A certain country has $10 billion in paper currency in circulation, and each day $50 million comes into the country's banks. The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks. Let x(t) denote the amount of new currency (in billions of dollars) in circulation at time t (in days), with x(0) = 0. Then
dx/dt (fraction of currency that is old)(0.05 billion$/day)
so
dx/dt = ((10-x)/10)(0.05) = 0.005(10-x)
(a) Determine x(t) by solving the differential equation and using the initial condition x(0) = 0
x(t) = ____
(b) How long will it take for the new davs bills to account for 90% of the currency in circulation? Give your answer in decimal form in days, rounded to the nearest day.
Answers
(a) The given differential equation is dx/dt = 0.005(10-x) with x(0) = 0.
Solving the differential equation, we get:
dx/(10-x) = 0.005 dt
Integrating both sides, we get:
-ln(10-x) = 0.005t + C
Applying initial condition x(0) = 0, we get C = ln(10)
Thus, we have:
-ln(10-x) = 0.005t + ln(10)
ln[(10-x)/10] = -0.005t
(10-x)/10 = e^(-0.005t)
x(t) = 10(1-e^(-0.005t))
Therefore, x(t) = 10(1-e^(-0.005t))
(b) We need to find the time t such that x(t) = 0.9*10 = 9 billion dollars.
Substituting x(t) = 9 in the equation, we get:
9/10 = 1-e^(-0.005t)
e^(-0.005t) = 0.1
Taking natural logarithm on both sides, we get:
-0.005t = ln(0.1)
t = -ln(0.1)/0.005 = 138.63 days ≈ 139 days (rounded to the nearest day)
Therefore, it will take approximately 139 days for the new bills to account for 90% of the currency in circulation.
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Related Questions
Medical researchers have determined that for exercise to be beneficial, a person’s desirable heart rate, r, in beats per minute, can be approximated by the formulas r = 143 minus 0.65 a for women r = 165 minus 0.75 a for men, where a represents the person’s age. if the desirable heart rate for a man is 135 beats per minute, how old is he? a. 22.5 years old b. 40 years old c. 45 years old d. 42.5 years old
Answers
Age will be 40 years old.
R, in beats per minute, can be approximated by the formulas, where a represents the person's age.
where R= 143 - 0.65a for women.... (1)
and R= 165 - 0.75a for men,..... (2)
So by implementing these two equation we get a = 40
If the sum of ages is X and Y, and the ratio of their ages is p:q, then the age of Y can be calculated using the formula shown below: Y's age = Y's ratio/Sum of ratios x sum of ages The age dependency ratio is the ratio of dependents (people under the age of 15 or over the age of 64) to the working-age population (people between the ages of 15 and 64). The proportion of dependents per 100 working-age population is shown in the data.
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Your local garden shop has plants on sale for $1.50 each. You are planning a vegetable
garden. You have $20 to spend on tomato plants
a.) write an inequality to find the greatest number of plants you can buy.
60 points please helppp
Answers
The inequality that can be used to find the greatest number of plants you can buy is p ≤$20 / 1.50.
What inequality can be used to find the greatest number of plants you can buy?Here are inequality signs and what they mean:
> means greater than
< means less than
≥ means greater than or equal to
≤ less than or equal to
The inequality sign that would be used in this question is ≤.
(cost of plant x number of plants) ≤ total spending money
$1.50p ≤ $20
p ≤ $20 / 1.50
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pls help asap thanks
Answers
Answer:
(x - 8) (x - 7)
Step-by-step explanation:
x² - 15x + 56
Consider the form x² + bx + c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is 56 and whose sum is −15.
-8, -7
Write the factored form using these integers.
(x - 8) (x - 7)
What is the smallest of 3 consecutive positvie integers if the product of the smaller two integers is 8 less than 4 times the largest integer?
Answers
Then the other two integers are x + 1 and x + 2.
According to the problem, the product of the smaller two integers is 8 less than 4 times the largest integer. This can be written as:
x(x + 1) = 4(x + 2) - 8
Simplifying and solving for x, we get:
x^2 + x = 4x + 8 - 8
x^2 - 3x - 0 = 0
Using the quadratic formula, we get:
x = [3 ± sqrt(3^2 - 4(1)(0))] / 2
x = [3 ± sqrt(9)] / 2
x = (3 + 3) / 2 or x = (3 - 3) / 2
x = 3/2 or x = 0 (reject)
Since x must be a positive integer, the smallest of the three consecutive positive integers is x = 3/2, which is not an integer. Therefore, there is no solution to the problem as stated. Your welcome
what is the variance of the number of heads that come up when a fair coin is flipped 13 times? (enter the final answer in decimal format and round to one decimal place.)
Answers
The variance of the number of heads that come up when a fair coin is flipped 13 times is 3.25
What is probability ?Mathematical representations of the likelihood of an event occurring or of a statement being true are dealt with in the area of probability. An outcome's probability is a number between 0 and 1, where 1 denotes certainty and 0 denotes the event's impossibility.
CalculationHere, flipping a coin is a Bernoulli trial, with n =13, with success as getting a head with probability p = 1/2.
We have to find the variance of the number of success in n Bernoulli trials. The variance of the number of successes in n Bernoulli trials is np(1-p).
the variance np \((1-p)=13\cdot (1/2)\cdot (1-1/2)=13\cdot (1/2)\cdot (1/2)= 3.25\)
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Of the range, the interquartile range, and the variance, the interquartile range is least influenced by an outlying value in the data set.
Is that true or false? please explane
Answers
The statement "the interquartile range is least influenced by an outlying value in the data set" is true.
The range is simply the difference between the maximum and minimum values in a data set.
The variance is a measure of the spread of a data set, and is calculated by finding the average of the squared differences between each data point and the mean of the data set.
The interquartile range, on the other hand, is a measure of the spread of the middle 50% of the data set. It is calculated by finding the difference between the upper quartile and the lower quartile.
Because the interquartile range only takes into account the middle 50% of the data, it is less influenced by outliers than the range and variance. This is because outliers can greatly impact the range and variance, but have little effect on the interquartile range.
In conclusion, the interquartile range is a more robust measure of spread for a data set, as it is less influenced by outliers.
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Let the C.P. be Rs x. If a shopkeeper makes 12% loss by selling a watch for Rs 484, find C.P.
Answers
Answer:
To find the cost price (C.P.) of the watch, we need to use the following formula:
C.P. = Selling Price / (1 - Loss Percentage)
In this case, we know that the Selling Price is Rs 484 and the Loss Percentage is 12%. To use this formula, we first need to convert the Loss Percentage to a decimal:
12% = 0.12
Now we can plug in the values into the formula:
C.P. = 484 / (1 - 0.12)
Solving for C.P., we get:
C.P. = 484 / 0.88
C.P. = Rs 550
Therefore, the cost price (C.P.) of the watch is Rs 550.
Make p the subject of the formula q-2p=p+4
Answers
Answer:
q-2p=p+4
q-4=p+3p
p=(q-4)/3is answer
hellppppp plzzzzzz........
Answers
1. Two crucial tasks inherent in the initial stage of group therapy are orientation and ______________.
2. Ambiguity and lack of a structured approach in groups often lead to:
Answers
Two crucial tasks inherent in the initial stage of group therapy are orientation and establishing group norms.Ambiguity and lack of a structured approach in groups often lead to confusion, inefficiency, and potential challenges.
Orientation involves providing essential information to group members about the purpose, goals, and guidelines of the therapy group.
It helps individuals understand what to expect, builds trust, and creates a sense of safety and predictability within the group. Orientation may include discussing confidentiality, group rules, expectations, and addressing any questions or concerns.
Establishing group norms involves collaboratively developing shared guidelines and expectations that govern the behavior and interactions within the group. This process allows group members to contribute to the creation of a supportive and respectful group climate. Group norms help set boundaries, encourage open communication, and foster a sense of cohesion among members.
Without clear structure and guidance, group members may struggle to understand their roles, goals, or the process of the group therapy. Ambiguity can hinder progress, create frustration, and impede meaningful communication.
Lack of structure may also result in difficulty managing conflicts, decision-making, or time management within the group. It can lead to unequal participation, power struggles, and a lack of accountability.
To address these issues, it is important for group therapy to provide a clear framework, establish ground rules, and facilitate structured activities or interventions that promote clarity, engagement, and progress. A structured approach helps create a supportive environment, enhances group dynamics, and maximizes the therapeutic benefits of the group process.
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Please help me with these questions. Thank you.
Answers
Answer:
36 x⁸y⁶
√2/2
Step-by-step explanation:
Factor 1296 into its prime factors
1296 = 24 • 34
To simplify a square root, we extract factors which are squares, i.e., factors that are raised to an even exponent.
Factors which will be extracted are :
1296 = 24 • 34
No factors remain inside the root !!
To complete this part of the simplification we take the squre root of the factors which are to be extracted. We do this by dividing their exponents by 2 :
36 = 22 • 32
At the end of this step the partly simplified SQRT looks like this:
36 √(x¹⁶y¹²)
STEP2:Simplify the Variable part of the SQRT
Rules for simplifing variables which may be raised to a power:
(1) variables with no exponent stay inside the radical
(2) variables raised to power 1 or (-1) stay inside the radical
(3) variables raised to an even exponent: Half the exponent taken out, nothing remains inside the radical. examples:
√(x⁸)=x⁴
√(x^-6)=x^-3
Applying these rules to our case we find out that
SQRT(x¹⁶y¹²) = x⁸y⁶
Combine both simplifications
sqrt (1296x¹⁶y¹²) = 36 x⁸y⁶
Simplified Root : 36 x⁸y⁶
number 2)
3√2/3√4=
=3√2/3×2
=3√2/6
=3√2/6 ×3/3
=1√2/2
=√2/2
which is a better buy. 5 grapfruit for 1.99 or 8 grape fruit for 2.99
Answers
The better buy is 8 grapefruit for 2.99. '
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that 5 grapefruit for 1.99 or 8 grapefruit for 2.99. The better price for the grapefruit will be calculated as,
5 grapefruit for 1.99
1 grapefruit = $1.99 /5 = $0.39
8 grapefruit for 2.99.
1 grapefruit = 2.99 / 8 = $0.37
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Explain how you would find 36×3/4 mentally
Answers
Answer:
= 27
Step-by-step explanation:
36×3/4
36 × 3
--
4
(cross multiplication)9 × 3
= 27
Select the correct answer. How many meters is one hectometer? ОА. 1,000 OB. 100 Ос. OD. O E. 0.01
Answers
Answer:
100
Step-by-step explanation:
1 hectometer = 100 meters
determine the value of k for which the system
Answers
For the system of equations to have a unique solution the value of k must not be 6
Now, According to the question:
The given equations are,
kx + 2y = 5
3x + y = 1
The above equations can be written as,
kx + 2y – 5 = 0
3x + y – 1 = 0
We need to find the value of k.
So, we know that if the two equations are \(a_1x+b_1y+c_1=0\), \(a_2x+b_2y+c_2=0\) Then we will compare the coefficients such that
\(\frac{a_1}{a_2},\frac{b_1}{b_2} and \frac{c_1}{c_2}\).
If \(\frac{a_1}{a_2}\neq \frac{b_1}{b_2}\) then the equations have unique solution, if \(\frac{a_1}{a_2}=\frac{b_1}{b_2} = \frac{c_1}{c_2}\)
then the equations have infinitely many solutions and if \(\frac{a_1}{a_2}=\frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)
then the equations have no solutions.
Here we can clearly see that,
\(a_1=k,b_1=2,c_1=-5\\\\\\a_2=3,b_2=2,c_2=-1\)
So, \(\frac{a_1}{a_2},\frac{b_1}{b_2} , \frac{c_1}{c_2}\)
\(\frac{a_1}{a_2}=\frac{k}{3}, \frac{b_1}{b_2}=\frac{2}{1} , \frac{c_1}{c_2}=\frac{5}{1}\)
We know that if the system of equation has unique solution then \(\frac{a_1}{a_2}\neq \frac{b_1}{b_2}\)
So, we solve on putting their values and we get,
\(\frac{k}{3}\neq \frac{2}{1}\)
\(k\neq 6\)
Hence, for the system of equations to have a unique solution the value of k must not be 6.
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The given question is incomplete, complete question is:
Find the value of k for which the following system of equation has the unique solution:-
kx + 2y = 5
3x + y = 1.
Which one is the correct answer, a b c d
Answers
\(g(f(x))=\sqrt{\frac{2301+100x^2}{23+x^2} }\)
Step-by-step explanation:\(Substitute\ and\ calculate:\\ \downarrow\\ g(f(x))=\sqrt{\frac{2301+100x^2}{23+x^2} }\)
I hope this helps you
:)
Toda expresión de la forma xn + yn tiene un factor con los signos alter nados, uno positivo y uno negativo.
Answers
????????????
Step-by-step explanation:
?????????
Which of the following arrow diagrams shows the one-to-one correspondence between set a and set b
#please help me
#nonsense will be reported
...........
Answers
Step-by-step explanation:
In set a, every element in the domain has only 1 image in the range. This is called a one-to-one correspondence.
However in set b, there is a many-to-one correspondence as more than 1 element in the domain has the same image.
What is 24x31? Can you show the explanation
Answers
Answer:
744
Step-by-step explanation:
2 4
x 3 1
_____
2 4
7 2 0
+ ______
7 4 4=744
Hope it helps have a great day:)
Answer:
744
Step-by-step explanation:
24
x31
744
To explain what 24 times 31 means, look at it as 24 added together 31 times. To get the answer, you could just write down the number 24, 31 times and then add the 24 numbers together.
It takes Amelia one minute to swim 1/60 of a kilometer.how far can she swim in 12 minutes
Answers
take 1/60 and multiply by 12, then simplify
If 40% of x + 20% of x + 1/10 = 75% of x, then find the value of x. Please give answers with steps
Answers
Answer:
x = 13/12
Step-by-step explanation:
explain on image
How much greater is A than B, give your answer correct to the nearest whole number
Answers
The area of concert B is 840,000 m² greater than the area allowed for concert A.
To determine how much greater the greatest number of people allowed to attend concert B is compared to concert A, we need to calculate the area of the rectangular field for concert B and compare it to the area of concert A.
The area of a rectangle is given by the formula: Area = length × width
For concert B, the rectangular field has a length of 1800 m and a width of 500 m.
Therefore, the area of concert B is:
Area of concert B = 1800 m × 500 m
= 900,000 m².
Now we can compare the area of concert B with the greatest number of people allowed to attend concert A, which is 60,000 m²
To determine how much greater the area of concert B is compared to concert A, we subtract the area of concert A from the area of concert B:
Area difference = Area of concert B - Area of concert A
= 900,000 m² - 60,000 m²
= 840,000 m².
Therefore, the area of concert B is 840,000 m² greater than the area allowed for concert A.
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3. The Alvarez family bought a car for $2,000. They made a down
payment of $500. If they want to pay the balance in 5 equal
payments, how much will each of these payments be?
Answers
The first step is to subtract $500 from $2000 and then you end up with $1500 which you then divide by 5.
Each payment will have to be $300
describe what it means to simplify an algebraic expression.
Answers
Find the area of the triangle.
Answers
Answer:
86.1yd
Step-by-step explanation:
area of a triangle= base*height/2
21*8.2=172.2
172.2/2=86.1
Answer:
86.1yd2
Step-by-step explanation:
area=1/2×b×h
a=1/2×21×8.2
a=86.1yd2
Complete the square
x^2 - 2x - 14 = 0
Answers
To complete the square for the equation x^2 - 2x - 14 = 0, follow these steps:
1. Move the constant term to the right side of the equation: x^2 - 2x = 14
2. Take half of the coefficient of x, square it, and add it to both sides of the equation: x^2 - 2x + 1 = 15
3. Factor the left side of the equation as a perfect square: (x - 1)^2 = 15
4. Take the square root of both sides of the equation, remembering to include both the positive and negative square roots: x - 1 = ±√15
5. Add 1 to both sides of the equation to isolate x: x = 1 ± √15
Therefore, the solutions to the equation x^2 - 2x - 14 = 0 are x = 1 + √15 and x = 1 - √15.
Solve for 5x+20=x-10
Please please explain it step by step it’s the only way I will get credit for my assignment
I will give brain whoever solves
Answers
Answer:
-7.5
Step-by-step explanation:
5x+20=x-10
Subtract x from both sides to have the variable on one side of the equation:
4x+20=-10
Subtract 20 from both sides of the equation to isolate the variable:
4x=-30
Divide both sides by 4 to find x:
x=-30/4=-7.5
1. Subtract x from both sides which will give you 4x+20=-10
2. Subtract 20 on each side of the equation which will give you 4x=-30
3. Divide 4 on both sides to leave x by itself and you will get
5x+20=x-10
-x -x
———————
4x+20=-10
-20 -20
———————
4x=-30
4x/4=-30/4
X=-30/4
-30/4 can reduce to -15/2 so x=-15/2
Hope that helps
What is the slope in the equation y = 2/5x - 6?
Answers
Answer:
2/5
Step-by-step explanation:
y=2/5x-6
m=2/5
y=mx+b
Slope: 2/5
y-intercept: 6
Step-by-step explanation:
Just Did Now, Am not Really good at asking questions
A moving company drove one of its trucks 100,042 miles one year. A second truck was
driven 98,117 miles, and a third truck was driven 120,890 miles. How many miles
were driven by all three trucks?
HELP ME PLEASE IVE BEEN STUCK FOR HOURS ON THIS!!!!!!
Answers
319049
as 100,042 + 98,117 + 120,890 = 319,049
The miller family's total bill was $67.50, if they want to give the waitress a 20% tip based on total bill, how much money will they need to leave the waitress? (tip money only)
Answers
If the waitress's tip was 20% based on the $67.50 bill, then her tip was $13.5
What is a percentage?The percentage is a mathematical operation that is expressed with the symbol "%" is a relationship that allows us to know the portion of an initial value to represent a set.
It could also be defined as a representation of an original value that lets us know a certain number of elements that we want from that original value or set.
The easiest way to solve this problem is with a rule of three implement, as follows:
100% ------ $67.5 20% ----- xx = (20% * $67.5 ) / 100%
x = $13.5
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