 12(100 - t) ([tex]\frac{t}{100}[/tex])
= [tex]\frac{3t(100 - t)}{25}[/tex]
⇒A(t) = [tex]\frac{3t(100 - t)}{25}[/tex]
∴ we get
The number A(t) of pounds of salt in the tank at time t = [tex]\frac{3t(100 - t)}{25}[/tex]
Divide round your answer to the nearest hundredth 5608.67÷7
Answer:
5608.67 ÷ 7 = 801.24
Steve watched television for 3/4 hour on Monday and 5/6 hour on Tuesday. How much longer did he watch television on Tuesday than on Monday?
_______ hour
Please do not send me a link to the answers I will have u reported but please answer this in feet
Answer:
24.5
Step-by-step explanation:
a = 3 and c = 12 c/a =
Answer:
4, because 12/3 is 4 cause 3×4=12
Answer: 4
Step-by-step explanation: you need to divide 12 and 3 to get 4.
"Death Valley Hike"
A hiker in Death Valley begins the climb out of the valley from 150 feet below sea level. 5
minutes later, the hiker is 87.5 feet below sea level.
a) Write an equation that models this situation.
b) What is the y-intercept and what does it represent?
c) What is the x-intercept and what does it represent?
d) What is the slope and what does it represent?
e) What is the hiker's distance below sea level after 10 minutes?
Answer:
I'm pretty sure E
Step-by-step explanation:
think about it he traveled 70 feet in 5 minutes so he would still be below sea level
Write a equation that can be used to find y, the number of gallons water used for showering for x minuets
Answer:
y = 6x
Step-by-step explanation:
To write an equation, find the unit rate or constant of proportionality (k).
Using a point on the line, (3, 18):
Constant of proportionality, k = y/x = 18/3
k = 6
To write an equation, substitute k = 6 into y = kx.
Thus:
y = 6x
What is the solution of the equation 4 1/5 + x = 4 3/5
[tex]4 \frac{1}{5} + x = 4 \frac{3}{5} \\ \\ x = 4 \frac{3}{5} - 4 \frac{1}{2} \\ \\ x = \frac{2}{5} [/tex]
What is the y-intercept of the line?
1
1/2
-2
0
Answer:its 1
Step-by-step explanation:
ooooooof i need some help; please explain
Answer:
just say i did it and that it got undone
Step-by-step explanation:
In ΔDEF, the measure of ∠F=90°, FD = 78 feet, and EF = 11 feet. Find the measure of ∠D to the nearest tenth of a degree.In ΔDEF, the measure of ∠F=90°, FD = 78 feet, and EF = 11 feet. Find the measure of ∠D to the nearest tenth of a degree.
Answer:
8 degrees
Step-by-step explanation:
Answer:
8
Step-by-step explanation:
BRAINIEST FOR CORRECT ANSWER!! :))) ——————-
Determine the possible side lengths of the third side of a triangle with known side
lengths of 5 and 8.
A) 5 < c < 8
B) 3 < c < 13
C-5 < c <-8
D) -3 < C < -13
I need help look in screenshot plzz
Answer:
45
Step-by-step explanation:
3 to the power of 2=9
and 9x5 =45
x-8=17
plssssss help me
Answer:
25
Step-by-step explanation:
17+8
Can someone help fast to find a and c
Answer: 15 10
Step-by-step explanation:
According to this diagram, what is cos 28°? 62° 17 8 28° 90° 15 O A. 17 15 O B. 8 17 O c. 15 17 O D. O E. 8 15 15 OF.
Answer:
C. [tex] Cos(28) = \frac{15}{17} [/tex]
Step-by-step explanation:
Recall: SOHCAHTOA
Thus,
[tex] Cos(\theta) = \frac{Adjacent}{Hypotenuse} [/tex]
Reference angle = 28°
Adjacent = 15
Hypotenuse = 17
Plug in the values
[tex] Cos(28) = \frac{15}{17} [/tex]
A rocket is launched from a tower. The height of the rocket, y in feet, is rel
time after launch, x in seconds, by the given equation. Using this equation
time that the rocket will hit the ground, to the nearest 100th of second.
y = -16x² + 147x + 80
9514 1404 393
Answer:
9.70 seconds
Step-by-step explanation:
The zero of the equation can be found using the quadratic formula.
ax² +bx +c = 0
x = (-b ±√(b² -4ac))/(2a)
We are only interested in the positive value of x, so this will be ...
x = (-147 -√(147² -4(-16)(80)))/(2(-16)) = (147 +√2679)/32
x ≈ 9.7028 ≈ 9.70 . . . . seconds
The rocket will hit the ground after about 9.70 seconds.
A veterinarian sampled 235 sheep from a large herd to obtain a 90% confidence interval for the mean weight of wool from the sheep. The interval was (23.7 lbs., 33.6 lbs.). If the veterinarian had used a 95% confidence interval instead, the confidence interval would be:_____.A. Wider and would have a smaller chance of missing the true mean. B. Narrower and would have a larger chance of missing the true mean. C. Narrower and would have a smaller chance of missing the true mean. D. Wider and would have a larger chance of missing the true mean. E. Wider but the chance of missing the true mean cannot be determined.
Answer:
A. Wider and would have a smaller chance of missing the true mean.
Step-by-step explanation:
Margin of error for a confidence interval:
The margin of error for a confidence interval is given by:
[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]
In which z is related to the confidence level(the higher the confidence level, the higher the value of z), [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.
95% confidence interval
Would be wider than the 90% confidence interval, due to the higher value of z.
Also, a x% confidence interval has a x% chance of having the true mean, and a (100-x)% probability of missing the true mean.
Then, a 95% confidence interval has a smaller(5%) chance of missing the true mean compared to the 90% confidence interval(10%). This means that the correct answer is given by option A.
I need help with this due today
Answer: P-22 A-30
P=28 A-28
P=12 A=9
P=18 A=18
Step-by-step explanation:
1. Side lengths are 5,6,5,6- perimeter is 22 Area is 5*6= 30
2. Side lengths are 10,4,4,2,6,2- perimeter is 28 Area can be found by finding the area of the whole region (4*10) and subtracting the missing piece (2*6) so 40-12=28
3. sides are all 3- perimeter is 3*4= 12 and area is 3*3=9
4. sides are 6,3,6,3- perimeter is 18 and area is 3*6=18
3. Write in fraction form: 15.43
(5 Points)
15 43/100
O 154 3/100
O 1543/100
O 15 43/10
Answer:
1543/100 is the answer
please answer the following questions!
Answer:
3 cm cube: 27
6 cm cube: 216
Step-by-step explanation:
First, you would need to find the volume of the 3 cm cube: 3*3*3=27
To build this cube you would need 27 1 cm cubes
For the 6 cm side cube you would need to do 6*6*6=216
So you would need 216 1 cm cubes
Hope this helps!
What is the solution to the equation 4x + 2(x − 2) = 4x + x − 12? (1 point)
Question 13 options:
1)
−8
2)
−1
3)
1
4)
8
Answer:
It's -8
Step-by-step explanation:
DISTRIBUTE :
4x + 2( x - 2 ) = 4x + x - 12
4x + 2x - 4 = 4x + x - 12
COMBINE LIKE TERMS :
4x + 2x - 4 = 4x + x - 12
6x - 4 = 4x + x - 12
COMBINE LIKE TERMS :
6x - 4 = 4x + x - 12
6x - 4 = 5x - 12
ADD 4 TO BOTH SIDES :
6x - 4 = 5x - 12
6x - 4 + 4 = 5x - 12 + 4
ADD THE NUMBERS/ SIMPLIFY :
6x = 5x - 8
SUBTRACT 5x FROM BOTH SIDES :
6x = 5x - 8
6x - 5x = 5x - 8 - 5x
SIMPLIFY :
x = -8
On a youth soccer team 3 out of 12 team members have played in previous years. Based on this information, if 150 kids are in the youth soccer league, then how many could be expected to have played the year before?
how does the digit 5 in the number 357 compare with the digit 5 in number 465
Answer:
The digit 5 in 357 is in the tens place while the digit in the number 465 is in the ones place
Step-by-step explanation:
Good luck
What is the some of the finite arithmetic series?
Answer:
455
Step-by-step explanation:
This one is the one I need help with my final question
Answer:
vertex C could be at (6,0) or (2,8)
(I'm sorry if I'm wrong, but if not I hope this helps!)
Step-by-step explanation:
vertex C needs to be placed somewhere on the graph where it would form a right angle (and therefore a right-angled triangle as well). vertex C could be at (6,0) or (2,8). this is because at these two points vertex C will form a right angle, therefore making it a right-angled triangle.
what property of equality is used to isolate the term with variable in the equation 2 x + 2 = 4?
A. Addition property
B. Reflexive property
C. Multiplication property
D. Transitive property
(it will help me a lot)
Answer:
I think it is addition property
HELP WILL MARK BRAINLIEST IF RIGHT!!!
Answer:
im so sorry bt I dont know this one
Step-by-step explanation:
what is the graph of the system? y < -2
Answer: The answer is -2 it's just going to stay as -2 because y has no value.
Answer: this is the graph i wish that could help :)
Which transformation could NOT be used to prove that two circles are congruent to one another?
Answer: D
Step-by-step explanation: