Answer:
m<4 = 44°
Step-by-step explanation:
Given:
m<8 = 34°
m<9 = 40°
m<3 = 62°
Required:
m<4
Solution:
m<2 = m<8 + m<9 (exterior angle theorem of a ∆)
m<2 = 34° + 40° (substitution)
m<2 = 74°
m<2 + m<3 + m<4 = 180° (sum of ∆)
74° + 62° + m<4 = 180° (substitution)
136 + m<4 = 180
Subtract 136 from each side
m<4 = 180 - 136
m<4 = 44°
A square with a perimeter of 20 units is graphed on a coordinate grid. The square is dilated by a scale factor of 0.4 with the origin as the center of dilation.
If ( x , y ) represents the location of any point on the original square, which ordered pair represents the coordinates of the corresponding point on the resulting square?
If (x, y) represents the location of any point on the original square, the ordered pair which represents the coordinates of the corresponding point on the resulting square is: (0.4x, 0.4y).
What is dilation?In Geometry, dilation simply refers to a type of transformation which changes the size of a geometric object and the location of its points based on the scale factor, but not its shape.
What is scale factor?In Geometry, a scale factor can be defined as the ratio of two corresponding side lengths or diameter in two similar geometric objects (shapes) such as equilateral triangles, square, quadrilaterals, polygons, etc., which can be used to either enlarge or reduce their size.
Generally speaking, the transformation rule for the dilation of a geometric object (square) based on a specific scale factor is given by this mathematical expression:
(x, y) → (SFx, SFy) = (0.4x, 0.4y)
Where:
x and y represents the data points.SF represents the scale factor.Read more on dilation here: brainly.com/question/20482938
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Complete Question:
A square with the perimeter of 20 units is graphed on a coordinate grid. The square is dilated by a scale factor of 0.4 with the origin as the center of dilation.
If (x, y) represents the location of any point on the original square, which ordered pair represents the coordinates of the corresponding point on the resulting square?
answer choices
(0.4x, 0.4y)
(x + 20, y + 20)
(x + 0.4, y + 0.4)
(20x, 20y)
Use integration to find the position function for the given velocity function and initial condition. (Rubric 10 marks) \[ v(t)=3 t^{3}+30 t^{2}+5 ; s(0)=3 \]
Answer:
\(\displaystyle s(t)=\frac{3}{4}t^3+10t^3+5t+3\)
Step-by-step explanation:
Integrate v(t) with respect to time
\(\displaystyle \int(3t^3+30t^2+5)\,dt\\\\=\frac{3}{4}t^4+10t^3+5t+C\)
Plug-in initial condition to get C
\(\displaystyle s(0)=\frac{3}{4}(0)^3+10(0)^3+5(0)+C\\\\3=C\)
Thus, the position function is \(\displaystyle s(t)=\frac{3}{4}t^3+10t^3+5t+3\) given the velocity function and initial condition.
im too tired for this please help me.
Answer:
-3
-2
-6
-4
-5
-1
Step-by-step explanation:
Algebra 2 Questions that i need help with
The graph has a horizontal intercept at (1, 0)
The line x = 0 (the y-axis) is a vertical asymptote; as x→0+,y→∞
The graph is decreasing if 0 < b < 1.
The domain of the function is x > 0, or (0, ∞)
The range of the function is all real numbers, or (−∞,∞)
What is function?
A function in mathematics from a set X to a set Y allocates precisely one element of Y to each element of X. The sets X and Y are collectively referred to as the function's domain and codomain, respectively.
The set of all positive real numbers serves as the function's domain. Assume that base 10 is written when no base is specified for the log. The equation y=logb(x+h)+k shifts the logarithmic function, y=logb(x), by k units vertically and h units horizontally. The graph would be moved upward if k>0.
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gelo bought a wallet which costs php250at 12% discount and a belt which costs php230 at 25% discount.how much did he pay in all?
a.php87.50
b.php392.50
c.php3925
d. php8750
Option B : Gelo bought a wallet for PHP 250 with a 12% discount and a belt for PHP 230 with a 25% discount. Gelo paid a total of PHP 392.50 for both items.
First, we need to calculate the amount of the discount for the wallet and the belt. The discount for the wallet is 250 * 12/100 = 30. The discount for the belt is 230 * 25/100 = 57.5.
Next, we subtract the discount from the original price to find the amount paid for each item. The amount paid for the wallet is 250 - 30 = 220. The amount paid for the belt is 230 - 57.5 = 172.5.
Finally, we add up the amount paid for both items to find the total amount paid: 220 + 172.5 = 392.5.
Therefore, Gelo paid a total of 392.5, so the correct answer is (b) PHP 392.50.
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Option B : Gelo bought a wallet for PHP 250 with a 12% discount and a belt for PHP 230 with a 25% discount. Gelo paid a total of PHP 392.50 for both items.
First, we need to calculate the amount of the discount for the wallet and the belt. The discount for the wallet is 250 * 12/100 = 30. The discount for the belt is 230 * 25/100 = 57.5.
Next, we subtract the discount from the original price to find the amount paid for each item. The amount paid for the wallet is 250 - 30 = 220. The amount paid for the belt is 230 - 57.5 = 172.5.
Finally, we add up the amount paid for both items to find the total amount paid: 220 + 172.5 = 392.5.
Therefore, Gelo paid a total of 392.5, so the correct answer is (b) PHP 392.50.
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Stock Standard Deviation Beta A 0.25 0.8 В 0.15 1.1 Which stock should have the highest expected return? A. A because it has the higher standard deviation B. B because it has the higher beta C. Not enough information to determine.
The answer is C. Not enough information to determine.
To understand which stock should have the highest expected return, we need more information about the stocks and the market. Standard deviation and beta are risk measures but do not directly provide information about expected return.
Standard deviation measures the dispersion of a stock's returns, with a higher standard deviation indicating greater volatility. Beta measures a stock's sensitivity to market movements, with a higher beta indicating greater responsiveness to market changes.
While risk and return are often positively correlated, meaning that higher risk investments typically offer higher potential returns, we cannot determine the expected return of these stocks based solely on their standard deviation and beta values. We would need additional information about the stocks, such as their historical returns or dividend yields, as well as the overall market conditions, to make an informed decision on which stock has the highest expected return.
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On a coordinate plane, 2 lines intersect at (0, 5).
Complete the statement.
The lines shown in the graph have the same
x-intercept.
y-intercept.
equation.
slope.
Answer:
y - intercept
Step-by-step explanation:
I took the assignment
Answer: b
y-intercept.
Step-by-step explanation:
did the assessment
I NEED HELP WITH #17
HELP PLEASEE
By complex and real number properties, the resulting electric resistance is equal to R' = 47 / 270 + i 4 / 45 ohms.
How to determine the total resistence of circuit by operation with complex numbers
Complex numbers are numbers of the form z = a + i b, where a, b are real numbers. The first component is the real component and the second one is the complex one. In this problem we find the case of a combination of two electric resistances described by two complex numbers. This expression must be simplified by means of real number and complex number properties:
R' = 1 / R₁ + 1 / R₂
If we know that R₁ = 4 + i 6 and R₂ = 2 - i 4, then the resulting electric resistance is:
1 / (4 + i 6) + 1 / (2 - i 4)
(4 - i 6) / [(4 + i 6) · (4 - i 6)] + (2 + i 4) / [(2 - i 4) · (2 + i 4)]
(4 - i 6) / (16 - i² 36) + (2 + i 4) / (4 - i² 16)
(4 - i 6) / (16 + 36) + (2 + i 4) / (4 + 16)
(4 - i 6) / 52 + (2 + i 4) / 20
(2 - i 3) / 27 + (1 + i 2) / 10
47 / 270 + i 4 / 45
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0.00000327 what is the proper scientific notation of this number?
Answer:
3.27 x 10^-6
Step-by-step explanation:
Hope this helps
Calculate the interquartile range of the given data. 44, 43, 23, 44, 48, 20, 30, 55, 45, 11, 9, 49, 4, 48, 7
To calculate the interquartile range (IQR) of the given data, we need to first find the first quartile (Q1) and the third quartile (Q3).
Step 1: Arrange the data in ascending order:
4, 7, 9, 11, 20, 23, 30, 43, 44, 44, 45, 48, 48, 49, 55
Step 2: Find the median (Q2):
Q2 is the middle value of the data set. In this case, we have an odd number of data points, so Q2 is the 8th value, which is 43.
Step 3: Find Q1:
Q1 is the median of the lower half of the data set. It is the middle value of the data points to the left of Q2. In this case, we have 7 data points to the left of Q2, so Q1 is the 4th value, which is 11.
Step 4: Find Q3:
Q3 is the median of the upper half of the data set. It is the middle value of the data points to the right of Q2. In this case, we have 7 data points to the right of Q2, so Q3 is the 4th value, which is 48.
Step 5: Calculate the interquartile range (IQR):
IQR = Q3 - Q1
IQR = 48 - 11
IQR = 37
Therefore, the interquartile range (IQR) of the given data is 37.
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like terms and distributive propertysimplify each expression by using the distributive property and by combining like terms
Given the expression:
\(\frac{1}{2}(12x-8)-\frac{1}{3}(18x-12)\)using the distributive property we get the following:
\(\begin{gathered} \frac{1}{2}(12x-8)-\frac{1}{3}(18x-12)+ \\ \frac{1}{2}(12x)+\frac{1}{2}(-8)-\frac{1}{3}(18x)-\frac{1}{3}(-12)= \\ 6x-4-6x+4=0 \end{gathered}\)therefore, the simplified expression is 0
Stacy is studying for a test. She thinks she
needs to study more than 2 hours in order to get
a good grade. Write an inequality to represent
this situation (you can use any variable).
Let x be the number of hours that Stacy needs to study to get a good grade. she needs to study for more than 2 hours to achieve a good grade.
Since she thinks she needs to study more than 2 hours to get a good grade, the inequality that represents this situation is:x > 2
This inequality says that x, the number of hours that Stacy needs to study, is greater than 2.
Based on the given information that Stacy thinks she needs to study more than 2 hours to get a good grade, we can write the following inequality:
h > 2
This inequality states that the number of hours 'h' that Stacy studies must be greater than 2.
In other words, she needs to study for more than 2 hours to achieve a good grade.
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You accepted a job for which the annual salary is $39236. This salary
includes a year-end bonus of $500. You will be paid once a month. What
will your gross pay be for each paycheck?
Answer:
$3228
Step-by-step explanation:
39236 - 500 = 38736
38736 / 12 = 3228
air decision a decision based on an event in which certain outcomes are favored or more likely than others 2. biases decision a decision based on an event in which the outcomes are equally likely
A fair decision is made when outcomes are equally likely, while a biased decision occurs when certain outcomes are favored or more likely than others.
It seems that you are describing two types of decision-making scenarios: "fair decision" and "biased decision."
Fair Decision:
A fair decision is made when the outcomes of an event are considered to be equally likely. In this case, there is no favoritism towards any particular outcome, and the decision is based on an even playing field where each outcome has an equal chance of occurring.
Biased Decision:
A biased decision occurs when certain outcomes of an event are favored or more likely than others. This means that there is a preference or inclination towards specific outcomes, which can influence the decision-making process. Biases can arise due to various factors such as personal beliefs, preferences, or external influences.
It's important to note that biases can affect decision-making in both positive and negative ways. Biased decisions may lead to unfair treatment or inaccurate judgments if they are not based on objective and unbiased evaluation of the available information.
In summary, a fair decision is made when outcomes are equally likely, while a biased decision occurs when certain outcomes are favored or more likely than others.
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In how many ways can five models line up to have their
photograph taken?
F. 120 G. 60 H. 25 I. 20
Answer:
120
Step-by-step explanation:
by using the basic counting rule, there are 5 positions first, then 4, then ,3 and so on down to one. so when you multiply them all together, you get 120. Also i took a test with the same qeustion, and that was the anwer :)
Answer:
120
Step-by-step explanation:
5 times 4 times 3 times 2 times 1= 120
5 choices for first
1 choice already taken so 4 for the other model
3 for the other
2 for the other
and 1 for the other
=120
Its correct, trust me!
And also write steps. e. If BAC = 90°, ABC= 65° and AD perpendicular to BC. Find x and y
Answer:
x=25⁰
y=25⁰
Step-by-step explanation:
ABC+BAC+ACB = 180⁰
65⁰+90⁰+y = 180⁰
y = 180⁰-155⁰
y = 25⁰
In triangle ABD,
ABD+ADB+BAD = 180⁰
65⁰+90⁰+x = 180⁰
x = 180⁰-155⁰
x = 25⁰
a property owner paid $25 per front foot for a lot 600 ft. x 1,452 ft. how many acres were in the lot that he bought?
A property owner paid $25 per front foot for a lot 600 ft. x 1,452 ft, The lot size is 600 ft. x 1,452 ft., which is equivalent to approximately 20 acres.
To determine the number of acres in the lot, we need to convert the dimensions from feet to acres.
The lot has a length of 600 ft and a width of 1,452 ft. To convert these dimensions to acres, we divide each dimension by the number of feet in an acre, which is 43,560.
Length in acres = 600 ft / 43,560 ft/acre
Width in acres = 1,452 ft / 43,560 ft/acre
Now, we can calculate the total area of the lot in acres by multiplying the length and width in acres:
Total area = Length in acres * Width in acres
After performing the calculations, the total area of the lot is obtained. The final answer represents the number of acres in the lot.
Please note that since the final answer is a numerical value, it can be provided directly without the need for an explanation.
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About 500 medicine cups are used daily at a long-term care facility nurse claims at approximately 4000 medicine cups are use a week what is the day to week use rate of medicine cups
Answer:
\(Rate = 0.125\)
Step-by-step explanation:
Given
\(Daily\ Cups = 500\)
\(Weekly\ Cups =4000\)
Required
Determine the day to week rate
This can be calculated by dividing the daily medicine cups by the weekly cups;
\(Rate = \frac{Daily}{Weekly}\)
Substitute 500 for Daily and 4000 for Weekly
\(Rate = \frac{500}{4000}\)
\(Rate = 0.125\)
Hence, the rate of the daily to weekly medicine cup is 0.125
Scout says the class is suspicious of their first grade teacher, Miss Caroline Fisher, because she comes from a county in North Alabama that _____ the secession of the Confederate States from the Union during the Civil War.
Scout says the class is suspicious of their first-grade teacher, Miss Caroline Fisher, because she comes from a county in North Alabama that supported the secession of the Confederate States from the Union during the Civil War.
The secession of the Confederate States from the Union during the Civil War refers to the act of Southern states, including Alabama, separating from the United States in order to form their own independent nation. Scout's statement suggests that Miss Caroline Fisher comes from a county in North Alabama that actively supported this secession.
During the Civil War, Alabama was a Confederate state, meaning it sided with the Southern states that seceded from the Union. Many counties within Alabama showed support for the Confederate cause and were known for their allegiance to the Confederacy.
Scout's statement implies that the class views Miss Caroline Fisher's background from a county in North Alabama as a reason for suspicion. The historical context of the Civil War and the divide between the Union and Confederate states likely contribute to the class's skepticism or concern regarding their teacher's origins and potential biases.
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Function or not a function
Answer:
no
Step-by-step explanation:
one x should only have one y for it to be a function
Find the surface area of the rectangular prism
The team go skirt pop pop pop pop pop pop...
someone designed a game that attracts children. in case of the child won they will get a gift otherwise the child will lose their money. There is a box filled with n balls
• The child along with the game owner switch turns such that in each turn a player
could draw k balls at once at two conditions:
√k ∈ Z ∧ 1 ≤ k ≤ n
• The child draws first.
• The player who draws the last ball, wins
I am trying to design a recursive algorithm in Python programming language that takes
input N
output: true if the child won otherwise false
The algorithm is designed to find out whether the child wins or loses in a game where two players take turns to pick balls. The algorithm takes the number of balls remaining in the box as input and returns True if the child wins, otherwise False.
The algorithm assumes that both players play optimally. The algorithm works by recursively picking balls from the box and checking whether the child or the game owner wins the game.
This is a recursive problem where each player plays optimally. There are two players in the game. The child will pick the balls first and then the game owner takes turns. Both will pick the balls optimally in order to win.
In each turn, k balls can be picked from the box. The winner is the one who picks the last ball. If the child picks the last ball, they win otherwise the game owner wins.
The algorithm is as follows:
Algorithm - Pick Balls
1. Define a recursive function called PickBalls(n). The function takes one parameter as input, which is the number of balls remaining in the box.
2. Check if the number of balls remaining is less than or equal to k. If yes, then return True as the game has ended and the child has won.
3. If the number of balls is greater than k, then pick k balls from the box.
4. If the child picks the last ball, then return True.
5. If the child does not pick the last ball, then the game owner picks the balls recursively.
6. If the game owner picks the last ball, then return False. Otherwise, return True.
Explanation: We are designing a recursive algorithm to find out whether the child will win or lose in a game where two players take turns to pick balls. The algorithm takes the number of balls remaining in the box as input and returns True if the child wins, otherwise False. We are assuming that both players play optimally. The algorithm works as follows:
If the number of balls remaining in the box is less than or equal to k, then the child can pick all the balls and win the game. Therefore, the function returns True.
If the number of balls remaining in the box is greater than k, then the child picks k balls from the box. If the child picks the last ball, then they win the game. Therefore, the function returns True.
If the child does not pick the last ball, then the game owner picks the balls recursively. If the game owner picks the last ball, then they win the game. Therefore, the function returns False. If the game owner does not pick the last ball, then the child picks again and the process repeats itself. If the child eventually picks the last ball, then they win the game. Therefore, the function returns True.
Conclusion: The algorithm is designed to find out whether the child wins or loses in a game where two players take turns to pick balls. The algorithm takes the number of balls remaining in the box as input and returns True if the child wins, otherwise False. The algorithm assumes that both players play optimally. The algorithm works by recursively picking balls from the box and checking whether the child or the game owner wins the game.
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Find the best buy for the item. Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or decimal rounded to the nearest thousandth as needed.) OA. The best buy for granulated sugar is the 10lb size and its unit price is $ B. The best buy for granulated sugar is the 4lb size and its unit price is $ GRANULATED SUGAR Size Price 4 lb 10 lb $2.93 $8.45
The best buy for granulated sugar is the 4lb option since it is lesser.
To find the best buy for the item , we calculate the cost per pound of the item :
Cost of item4lb = $2.93
10lb = $8.45
Cost per lb for 4lb :
2.93/4 = $0.7325 per lb
Cost per lb for 10lb :
8.45/10 = $0.845 per lb
Since the cost per lb for 4lb granulated sugar is lesser, then it is the best buy.
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Solve for x: -9 = 2x - 5 help plsss
Answer:
x = -2
Step-by-step explanation:
Answer:
x=-2
Step-by-step explanation:
An airplane is flying on a bearing of 170 degrees at 495 mph. Find the component form of the velocity of the airplane. Show and explain all work.
Answer:
(-487.48, 85.96)
Explanation:
The component form of the velocity can be found as
(vx, vy)
Where
vx = vcos(170)
vy = vsin(170)
So, replacing the velocity v by 495 mph, we get:
vx = (495)cos(170)
vx = -487.48
vy = 495sin(170)
vy = 85.96
Therefore, the answer is
(-487.48, 85.96)
How many ways can a person toss a coin 11 times so that the number of tails is between 4 and 8 inclusive
In order to solve this problem, we'll use the Combinations Formula. This is because the order in which the coins are flipped is unimportant; all that matters is the total number of heads and tails. We'll begin by counting the number of ways to get exactly 4 tails and exactly 8 tails, then the number of ways to get 5 tails and 6 tails, and finally the number of ways to get 7 tails. We will then add all of these numbers together to get the final answer.
When you toss a coin, there are only two possibilities: heads or tails.
Let's call the number of times you get tails "t." We want to find the number of ways to toss the coin 11 times so that
4 <= t <= 8.
Here's how we can do it:
First, we'll find the number of ways to get exactly 4 tails. We know that we're flipping the coin 11 times, so there are 11 choose 4 ways to choose which 4 of those tosses will be tails. The other 7 tosses will be heads. Using the Combinations Formula, this gives us:
11 choose 4 = 330
Next, we'll find the number of ways to get exactly 8 tails. This is similar to the previous step, except that we're choosing 8 of the 11 tosses to be tails. The other 3 tosses will be heads. Using the Combinations Formula, this gives us:
11 choose 8 = 330
Again, using the Combinations Formula, we can find the number of ways to get exactly 5 tails and 6 tails:
11 choose 5 = 462
11 choose 6 = 462
Finally, we need to find the number of ways to get exactly 7 tails. This is a little trickier, but we can do it by splitting the problem into two cases: when the first toss is a tail and when the first toss is a head.
Case 1: The first toss is a tail. Then we need to get 6 tails in the remaining 10 tosses. There are 10 choose 6 ways to choose which 6 of the remaining 10 tosses will be tails. Using the Combinations Formula, this gives us:
10 choose 6 = 210
Case 2: The first toss is a head. Then we need to get 7 tails in the remaining 10 tosses. There are 10 choose 7 ways to choose which 7 of the remaining 10 tosses will be tails.
Using the Combinations Formula, this gives us:
10 choose 7 = 120
Adding up all of these numbers, we get:
330 + 330 + 462 + 462 + 210 + 120 = 1914
So there are 1914 ways to toss a coin 11 times so that the number of tails is between 4 and 8 inclusive.
Thus, the required answer is 1914.
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shirabi spent $208 on a sewing machine to make purses. she spends a total of $10 on thread, fabric, and accessories for each purse and plans to charge $36 for each purse. the equation represents her break-even point, when x represents the number of purses sold. 208 10x
Shirabi needs to sell a minimum of 20 purses in order to reach her break-even point, where her total revenue equals her total costs.
To find the break-even point, we need to determine the number of purses that Shirabi needs to sell in order to cover her costs and not incur any loss. The equation representing her break-even point is given as:
208 + 10x
Here, 208 represents the cost of the sewing machine, and 10x represents the total cost of thread, fabric, and accessories for each purse, multiplied by the number of purses sold.
To find the break-even point, we need to set the equation equal to zero:
208 + 10x = 0
Now, let's solve for x:
10x = -208
x = -208/10
x = -20.8
Since the number of purses sold cannot be negative, we can disregard the negative solution. Therefore, Shirabi's break-even point is when she sells 20 purses.
Shirabi needs to sell a minimum of 20 purses in order to reach her break-even point, where her total revenue equals her total costs. Selling fewer than 20 purses would result in a loss, while selling more than 20 purses would generate a profit.
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Compare the budgets of Hong Kong, United States of America, and
Korea based on your definition of a budget, in terms of contents,
formats, advantages, and disadvantages, etc.
The budgets of Hong Kong, the United States of America, and Korea differ in contents, formats, advantages, and disadvantages. While each budget has its strengths and weaknesses, they all aim to provide a clear and transparent financial plan for their respective countries.
A budget is a financial plan that estimates expected income and expenditure for a specific period. It may include income, expenses, debts, and savings. Budgets may vary from country to country and can be analyzed by comparing their contents, formats, advantages, and disadvantages. Here are the budgets of Hong Kong, the United States of America, and Korea:
Hong Kong Budget:United States Budget:
Contents: The US budget comprises revenue, expenditures, and deficit or surplus. It includes an analysis of taxes, social security, and Medicare.Format: The US budget is presented in a complex and lengthy format, including tables, graphs, and other financial documents.Advantages: The budget provides detailed information on tax expenditures and encourages public participation in the budget process.Disadvantages: The budget can be challenging to understand due to its complexity, and it may not provide an accurate depiction of federal spending.Korean Budget:
Contents: The Korean budget comprises revenue, expenditures, and surplus or deficit. It includes detailed information on taxes, social security, and public welfare.Format: The Korean budget is presented in a clear and concise format, including tables and charts to aid understanding.Advantages: The budget is easy to understand, and it promotes transparency and accountability. It also provides detailed information on social welfare expenditures.Disadvantages: The budget may not provide an accurate depiction of government spending, and it may not include information on hidden expenditures.Learn more about Budget:
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i know it’s kinda all hard to see. i have many more questions like this and i can’t do graphs to save my life.. pls help !!
Answer: D
Step-by-step explanation:
a tank holds 4000 liters of water in which 100 grams of salt have been dissolved. saltwater with a concentration of 1 grams/liter is pumped in at 10 liters/minute and the well mixed saltwater solution is pumped out at the same rate. write initial the value problem for:
The mass of salt in the tank at time t.
dS/dt = 10 - S/400
S(0) = 100 grams
The solution is S(t) = 4000 - 3900\(e^{\frac{-t}{400}}\)
A tank holds water V(0) = 4000 liters in which salt S(0) = 100 grams.
So dS/dt = S(in) - S(out)
S(in) = 1 × 10 = 10 gram/liters
S(out) = S/V × 10 = 10S/V gram/liters
V = V(0) + q(in) - q(out)
V = 4000 + 10t - 10t
V = 4000 liters
dS/dt = 10 - 10S/V
dS/dt = 10 - 10S/4000
dS/dt = 10 - S/400
Now given; S(0) = 100.
Here, p(t) = 1/400, q(t) = 10
\(\int p(t)dt = \int\frac{1}{400}dt\)\(\int p(t)dt = \frac{1}{400}t\)
\(\mu=e^{\int p(t)dt}\)
\(\mu=e^{\frac{t}{400}}\)
So, S(t) = \(\frac{\int\mu q(t)dt+C}{\mu}\)
S(t) = \(\frac{\int e^{\frac{t}{400}} \cdot10dt+C}{e^{\frac{t}{400}}}\)
S(t) = \(e^{\frac{-t}{400}} \left({\int e^{\frac{t}{400}} \cdot10dt+C}\right)\)
S(t) = \(e^{\frac{-t}{400}} \left({10\times\frac{e^{\frac{t}{400}}}{1/400} +C}\right)\)
S(t) = \(e^{\frac{-t}{400}} \left({4000\times{e^{\frac{t}{400}} +C}\right)\)
Now solving the bracket
S(t) = 4000 + \(e^{\frac{-t}{400}}\)C.....(1)
At S(0) = 100
100 = 4000 + \(e^{\frac{-0}{400}}\) C
100 = 4000 + \(e^{0}\) C
100 = 4000 + C
Subtract 4000 on both side, we get
C = -3900
Now S(t) = 4000 - 3900\(e^{\frac{-t}{400}}\)
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The complete question is:
A tank holds 4000 liters of water in which 100 grams of salt have been dissolved. Saltwater with a concentration of 1 grams/liter is pumped in at 10 liters/minute and the well mixed saltwater solution is pumped out at the same rate. Write initial the value problem for:
The mass of salt in the tank at time t.
dS/dt =
S(0) =
The solution is S(t) =