The monthly payment for a $17,500 loan at 6% APR over 25 years is about $113.36. The total amount paid over the duration of the loan is $34,008, of which about 51.46% goes towards the principal and 48.54% towards interest.
Explanation:To answer this question, we need to use the formula for calculating the monthly payment for a loan, which is P[r(1 + r)^n]/[(1 + r)^n - 1], where P is the principal loan amount, r is the monthly interest rate (annual rate divided by 12), and n is the number of payments (years times 12).
a. Monthly payment:
First, we have to calculate r: 6% APR implies a yearly interest rate of 6%/12 = 0.005 per month. Plugging P = $17,500, r = 0.005, and n = 25*12 = 300 into the formula, we get the monthly payment of approximately $113.36.
b. Total amount paid:
The total amount paid over the duration of the loan is simply the monthly payment times the total number of payments, so $113.36*300 = $34,008.
c. Percentages of principal and interest:
The percentage paid towards the principal is the original loan amount divided by the total payment amount times 100, so $17,500/$34,008*100 = approximately 51.46%. Therefore, the percentage paid for interest is 100 - 51.46 = 48.54%.
Learn more about Student Loan here:https://brainly.com/question/32329996
#SPJ3
The monthly payment for the loan is approximately $114.08. Over 25 years, the total amount paid will be about $34,224. About 51.11% of this amount goes toward the principal and 48.89% goes toward interest.
To solve this problem, we need to calculate the monthly payment, total amount paid, and the percentages of principal and interest paid for a student loan of $17,500 at a fixed APR of 6% for 25 years.
a. Calculate the monthly payment
We use the formula for the monthly payment on an amortizing loan:
→ [tex]M = P [r(1 + r)^n] / [(1 + r)^{n - 1}][/tex]
where:
→ P = loan principal ($17,500)
→ r = monthly interest rate (annual rate / 12)
= 6% / 12
= 0.005
→ n = total number of payments (years * 12)
= 25 * 12
= 300
Substituting the values into the formula:
→ [tex]M = 17500 [0.005(1 + 0.005)^{300}] / [(1 + 0.005)^{300 – 1}][/tex]
→ M = 17500 [0.005(4.2918707)] / [4.2918707 – 1]
→ M = 17500 [0.02145935] / [3.2918707]
→ M = 375.53965 / 3.2918707
→ M ≈ $114.08
b. Determine the total amount paid over the term of the loan
→ Total amount paid = monthly payment * total number of payments
→ Total amount paid = 114.08 * 300
≈ $34,224
c. Of the total amount paid, what percentage is paid toward the principal and what percentage is paid for interest
→ Principal: $17,500
→ Interest: Total amount paid - Principal
→ Interest = 34,224 - 17,500
≈ $16,724
→ Percentage toward principal = (Principal / Total amount paid) * 100
≈ (17500 / 34224) * 100
≈ 51.11%
→ Percentage toward interest = (Interest / Total amount paid) * 100
≈ (16724 / 34224) * 100
≈ 48.89%
â(Future valueâ) Sarah Wiggum would like to make a singleâ lump-sum investment and have â$ 1.7 million at the time of her retirement in 34 years. She has found a mutual fund that expects to earn 8 percent annually. How much must Sarah investâ today? If Sarah earned an annual return of 16 16 âpercent, how much must she investâ today? a. If Sarah can earn 8 percent annually for the next 34 âyears, how much will she have to investâ today? â $ nothing â(Round to the nearestâ cent.)
Answer:
a) at 16%: $10,936.47
b) at 8%: $124,177.02
Step-by-step explanation:
At annual rate of return "r", the multiplier of Sarah's initial investment will be ...
k = (1+r)^34
For r = 0.16, k ≈ 155.433166, and Sarah's investment needs to be ...
$1.7·10^6/k ≈ $10,936.47
__
For r= 0.08, k ≈ 13.6901336, and Sarah's investment needs to be ...
$1.7·10^6/k ≈ $124,177.02
You can afford monthly deposits of 140 into an account that pays 3.8% compounded monthly. How long will be untl you have $11.300 to buy a boat? Type the number of months: (Round to the next higher month it not exact Question He Check Answer Enter your answer in the answer box and then click Check Answer All parts showing
Answer:
72 months approx.
Step-by-step explanation:
Monthly deposit = m = $140
r = 3.8% or 0.038
Amount needed in the account = A = $11300
The formula will be :
[tex]11300=140(\frac{(1+0.038/12)-1}{0.038/12} )[/tex]
[tex]11300=140(\frac{(1+0.038/12)-1}{0.003166})[/tex]
[tex]11300=44219.83((1.003166)^{m}-1)[/tex]
[tex]0.2555=(1.003166)^{m}-1[/tex]
[tex]1.2555=(1.003166)^{m}[/tex]
m=log1.2555/log1.003166
m =71.98 ≈ 72
Hence, it will take 72 months approx.
Is the given function phi(x) = x^2 - x^-1 an explicit solution to the linear equation d^2y/dx^2 - 2/x^2 y = 0? Circle your answer. (a) yes (b) no
Answer:
Yes
Step-by-step explanation:
We are given that a function [tex]\phi(x)=x^2-x^{-1}[/tex]
We have to find that given function is an explicit solution to the linear equation
[tex]\frac{d^2y}{dx^2}-\frac{2}{x^2}y=0[/tex]
If given function is an explicit solution of given linear equation then it satisfied the given differential equation
Differentiate w.r.t x
Then we get [tex]\phi'(x)=2x+x^{-2}[/tex]
Again differentiate w.r.t x
Then we get
[tex]\phi''(x)=2-\frac{2}{x^3}[/tex]
Substitute all values in the given differential equation
[tex]2-\frac{2}{x^3}-\frac{2}{x^2}(x^2-x^{-1})[/tex]
=[tex]2-\frac{2}{x^3}-2+\frac{2}{x^3}=0[/tex]
Hence, given function is an explicit solution of given differential equation.
Therefore, answer is yes.
Express as the sum or difference of logarithms. log311y
The function log311y can be expressed as the sum of two logarithms, log3(11) + log3(y), according to the product rule of logarithms.
Explanation:The function log311y represents the logarithm base 3 of the product of the numbers 11 and y. Using the rules of logarithms, we can rewrite this expression as a sum of two logarithms.
According to the product rule of logarithms, the logarithm of a product is the sum of the logarithms of the component numbers. Applying this rule to our expression, log311y becomes:
log3(11) + log3(y)
This is the sum of the logarithm base 3 of 11 and the logarithm base 3 of y. In conclusion, the function log311y can be expressed as the sum of the separate logarithms: log3(11) + log3(y).
Learn more about Logarithms here:https://brainly.com/question/37245832
#SPJ6
The logarithm log3(11y) can be broken down into two simpler logarithms, log3(11) and log3(y), by using properties of logarithms. This is the sum of the two simpler logarithms.
Explanation:To express the logarithm log3(11y) as the sum or difference of logarithms, we will utilize the properties of logs:
The logarithm of a product is the sum of the logarithms of the numbers (log(xy) = log(x) + log(y)).The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number.The logarithm of a quotient is the difference of the logarithms of the numbers.Applying these properties to the given logarithm, we find:
log3(11y) = log3(11) + log3(y)
Thus, the original logarithm has been expressed as a sum of two simpler logarithms.
Learn more about Logarithms here:https://brainly.com/question/37287488
#SPJ6
Assume that when adults with smartphones are randomly selected, 51% use them in meetings or classes. If 11 adult smartphone users are randomly selected, find the probability that fewer than 5 of them use their smartphones in meetings or classes.
Answer:
The probability is 0.2356.
Step-by-step explanation:
Let X is the event of using the smartphone in meetings or classes,
Given,
The probability of using the smartphone in meetings or classes, p = 51 % = 0.51,
So, the probability of not using smartphone in meetings or classes, q = 1 - p = 1 - 0.51 = 0.49,
Thus, the probability that fewer than 5 of them use their smartphones in meetings or classes.
P(X<5) = P(X=0) + P(X=1) + P(X=2) + P(X=3)+P(X=4)
Since, binomial distribution formula is,
[tex]P(x) = ^nC_r p^x q^{n-x}[/tex]
Where, [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]
Here, n = 11,
Hence, the probability that fewer than 5 of them use their smartphones in meetings or classes
[tex]=^{11}C_0 (0.5)^0 0.49^{11}+^{11}C_1 (0.5)^1 0.49^{10}+^{11}C_2 (0.5)^2 0.49^{9}+^{11}C_3 (0.5)^3 0.49^{8}+^{11}C_4 (0.5)^4 0.49^{7} [/tex]
[tex]=(0.5)^0 0.49^{11}+11(0.5)0.49^{10} + 55(0.5)^2 0.49^{9}+165 (0.5)^3 0.49^{8} +330(0.5)^4 0.49^{7} [/tex]
[tex]=0.235596671797[/tex]
[tex]\approx 0.2356[/tex]
Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. (Assume that n begins with 1.){1/2,1/4,1/6,1/8,1/10,...}a_n = ?
Answer:
[tex]a_{n}=\frac{1}{2n}[/tex] [Where a ≥ 1 ]
Step-by-step explanation:
The pattern of the given sequence is {[tex]\frac{1}{2},\frac{1}{4},\frac{1}{6},\frac{1}{8},\frac{1}{10},......[/tex]
We have to find a formula for the general term [tex]a_{n}[/tex] of the given sequence.
We can rewrite the terms of the sequence as
[tex]\frac{1}{2}=\frac{1}{(2)(1)}[/tex]
[tex]\frac{1}{4}=\frac{1}{(2)(2)}[/tex]
[tex]\frac{1}{6}=\frac{1}{(2)(3)}[/tex]
[tex]\frac{1}{8}=\frac{1}{(2)(4)}[/tex]
[tex]\frac{1}{10}=\frac{1}{(2)(5)}[/tex]
Now we can write the term [tex]a_{n}[/tex] as
[tex]a_{n}=\frac{1}{2n}[/tex]
Where n = 1, 2, 3, 4, 5......
List the different combinations of heads and tails that can occur when 3 ordinary coins are tossed. Use h for heads and t for tails. One combination is hht. List the other combinations, taking order into account (Use a comma to separate answers) More i () a Enter your answer in the answer box ere to search
[tex]HTT,HTH,HHH,TTT,THT,THH,TTH[/tex]
A new drug on the market is known to cure 20% of patients with breast cancer. If a group of 20 patients is randomly
selected, what is the probability of observing, at most, one patient who will be cured of breast cancer?
A• (20)/1 (0-20)^1 (0.80)^19
B. 1-(20)/1 (.20)'(0.230)
C. (20)/0( .80)^20+(20/1)(.20)^1(.80)^19
D • (20)/0 (.80)^20
E. 1-(20/0)(.80)^20
01-(20)10.2010
Answer:
It's actually C
Step-by-step explanation:
don't forget about the probability of 0 too
you have to add the two probability formulas
The probability of observing, at most, one patient who will be cured of breast Cancer will be P = (20 / q²⁰) + 20 / (0.20 x 0.80¹⁹). Then the correct option is C.
What is probability?Its basic premise is that something will almost certainly happen. The percentage of favorable events to the total number of occurrences.
A new drug on the market is known to cure 20% of patients with breast cancer.
p = 0.20
q = 1 – 0.20
q = 0.80
If a group of 20 patients is randomly selected.
The probability of observing, at most, one patient who will be cured of breast Cancer will be
P = (20 / q²⁰) + 20 / (0.20 x 0.80¹⁹)
Then the correct option is C.
More about the probability link is given below.
https://brainly.com/question/795909
#SPJ2
Write the equation that passes through the given point and has the slope indicated: (1, -3); with slope (-3/5)
Answer:
y = -3/5x - 12/5
Step-by-step explanation:
The equation I'm going to give is going to be in slope-intercept form. If you need it in point-slope, I can do so in an edit or the comments.
Slope-intercept form is: y = mx + b where m is the slope, b is the y-intercept.
So let's plug in our given slope:
y = -3/5x + b
Using this, we now plug in our x- and y-coordinates from the given point to solve for b.
-3 = -3/5(1) + b
-3 = -3/5 + b
Add 3/5 to both sides to isolate variable b.
-3 + 3/5 = b
-15/5 + 3/5 = b
-12/5 = b
Plug this new info back into the original equation and your answer is
y = -3/5x - 12/5
Use set notation to write the members of the following set, or state that the set has no members
Odd numbers between 22 and 66 that are multiples of 5.
Answer:
Odd numbers that are multiple of 5 and are in between 22 and 66 are-
25, 35, 45, 55, 65
Let this set be represented by A
A= {25, 35, 45, 55, 65}
the above form represents the set in its roster form
The set notation for the odd numbers between 22 and 66 that are multiples of 5 is { x ∈ N | x is odd, 22 < x < 66, x ≡ 0 (mod 5) }.
Explanation:The set notation for the odd numbers between 22 and 66 that are multiples of 5 can be written as:
{ x ∈ N | x is odd, 22 < x < 66, x ≡ 0 (mod 5) }
Identify the set as a set of natural numbers, denoted as N.Define the condition for the set membership using a vertical bar |.List the conditions for the set, which include being odd, between 22 and 66, and a multiple of 5.Use the congruence notation x ≡ 0 (mod 5) to represent the requirement of being a multiple of 5.Learn more about Set notation here:https://brainly.com/question/32799768
#SPJ2
Use a substitution method to solve both of the following DEs, stating the general solution clearly, and showing all work clearly. a. dy/dx - y = e^xy^2 (Solve explicitly for y.) b. dy/dx = x + y/x - y (You can leave the General Solution in implicit form.)
[tex]1.\rightarrow \frac{dy}{dx}-y=e^x y^2\\\\\rightarrow \frac{1}{y^2}\frac{dy}{dx}-\frac{1}{y}=e^x\\\\ \text{put},\frac{-1}{y}=z\\\\ \frac{dy}{y^2} =d z\\\\ \frac{dy}{dx} \times \frac{1}{y^2}=\frac{dz}{dx}\\\\\frac{dz}{dx} +z=e^x\\\\ \text{Integrating factor}=e^{\int {1} \, dx}\\\\=e^x \\\\ \text{Multiplying both sides by }e^x\\\\e^x(\frac{dz}{dx} +z)=e^{2x}\\\\ \text{Integrating both sides}\\\\z e^x=\frac{e^{2x}}{2}+C\\\\ \frac{-e^x}{y}=\frac{e^{2x}}{2}+C[/tex]
-----------------------------------------------------------------------------------------------------------------
[tex]\rightarrow \frac{dy}{dx}=x+\frac{y}{x}-y\\\\\rightarrow \frac{dy}{dx}-x=\frac{y}{x}-y\\\\\rightarrow \frac{dy}{dx}+y(1-\frac{1}{x})=x\\\\\text{Integrating factor}=e^{\int{1-\frac{1}{x}}\,dx}\\\\=e^{x-\log x}\\\\ \text{Multiplying both sides by} e^{x-\log x}\\\\e^{x-\log x}\times[\frac{dy}{dx}+y(1-\frac{1}{x})]=x \times e^{x-\log x}\\\\y\times e^{x-\log x} =\int x \times e^{x-\log x} \, dx}\\\\y\times e^{x-\log x}=\int x \times \frac{e^x}{e^{\log x}}\,dx\\\\y\times e^{x-\log x}=\int x \times \frac{e^x}{x} \, dx\\\\y\times e^{x-\log x}=e^x+K[/tex]
Find the future value (FV) of the annuity due. (Round your answer to the nearest cent.) $175 monthly payment, 7% interest, 11 years
Answer:
The future value of the annuity due to the nearest cent is $2956.
Step-by-step explanation:
Consider the provided information:
It is provided that monthly payment is $175, interest is 7% and time is 11 years.
The formula for the future value of the annuity due is:
[tex]FV of Annuity Due = (1+r)\times P[\frac{(1+r)^{n}-1}{r}][/tex]
Now, substitute P = 175, r = 0.07 and t = 11 in above formula.
[tex]FV of Annuity Due = (1+0.07)\times 175[\frac{(1+0.07)^{11}-1}{0.07}][/tex]
[tex]FV of Annuity Due = (1.07)\times 175[\frac{1.10485}{0.07}][/tex]
[tex]FV of Annuity Due = 187.5(15.7835)[/tex]
[tex]FV of Annuity Due = 2955.4789[/tex]
Hence, the future value of the annuity due to the nearest cent is $2956.
Subjects for the next presidential election poll are contacted using telephone numbers in which the last four digits are randomly selected (with replacement). Find the probability that for one such phone number, the last four digits include at least one 0.
Answer: 0.3439
Step-by-step explanation:
Total number of digits : 10
The number of digits except zero =9
The probability of selecting non-zero number :-
[tex]\dfrac{9}{10}=0.9[/tex]
Then, the probability that one such phone number , the last four digits do not include any zero:-
[tex]\text{P(no zero )}=(0.9)^4[/tex]
Then , the probability that for one such phone number, the last four digits include at least one 0:-
[tex]\text{P(at-least one zero )}=1-(0.9)^4=0.3439[/tex]
Hence, the probability that for one such phone number, the last four digits include at least one 0 is 0.3439 .
To find the probability of at least one 0 in a four-digit number, one can use the complement of the probability of no 0s occurring in any of the digits. The probability of at least one 0 is approximately 34.39%.
Explanation:The student is inquiring about the probability of a certain event, specifically related to randomly generating telephone numbers. To find the probability that at least one digit is a 0 in a randomly generated four-digit number, we can use the complementary probability approach. The probability of not getting a 0 in one digit is 9/10 since there are 9 other possible digits (1-9). Hence, the probability of not getting a 0 in any of the four digits is (9/10)^4. Subtracting this from 1 will give the probability of having at least one 0 in the four-digit sequence:
Probability of at least one 0 = 1 - (Probability of no 0 in any digit)
Probability of at least one 0 = 1 - (9/10)^4
After calculating, we have:
Probability of at least one 0 = 1 - 0.6561 = 0.3439
Therefore, the probability of having at least one 0 in a randomly chosen telephone number with four digits is approximately 34.39%.
. CAR WASH Shea and Tucker are washing their father's car. Shea can wash it by herself in 20 minutes. Tucker can wash it by himself in 30 minutes. 19 How long does it take them to wash the car if they work together?
Answer:
12 minutes
Step-by-step explanation:
First we figure out how much of a car each person can wash in 1 minute.
Shea can wash the car by herself in 20 minutes
Therefore, she can wash a car in 1 minute = [tex]\frac{1}{20}[/tex]
Tucker can wash a car in 30 minutes.
Tucker can wash the car in 1 minute = [tex]\frac{1}{30}[/tex]
Thus, in one minute together they wash a car
[tex]\frac{1}{20}[/tex] + [tex]\frac{1}{30}[/tex] = [tex]\frac{50}{600}[/tex] = [tex]\frac{1}{12}[/tex]
In one minute together they can wash = [tex]\frac{1}{12}[/tex] of a car
Time needed to wash entire car together = 12 minutes.
It takes them 12 minutes to wash the car.
Final answer:
Shea and Tucker can wash their father's car in 12 minutes if they work together.
Explanation:
To find out how long it takes Shea and Tucker to wash the car together, we can use the concept of work rates.
Shea can wash the car by herself in 20 minutes, so her work rate is 1/20 of the car per minute.
Tucker can wash the car by himself in 30 minutes, so his work rate is 1/30 of the car per minute.
When two people work together, their work rates are additive. So, if Shea and Tucker work together, their combined work rate is 1/20 + 1/30 = 1/12 of the car per minute.
Since the work rate is the reciprocal of the time taken, the time it takes them to wash the car together is 12 minutes. Therefore, it takes Shea and Tucker 12 minutes to wash their father's car if they work together.
Find X.
A.124
B.138
C.282
D.69
To find x, find the difference between the outer angle and middle angle, then divide by two.
X = (210 - 72) / 2
x = 138 / 2
x = 69
The answer is D.
Answer
subtract 210 with 72 and then divide that by 2 and you get 138. so x=138.
Answer all questions: 1) The electric field of an electromagnetic wave propagating in air is given by E(z,t) = 4cos(6 x 10^8 t - 2z) +3 sin(6 x 10 t -2z) (V/m). Find the associated magnetic field H(z,t)
The magnetic field H(z,t) of an electromagnetic wave is related to the electric field E(z,t) by a factor of the speed of light. Therefore, if E(z,t) = 4cos(6 x 10^8 t - 2z) +3 sin(6 x 10^8 t -2z), the associated magnetic field would be H(z,t) = (4/c) cos(6 x 10^8 t - 2z) +(3/c) sin(6 x 10^8 t -2z), where c is speed of light, approximately 3 x 10^8 m/s.
Explanation:The question is asking for the associated magnetic field H(z,t) of an Electromagnetic wave given the electric field E(z,t). A crucial fact to know for this question is that the electric and magnetic fields in an electromagnetic wave are perpendicular to each other and the direction of propagation. They also have a constant ratio of magnitudes in free space or air, which is the speed of light given by c = 1/√εOMO. Because of these relations, we know that we can find the magnetic field by simply dividing the given electric field by the speed of light in units that match the given Electric field.
So, if E(z,t) = 4cos(6 x 10^8 t - 2z) +3 sin(6 x 10^8 t -2z), then the associated magnetic field would be H(z,t) = (4/c) cos(6 x 10^8 t - 2z) +(3/c) sin(6 x 10^8 t -2z), where c is the speed of light, approximately 3 x 10^8 m/s.
Learn more about Electromagnetic Waves here:https://brainly.com/question/29774932
#SPJ12
To find the associated magnetic field H(z, t), you can use Faraday's law of electromagnetic induction. This law states that the rate of change of magnetic flux through a surface is equal to the induced electromotive force (EMF) along the boundary of the surface. By following a step-by-step process, you can find the magnetic field B(z, t) using the given electric field E(z, t).
Explanation:The associated magnetic field H(z, t) can be found by using Faraday's law of electromagnetic induction. Faraday's law states that the rate of change of magnetic flux through a surface is equal to the electromotive force (EMF) along the boundary of the surface. In this case, the magnetic field is changing due to the time-dependent electric field, so we can use Faraday's law to find the magnetic field.
Start by finding the magnetic flux through a surface with an area A in the z-direction.The magnetic field B is perpendicular to the surface, so the magnetic flux is given by Φ = B * A.By Faraday's law, the rate of change of magnetic flux is equal to the induced EMF around the boundary of the surface. In this case, the induced EMF is caused by the changing electric field.From the given electric field E(z, t), we can differentiate it with respect to time to find the rate of change, which gives us the induced EMF.Equating the rate of change of magnetic flux to the induced EMF, we can solve for the magnetic field B(z, t).By following these steps, you can find the associated magnetic field H(z, t) using Faraday's law of electromagnetic induction.
Learn more about Faraday's law here:https://brainly.com/question/1640558
#SPJ12
The recommended single dose for a particular drug is 100 mg. How many doses can be obtained from 20 grams of the drug?
A. 2,000 doses
B. 200 doses
C. 2 doses
D. 20 doses
Answer:
B. 200 doses
Step-by-step explanation:
Given,
1 dose is required for 100 mg,
Since, 1 mg = 0.001 g,
⇒ 100 mg = 0.1 g
⇒ 1 dost is required for 0.1 g,
Thus, the ratio of doses and quantity ( in gram ) is [tex]\frac{1}{0.1}=10[/tex]
Let x be the doses required for 20 grams,
So, the ratio of doses and quantity is [tex]\frac{x}{20}[/tex]
[tex]\implies \frac{x}{20}=10[/tex]
[tex]\implies x=200[/tex]
Hence, 200 doses can be obtained from 20 grams of the drug.
Option 'B' is correct.
To solve this question, we will follow these steps:
1. We need to ensure that we use the same units for both the drug amount and the dose. Since the drug amount is given in grams and the dose in milligrams, we will convert the drug amount from grams to milligrams.
2. We know that 1 gram is equivalent to 1000 milligrams.
3. Now, let's convert the drug amount from grams to milligrams:
We have 20 grams of the drug, so the conversion to milligrams is:
\(20 \text{ grams} \times \dfrac{1000 \text{ milligrams}}{1 \text{ gram}} = 20,000 \text{ milligrams}\)
4. Next, we will divide the total milligrams of the drug by the milligram dosage that is recommended per dose to find out how many doses we can get from the drug amount.
5. Given that each dose is 100 mg, we now divide the total drug amount in milligrams by the dose in milligrams:
\(20,000 \text{ milligrams} \div 100 \text{ milligrams per dose} = 200 \text{ doses}\)
Therefore, from 20 grams of the drug, we can obtain 200 doses.
The correct answer is:
B. 200 doses
Give the largest interval I over which the general solution is defined. PLEASE EXPLAIN HOW!!!
(x^2-1)dy/dx+2y=(x+1)^2
Divide both sides by [tex]x^2-1[/tex] to get a linear ODE,
[tex]\dfrac{\mathrm dy}{\mathrm dx}+\dfrac2{x^2-1}y=\dfrac{x+1}{x-1}[/tex]
In order for this operation to be valid in the first place, we require that [tex]x\neq\pm1[/tex] (since that would make [tex]\dfrac1{x^2-1}[/tex] undefined, which we don't want to happen). Then we are forcing any solution to the ODE to exist on any of the three intervals, [tex](-\infty,-1)[/tex], [tex](-1, 1)[/tex], or [tex](1,\infty)[/tex], and either the first or third of these can be chosen as the largest interval.
In case you also need to solve the ODE: Multiply both sides by [tex]\dfrac{1-x}{1+x}[/tex], so that
[tex]\dfrac{1-x}{1+x}\dfrac{\mathrm dy}{\mathrm dx}-\dfrac2{(1+x)^2}y=-1[/tex]
Then the left side can be condensed as the derivative of a product, since
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac{1-x}{1+x}\right]=-\dfrac2{(1+x)^2}[/tex]
and we have
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac{1-x}{1+x}y\right]=-1[/tex]
Integrate both sides:
[tex]\displaystyle\int\frac{\mathrm d}{\mathrm dx}\left[\frac{1-x}{1+x}y\right]\,\mathrm dx=-\int\mathrm dx[/tex]
[tex]\dfrac{1-x}{1+x}y=-x+C[/tex]
[tex]\implies\boxed{y=\dfrac{(-x+C)(1+x)}{1-x}}[/tex]
The largest interval over which the general solution is defined for the given differential equation is [-1, ∞).
Here's how:
Rewrite the differential equation in proper form.Analyze the coefficients to determine the interval of definition.In this case, the interval is determined by the denominator of the coefficient of dy/dx.Solve the Method of variation of Parameters. y" - 3y' + 2y = 4e^3t
Answer:
CF+PI=[tex]c_1e^{2x}+c_2e^{x}[/tex]+[tex]2e^{3t}[/tex]
Step-by-step explanation:
we have given y"-3y'=2y=[tex]4e^{3t}[/tex]
this differential equation solution have two part that CF and PI
CALCULATION OF CF :
[tex]m^2-3m+2=0[/tex]
[tex]m^2-2m-m+2=0[/tex]
[tex](m-1)(m-2)=0[/tex]
m=1 and m=2
so CF=[tex]c_1e^{2x}+c_2e^{x}[/tex]
CALCULATION OF PI :
PI = [tex]\frac{4e^{3t}}{(m-1)(m-2)}[/tex]
at m= 3 in PI
[tex]PI=\frac{4e^{3t}}{2}=2e^{3t}[/tex]
so the complete solution is
CF+PI=[tex]c_1e^{2x}+c_2e^{x}[/tex]+[tex]2e^{3t}[/tex]
point) Suppose that the trace of a 2 x 2 matrix A is tr(A)= -9 and the determinant is det(A) 18. Find the eigenvalues of A. The eigenvalues of A are (Enter your answers as a comma separated list.)
Answer with explanation:
Matrix A= (2 × 2) Matrix
Trace A= -9
Also,Determinant A= |A|=18
⇒Characteristics Polynomial is given by
Δ(A)=A² -A ×trace (A)+Determinant (A)
=A²+9 A+18
=(A+6)(A+3)
So, eigenvalues can be obtained by substituting :
Δ(A)=0
(A+6)(A+3)=0
A= -6 ∧ A= -3
Two Eigenvalues are = -6, -3
Find the? inverse, if it? exists, for the given matrix.
@MATX{{3;3;-1};{-12;-12;4};{2;6;0}}
Answer:
Step-by-step explanation:
The Inverse of the matrix doesn't exist because the determinant is equal to 0.
Answer:
The inverse of given matrix is not exist, since determinant is 0.
Step-by-step explanation:
The inverse of a square matrix [tex]A[/tex] is [tex]A^{-1}[/tex] such that
[tex]A A^{-1}=I[/tex] where I is the identity matrix.
Consider, [tex]A = \left[\begin{array}{ccc}3&3&-1\\-12&-12&4\\2&6&0\end{array}\right][/tex]
[tex]\mathrm{Matrix\:can\:only\:be\:inverted\:if\:it\:is\:non-singular,\:that\:is:}[/tex]
[tex]\det \begin{pmatrix}3&3&-1\\ -12&-12&4\\ 2&6&0\end{pmatrix}\ne 0[/tex]
[tex]\det \begin{pmatrix}3&3&-1\\ -12&-12&4\\ 2&6&0\end{pmatrix}[/tex]
[tex]\mathrm{Find\:the\:matrix\:determinant\:according\:to\:formula}:\quad \:[/tex]
[tex]\det \begin{pmatrix}a&b&c\\ d&e&f\\ g&h&i\end{pmatrix}=a\cdot \det \begin{pmatrix}e&f\\ h&i\end{pmatrix}-b\cdot \det \begin{pmatrix}d&f\\ g&i\end{pmatrix}+c\cdot \det \begin{pmatrix}d&e\\ g&h\end{pmatrix}[/tex]
[tex]=3\cdot \det \begin{pmatrix}-12&4\\ 6&0\end{pmatrix}-3\cdot \det \begin{pmatrix}-12&4\\ 2&0\end{pmatrix}-1\cdot \det \begin{pmatrix}-12&-12\\ 2&6\end{pmatrix}[/tex]
[tex]=3\left(-24\right)-3\left(-8\right)-1\cdot \left(-48\right)[/tex]
[tex]3\left(-24\right)-3\left(-8\right)-1\cdot \left(-48\right)=0[/tex]
Therefore, the inverse of given matrix is not exist, since determinant is 0.
Completion time (from start to finish) of a building remodeling project is normally distributed with a mean of 200 work-days and a standard deviation of 10 work-days. To be 99% sure that we will not be late in completing the project, we should request a completion time of _______ work-days.
Answer:
233 days.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 200, \sigma = 10[/tex]
To be 99% sure that we will not be late in completing the project, we should request a completion time of ...
This is the value of X when Z has a pvalue of 0.99. So X when Z = 2.325.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]2.325 = \frac{X - 200}{10}[/tex]
[tex]X - 200 = 10*2.325[/tex]
[tex]X = 232.5[/tex]
So the correct answer is 233 days.
d) neither one-to-one nor onto. 15. Determine whether each of these functions is a bijection from R to R. a) f(x)=2x+1 b) f(x)=x2+1 c) f(x) r3 d) f(x) (x2 +1)/(r2 +2) a function f(x)-ex from the set of real
Answer:
The only bijection is f(x)=2x+1.
I took r to be a constant.
Step-by-step explanation:
Bijections are both onto and one-to-one.
Onto means every element of the codomain gets hit. Here the codomain is the set of real numbers. So you want every y to get hit.
One-to-one means you don't want any y to get hit more than once.
f(x)=2x+1 is a linear function. It is diagonal line so every element of the codomain will get hit and hit only once so this function is onto and one-to-one.
f(x)=x^2+1 is a quadratic function. It is parabola so not every element of our codomain will get not get hit and of those that do get hit they get hit more than once. So this is neither onto or one-to-one.
f(x)=r^3 is a constant function. It is a horizontal line so not every y will get hit so it isn't onto. The same y is being hit multiple times so it isn't one-to-one.
f(x)=(x^2+1)/(r^2+2) is a quadratic. It is a parabola. Quadratic functions are not onto or one-to-one.
1000 mL of D5W is ordered to be infused over 5 hours. The drop factor is 10 gtt/mL. How many gtt/min should be given to infuse the 1000 mL over 5 hours?
Answer:
flow rate of to infuse the 1000 mL over 5 hours is 33.33 gtt/min
Step-by-step explanation:
Given data
volume = 1000 mL
time = 5 hours
drop factor = 10 gtt/min
to find out
flow rate of to infuse the 1000 mL over 5 hours
Solution
we know flow rate formula i.e.
flow rate = ( volume × drop factor ) / time .................1
here time will be in min so time = 5 hours = 5 × 60 = 300 min
put volume, drop factor and time value in equation 1 and we get flow rate
flow rate = ( volume × drop factor ) / time
flow rate = ( 1000 × 10 ) / 300
flow rate = 33.33 gtt/min
flow rate of to infuse the 1000 mL over 5 hours is 33.33 gtt/min
Write equations for the vertical and horizontal lines passing through the point . (7,5)
A horizontal line is a line where all of the [tex]y[/tex] values are the same. In this case, [tex]\boxed{y=5}[/tex], so that is the equation.
A vertical line is where all of the [tex]x[/tex] values are the same. Here, [tex]\boxed{x=7}[/tex], so that's the equation.
Answer:
see below
Step-by-step explanation:
A horizontal line has the same y value and has a constant y value
y=5
A vertical line has the same x value and has a constant x value
x=7
Find all values of x that are NOT in the domain of h.
If there is more than one value, separate them with commas
h(x) = x + 1 / x^2 + 2y + 1
Answer:
if x= -1 then its is NOT in the domain of h.
Step-by-step explanation:
Domain is the set of values for which the function is defined.
we are given the function
[tex]h(x) =\frac{x+1}{x^2 + 2x + 1}[/tex]
Solving the denominator by factorization
[tex]h(x) =\frac{x+1}{x^2 + x+x + 1}\\h(x) =\frac{x+1}{x(x+1)+1(x + 1)}\\h(x) =\frac{x+1}{(x+1)(x+1)}\\h(x) =\frac{1}{(x+1)}[/tex]
So, if x = -1 then its is NOT in the domain of h.
A civil service exam yields scores which are normally distributed with a mean of 81 and a standard deviation of 5.5. If the civil service wishes to set a cut-off score on the exam so that 15% of the test takers fail the exam, what should the cut-off score be? Remember to round your z-value to 2 decimal places.
Answer:
The cutoff score should be 75.8 marks
Step-by-step explanation:
The cut-off should be set as the value corresponding to an area 15% in the normal distribution diagram.
Using the standard distribution tables we have value of standard normal deviate (Z) corresponding to area of 15% = -1.04
Thus we have
[tex]-1.04=\frac{X-\bar{X}}{\sigma }\\\\X=-1.04\times 5.5+81\\X=75.8[/tex]
Find the amount of time needed for the sinking fund to reach the given accumulated amount. (Round your answer to two decimal places.) $295 monthly at 5.2% to accumulate $25,000.
Answer:
8.82 years.
Step-by-step explanation:
Since, the monthly payment formula is,
[tex]P=\frac{PV(r)}{1-(1+r)^{-n}}[/tex]
Where, PV is the present value of the loan,
r is the rate per month,
n is number of months,
Here,
PV = $ 25,000,
Annual rate = 5.2 % = 0.052 ⇒ Monthly rate, r = [tex]\frac{0.052}{12}[/tex]
( 1 year = 12 months )
P = $ 295,
By substituting the values,
[tex]295=\frac{25000(\frac{0.052}{12})}{1-(1+\frac{0.052}{12})^{-n}}[/tex]
By the graphing calculator,
We get,
[tex]n = 105.84[/tex]
Hence, the time ( in years ) = [tex]\frac{105.84}{12}=8.82[/tex]
what is the solution if the inequality shown below? a-1>11
Answer:
a > 12
Step-by-step explanation:
Isolate the variable a. Treat the > sign like the = sign. What you do to one side, you do to the other. Add 1 to both sides:
a - 1 (+1) > 11 (+1)
a > 11 + 1
a > 12
a > 12 is your answer.
~
Answer:
[tex]\huge \boxed{a>12}[/tex]
Step-by-step explanation:
Add by 1 from both sides.
[tex]\displaystyle a-1+1>11+1[/tex]
Simplify, to find the answer.
[tex]\displaystyle 11+1=12[/tex]
[tex]\displaystyle a>12[/tex], which is our answer.
The manufacturer of a certain engine treatment claims that if you add their product to your engine, it will be protected from excessive wear. An infomercial claims that a woman drove 33 hours without oil, thanks to the engine treatment. A magazine tested engines in which they added the treatment to the motor oil, ran the engines, drained the oil, and then determined the time until the engines seized. Determine the null and alternative hypotheses that the magazine will test.
Answer: [tex]H_0:\mu\geq33[/tex]
[tex]H_a:\mu<33[/tex]
Step-by-step explanation:
Let [tex]\mu[/tex] be the average number of hours a person drive without adding the product.
Given claim : An infomercial claims that a woman drove 33 hours without oil.
i.e. [tex]\mu\geq33[/tex]
It is known that the null hypothesis always contains equal sign and alternative hypothesis is just opposite of the null hypothesis.
Thus the null and alternative hypothesis for the given situation will be :-
[tex]H_0:\mu\geq33[/tex]
[tex]H_a:\mu<33[/tex]