The solution of the initial-value problem is y(x) = 2e-x + 2eⁿx.
The given differential equation is 4x²y'' + y = 0.
The substitution is t = -x.
Thus, x = -t, and therefore dx/dt = -1.
Solve for y' and y'' in terms of t instead of x. Using the chain rule,
y' = dy/dx × dx/dt = dy/dt × (-1) = -y'.y'' = d²y/dx² × (dx/dt)² + dy/dx × d²x/dt² = d²y/dt² + y′.
We have to replace y'' and y' in the differential equation and simplify it.
4x²y'' + y = 0 becomes 4t²(y'' - y') + y = 0.
We now have a first-order homogeneous linear differential equation. The characteristic equation of the differential equation is r² - 1 = 0.
Solving the characteristic equation, we get r = ±1.
Therefore, the general solution is y(t) = c₁et + c₂e-t.
This solution is for the interval (-[infinity], infinity). We need to solve for the given initial conditions to find the particular solution.
Using the first initial condition, we get4 = y(-1) = c₁e-1 - c₂e1 ⇒ c₁e-1 - c₂e1 = 4.
Using the second initial condition, we get-4 = y'(-1) = -c₁e-1 - c₂e1 ⇒ c₁e-1 + c₂e1 = 4.
The system of linear equations is given by[c₁e-1, -c₂e1] × [1, 1; -1, 1] = [4, -4]
Solving for c₁ and c₂, we getc₁ = 2e, c₂ = 2e-1.
The particular solution is
y(t) = 2eet + 2e-t.
Using the substitution x = -t, we gety(x) = 2e-x + 2eⁿx.
The solution of the initial-value problem is y(x) = 2e-x + 2eⁿx.
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A hemisphere with diamater 17 cm
Answer:
not enough information
Step-by-stepexplanation:
A running track consists of two parallel lines that are connected at each end but the curved boundary of a semicircle. The parallel lines are 30 meters long and 7 meters apart. Fine the area inside the running track.
Answer:
The area inside the running track is 248.5 m².
Step-by-step explanation:
The area inside the running track is given by the sum of the area of a rectangle and the area of a semicircle:
[tex] A_{t} = A_{r} + 2A_{c} [/tex]
[tex] A_{t} = x*(2r) + 2*(\frac{\pi r^{2}}{2}) [/tex]
Where:
x: is one side of the rectangle = 30 m
r: is the radius = half of the other side of the rectangle = 7/2 m = 3.5 m
Hence, the total area is:
[tex] A_{t} = 30*7 + \pi (3.5)^{2} = 248.5 m^{2} [/tex]
Therefore, the area inside the running track is 248.5 m².
I hope it helps you!
An assembly operation in a manufacturing plant requiresapproximately a 1- month training period for a new employee toreach maximum efficiency in assembling a device. A new method oftraining was suggested, and a test was conducted to compare the newmethod with the standard procedure. Two groups of nine newemployees were trained for a period of 3 weeks, one group using thenew method and the other following the standard procedure. Thelength of time (in minutes) required for each employee to assemblethe device was recorded at the end of the 3-week period. Thesemeasurements appear in the table below.
Do the stat present sufficient evidence to indicate that the meantime to assemble at the end of a 3 week training period is less forthe new training procedure?
Standard Procedure
New Procedure
32
35
37
31
35
29
28
25
41
34
44
40
35
27
31
32
34
31
a. Perform the test using the classical approach α=0.05.
b. Perform the above test assuming population variances areequal.
c. Perform the above test (#2) using the p-value approach.(You will not be able to find an exact p-value but you will be ableto find an interval within which the p-value falls. This will beenough for you to make a decision.)
P-value is less than the level of significance α = 0.05, we reject the null hypothesis and conclude that the mean time to assemble at the end of a 3-week training period is less for the new training procedure.
a. Performing the test using the classical approach α=0.05
Hypothesis test:
Null hypothesis H₀: The mean time for the new procedure = the mean time for the standard procedure
Alternative hypothesis H₁: The mean time for the new procedure < the mean time for the standard procedure (one-tailed test)
Level of significance α = 0.05
Since population standard deviations are unknown, we will use the t-test to conduct the hypothesis test.
Below is the calculation of the t-test.(For detailed calculation of t-test, please refer to the image attached)
Using a t-distribution table with df = 14 (n1 + n2 - 2), the critical value for a one-tailed test at α = 0.05 is -1.761.
The test statistic (-2.029) is less than the critical value (-1.761), so we reject the null hypothesis and conclude that the mean time to assemble at the end of a 3-week training period is less for the new training procedure.
b.Performing the above test assuming population variances are equal
Hypothesis test
Null hypothesis H₀: The mean time for the new procedure = the mean time for the standard procedure
Alternative hypothesis H₁: The mean time for the new procedure < the mean time for the standard procedure (one-tailed test)Level of significance α = 0.05
Since population variances are assumed to be equal, we will use the pooled t-test to conduct the hypothesis test.
Below is the calculation of the pooled t-test.(For detailed calculation of pooled t-test, please refer to the image attached)
Using a t-distribution table with df = 16 (n₁ + n₂ - 2), the critical value for a one-tailed test at α = 0.05 is -1.746.
The test statistic (-2.029) is less than the critical value (-1.746), so we reject the null hypothesis and conclude that the mean time to assemble at the end of a 3-week training period is less for the new training procedure.
c. Performing the above test (#2) using the p-value approach.
Hypothesis test:
Null hypothesis H₀: The mean time for the new procedure = the mean time for the standard procedure
Alternative hypothesis H₁: The mean time for the new procedure < the mean time for the standard procedure (one-tailed test)
Level of significance α = 0.05
Since population variances are assumed to be equal, we will use the pooled t-test to conduct the hypothesis test.
The test statistic is -2.029, which corresponds to a p-value of 0.0328.
Since the p-value is less than the level of significance α = 0.05, we reject the null hypothesis and conclude that the mean time to assemble at the end of a 3-week training period is less for the new training procedure.
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You just obtained a credit card. You immediately purchase a stereo system for $200. your credit limit is $1000. Let's assume that you make no payments and purchase nothing more and there are no other fees. The monthly interest rate is 1.42%.
What is the growth rate of your credit card balance?
A. 0.42
B. 0.0142
C. 14.2
D. 142
Answer:
its b
Step-by-step explanation:
have a great day <3
The growth rate of your credit card balance is approximately 0.0142.
Therefore, the correct option is B.
Given that you got a credit card, you purchased a stereo system for $200. your credit limit is $1000.
The monthly interest rate is 1.42%.
We need to determine the growth rate of your credit card balance,
To calculate the growth rate of your credit card balance, we need to consider the monthly interest rate and the initial purchase amount.
The monthly interest rate is 1.42%, which can be expressed as a decimal by dividing it by 100: 0.0142.
The initial purchase amount is $200.
Assuming no payments are made and no additional purchases are made, the credit card balance will increase each month due to the accrued interest.
To calculate the growth rate, we can divide the accrued interest by the initial purchase amount.
Accrued interest = Monthly interest rate x Initial purchase amount
= 0.0142 x $200
= $2.84
Growth rate = Accrued interest / Initial purchase amount
= $2.84 / $200
≈ 0.0142
Therefore, the growth rate of your credit card balance is approximately 0.0142, which corresponds to option B.
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Newborn babies weigh an average of 7.5 pounds with a standard deviation of 1.25 pounds. Find the 2-score for a baby who weighs 6.70 pounds. Round your answer to 2 decimal places. I
This indicates that the baby's weight is 0.64 standard deviations below the mean weight of newborn babies.
What is the sample size required to estimate the population mean with a given margin of error, confidence level, and population standard deviation?To calculate the z-score, we use the formula:
z = (x - μ) / σx is the observed value (weight of the baby),μ is the mean (average weight of newborn babies),σ is the standard deviation.In this case, the observed weight is 6.70 pounds, the mean weight is 7.5 pounds, and the standard deviation is 1.25 pounds.
Plugging these values into the formula, we get:
z = (6.70 - 7.5) / 1.25Calculating this, we find that the z-score is approximately -0.64.
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Cheryl moves houses. Her old house is 3 kilometers from her new house. How many meters is it from the old house to the new house?
Answer:
3000 here hope this helps you out
D. The cost to play the game is $2. The winner gets $6 for making purple. How much profit would the school make from 18 people playing the game?
Answer:
$30
Step-by-step explanation:
1 person = $ 2
18 person = 2 × 18 = $36
winner = $6
profit = 36 - 6 = $30
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I’m gonnna fail :(
What is the quotient of 1,234 and 4?
Answer:
308.5 lol
Step-by-step explanation:
Ben wants to buy a new car stereo and he has already saved some money. He used this inequality to represent the amount he still has to save to be able to buy the stereo, where a represents the amount still left to save.
a + 212 greater-than-or-equal-to 365
If Ben saves $15 a week for the next 10 weeks, will he be able to buy the stereo and why?
Answer:
Since Ben would have $362, he won't be able to buy the stereo as he needs to have an amount greater than or equal to 365.
Step-by-step explanation:
First, you need to determine the amount that Ben would have after saving $15 for the next 10 weeks:
$15*10=$150
Now, the inequality indicates that the amount left to save plus 212 should be greater than or equal to 365, so you have to add up 212 plus the amount Ben will save in the next 10 weeks to be able to determine if he would be able to buy the stereo:
150+$212=$362
Since Ben would have $362, he won't be able to buy the stereo as he needs to have an amount greater than or equal to 365.
In the following diagram, parallelogram ABCD contains triangle CED. Line CE and line ED intersect at point E. The measures of ∠CEA and ∠CDE are shown. What is the measure of x?______ degrees *
Answer:
∠x = 59°
Step-by-step explanation:
∠ECD = 52° because they are alternate angles and will be equal
∠x = 180° - 52° - 69° = 59° because the sum of the interior angles of a triangle = 180°
Jenna earns $8.75 per hour working at the mall. If Jenna worked 25 1/5 hours last week, how much did she earn before taxes were taken out?
Answer:
$220.5
Step-by-step explanation:
To answer this simply multiply the number of hours (25.2) times the hourly wage 8.75 and you'll get the answer
25.2 times 8.75 equals 220.5
can someone please help me with this problem?
Answer:
14
Step-by-step explanation:
First you figure out what the value of x is, and then you must add the numbers in the Participates in Sports column to get 14
Simplify:
5a+6a^2+2a^0-3a^2-9a
3a^2-4a+2
-a^3+2
-a^2+2
3a^2-4a+1
Answer:
I believe that the answer is '3a^2-4a+2
. A loan worth 500,000 pesos is payable in 10 years at an effective annual interest rate of 18%. At the end of each year, the borrower pays 50,000 pesos in principal, which is 1/10 of the loan amount, along with the interest due. Find a formula for the kth payment Pₖ. Then construct an amortization schedule.
The formula for the kth payment Pₖ is Pₖ = (1/10 * PV) + (PV * r).
To find a formula for the kth payment Pₖ, we can start by calculating the monthly interest rate and the total number of payments.
Given that the loan is payable in 10 years, we have a total of 10 payments. The loan amount PV is 500,000 pesos, and the effective annual interest rate r is 18%.
First, let's calculate the monthly interest rate:
Monthly interest rate = (1 + r)^(1/12) - 1
Substituting the values, we have:
Monthly interest rate = (1 + 0.18)^(1/12) - 1 ≈ 1.4337% or 0.014337
Now, let's find the kth payment Pₖ. Since the borrower pays 50,000 pesos in principal at the end of each year, which is 1/10 of the loan amount, we can modify the formula to:
Pₖ = (1/10 * PV) + (PV * r)
Substituting the given values, we have:
Pₖ = (1/10 * 500,000) + (500,000 * 0.014337)
Simplifying, we get:
Pₖ ≈ 50,000 + 7,168.5 ≈ 57,168.5 pesos
This formula gives the kth payment Pₖ for any specific year during the loan term.
Now, let's construct an amortization schedule for this loan:
Year | Payment | Principal | Interest | Balance
--------------------------------------------------
1 | 57,168.5 | 50,000 | 7,168.5 | 450,000
2 | 57,168.5 | 50,000 | 7,168.5 | 400,000
3 | 57,168.5 | 50,000 | 7,168.5 | 350,000
...
10 | 57,168.5 | 50,000 | 7,168.5 | 0
In each year, the principal payment remains constant at 50,000 pesos, and the interest payment gradually decreases as the outstanding balance decreases. The balance reaches zero after 10 years, indicating that the loan has been fully paid off.
Please note that the actual schedule may vary slightly due to rounding errors and the specific date of the first payment.
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What is the slope, m, of the line? Enter your answer as a decimal.
Round to the nearest hundredth.
Answer:
4.56
Step-by-step explanation:
The slope (m) of the line given in the graph is -0.332.
In the given graph coordinates of the line are given, that is (2,0) and (-4, 2).
How to find the slope of the line?Pick two points on the line and determine their coordinates. Determine the difference in y-coordinates of these two points (rise). Determine the difference in x-coordinates for these two points (run). Divide the difference in y-coordinates by the difference in x-coordinates (rise/run or slope).
Now, the slope of the line=m=0-2/2-(-4)=-2/6=-0.33333
-0.33333≈-0.332
Therefore, the slope (m) of the line given in the graph is -0.332.
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Given m
|| n, find the value of x.
т
n
(6x+19)°
(4x+11)
a
Help what is X
Answer:
x=15
supplementary angle equals 180
Write an equation for this problem ( tell if it’s linear or exponential)
The drawing shows a semicircular window separated into 3 sections of colored glass
Approximately how many square inches of the tinted glass are red
Answer:
141.3
Step-by-step explanation:
27.78% of the total area is covered with red.
What is the area of a sector? What is a expression? What is a mathematical equation?A part of a circle is called sector. The area of a sector is given by -(Ф/360) x πr²
A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions.Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.We have the given semicircle as shown in image.
Total green coloured area -
A[G] = 2 x {(Ф/360) x πr²}
A[G] = 2 x (40/360 x πr²}
A[G] = (2/9)πr²
Now -
A[R] = πr²/2 - (2/9)πr²
A[R] = πr²{1/2 - 2/9}
A[R]% = (πr²{1/2 - 2/9}/πr²) x 100
A[R]% = (1/2 - 2/9) x 100
A[R]% = (9 - 4)/18 x 100
A[R]% = 5/18 x 100
A[R]% = 500/18
A[R]% = 27.78%
Therefore, 27.78% of the total area is covered with red.
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The Zombie Apocalypse has come to Panama City Beach. It starts when 3 Spring Breakers become zombie after an encounter with a deadly fish. Waking up the next day, each Spring Breaker can infect 4 more people per day before needing a rest. Each of these zombies can, in turn, infect an additional 4 people per day. Write an Exponential Function to represent this situation. Then find out how many crazy Spring Break Zombies will be roaming the beach at the end of the week (Day 7)?
Answer:
By the end of the week there will be 49,152 zombies in Panama City Beach.
Step-by-step explanation:
Since the Zombie Apocalypse has come to Panama City Beach, starting when 3 Spring Breakers become zombie after an encounter with a deadly fish, and since waking up the next day, each Spring Breaker can infect 4 more people per day before needing a rest , and each of these zombies can, in turn, infect an additional 4 people per day, to determine how many crazy Spring Break Zombies will be roaming the beach at the end of the week the following exponential function has to be solved:
3 x 4 ^ 7 = X
3 x 16,384 = X
49.152 = X
Therefore, by the end of the week there will be 49,152 zombies in Panama City Beach.
The number of crazy Spring Break Zombies will be roaming the beach at the end of the week 7 are 49,152 and the equation for the situation is,
[tex]x=3\times(4)^7\\[/tex]
What is an exponential function?Exponential function is the function in which the function growth or decay with the power of the independent variable. The curve of the exponential function depends on the value of its variable.
The exponential function with dependent variable y and independent variable x can be written as,
[tex]y=ba^x+c[/tex]
Here, a, b and c are the real numbers.
The Zombie Apocalypse has come to Panama City Beach. It starts when 3 Spring Breakers become zombie after an encounter with a deadly fish.
In the next morning, each Spring Breaker can infect 4 more people per day before needing a rest. Each of these zombies can, in turn, infect an additional 4 people per day.
As total number of weeks are 7. Thus, for this situation, the exponential function for the variable x can be given as,
[tex]x=3\times(4)^7\\[/tex]
Now to find the number of crazy Spring Break Zombies will be roaming the beach at the end of the week 7, simplify the above equation as,
[tex]x=3\times(4)^7\\\\x=3\times16384\\x=49152[/tex]
Hence, the number of crazy Spring Break Zombies will be roaming the beach at the end of the week 7 are 49,152 and the equation for the situation is,
[tex]x=3\times(4)^7\\[/tex]
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need help with this one also
Answer:
scale factor 3 is the answer.
Step-by-step explanation:
Length = 18 in
Breadth = 3 in
height = 10 in
Volume of cuboid = l × b × h
V = 18 × 3 × 10
V = 540 in³
540 in³ is the original volume. The volume needs to be tripled, so multiply the original volume with 3
V = 540 × 3 = 1620 in³
∴ 1620 in³ will be the new volume if it's tripled and the scale factor is 3.
If your car gets 29 miles per gallon, how much does it cost to drive 300 miles when gasoline costs $3.20 per gallon?
If your car gets 29 miles per gallon, the cost to drive 300 miles when gasoline costs $3.20 per gallon is approximately $33.10.
One way to determine the cost of driving 300 miles is to divide the number of miles by the miles per gallon (MPG) and then multiply by the cost of gasoline per gallon.
Divide 300 miles by 29 MPG to get the total number of gallons of gasoline needed:
300 ÷ 29 ≈ 10.34 gallons
Round up to the nearest whole number to get 11 gallons
Multiply the number of gallons by the cost per gallon of gasoline:
$3.20/gallon x 11 gallons = $35.20
Therefore, it will cost $35.20 to drive 300 miles when gasoline costs $3.20 per gallon with a car that gets 29 miles per gallon.
We know that 1 US gallon equals 3.785411784 liters. Therefore, the car travels 29 x 3.785411784 = 109.724897 liters in 1 US gallon.
So, the cost of traveling 300 miles will be 300/29 = 10.34482759 gallons.
So, the cost of traveling 300 miles will be $3.2 x 10.34482759 = $33.10344828. Therefore, the cost of traveling 300 miles will be $33.10 approximately.
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A boat has a depreciation rate of 12% per year. If the original price of the boat was $15,000, what is the value of the boat 5 years later? Round your answer to the nearest penny (hundredths place value).
Answer: $7,915.98
Step-by-step explanation: This is an exponentially decaying problem, so that means we use the formula ab^x.
The a value is the original price or the starting point, which is 15,000. The b value is the ratio in which it decreases or increases. It is decreasing, so we subtract 12 from 100, which is 88. The ratio should be in decimal form, so it is 0.88. Finally, the x should be the exponent, the amount of times it is getting multiplied by the ratio. So the equation would be 15,000(0.88)^5.
Then, we solve.
0.88 * 0.88 * 0.88 * 0.88 * 0.88 = 0.5277319168
Then 0.5277319168 * 15,000 = 7,915.978752.
Of course, we round to the nearest hundredth, which is 7,915.98. Hope this helped.
HELP ME TEST QUESTION!! A bar graph is shown below. What percent of owners chose dog as their favorite pet?
The flywheel on a steam engine rotates 60 revolutions every 15 seconds what is the flywheel rate of revolution per second
Answer:
4 revolutions per second
Step-by-step explanation:
60 revs / 15 secs = 4 revs/sec
Answer:
4 revolutions per second
Step-by-step explanation:
60/15 = 4
16. Find the perimeter of AJKL.
J
8x - 35
N
5x - 8
77-9
K
Р
M M
2y + 11
o
32
L
Answer:
Parameter OF Δ JKL = 138 unit
Step-by-step explanation:
Given:
JN = 8x - 35
NK = 7y - 9
OK = 2y + 11
OL = 32 - [2y + 11]
JM = 5x - 8
Find:
Parameter OF Δ JKL
Computation:
We know that tangent from same points are always equal
So,
ML = OL
JM = JN
5x - 8 = 8x - 35
8x - 5x = 35 - 8
3x = 27
x = 9
So,
JN = 8x - 35
JN = 8(9) - 35
JN = 37 unit
JM = 5x - 8
JM = 5(9) - 8
JM = 37 unit
NK = OK
7y - 9 = 2y + 11
5y = 20
y = 4
NK = 7y - 9
NK = 7(4) - 9
NK = 19 unit
OK = 2y + 11
OK = 2(4) + 11
Ok = 19 unit
OL = 32 - [2(y) + 11]
OL = 32 - [2(4) + 11]
OL = 13 unit
So,
LM = OL = 13 unit
Parameter OF Δ JKL = JN + NK + OK + OL + LM + JM
Parameter OF Δ JKL = 37 + 19 + 19 + 13 + 13 + 37
Parameter OF Δ JKL = 138 unit
Answer:
Step-by-step explanation:
g
A combination lock with three dials, each numbered 1 through 8, is defective in that you only need to get two of the numbers right to open the lock. (For example, suppose the true combination is 4-2-7. Then 4-2-7 would open the lock, but so would 4-2-5, 4-2-2, 4-8-7 or 4-6-7. But not 2-4-7.)
Answer:
64 combination attempts
Step-by-step explanation:
Since each dial in the lock is numbered from 1-8 then there are 8 possibilities for each dial. Out of the three dials, only 2 actually need to be correct in order for the lock to open therefore, we simply raise the number of possibilities for each dial to the power of 2 which should give us the total number of tries we need in order to guarantee that it opens. Assuming that you are making the combinations in numerical order 1-1-1, 1-1-2, 1-1-3, etc.
8^2 = 64 combination attempts
The cone and cylinder above have the same radius, r, and height, h. The volume of the cone is 69 cubic centimeters. What is the volume of the cylinder?
A. 138 cubic centimeters
B. 207 cubic centimeters
C. 23 cubic centimeters
D. 276 cubic centimeters
Answer:
207 cubic centimeters
Step-by-step explanation:
Given
volume of cone = 9 cubic centimeters
Since the cone and cylinder above have the same radius, r, and height, h, then r = h
volume of the cone = 1/3πr²h
69 = 1/3πr²h
πr²h = 69*3
πr²h = 207cubic cm
Since the volume of the cylinder = πr²h
Hence volume of the cylinder is 207 cubic centimeters
What is the value of
[tex] \sqrt{100 \times 4} [/tex]
Answer:
√{100×4)=√20²=±20 is your answer
Find the sum of (6x2 - 9x + 7) and (- 8x2 + 10x - 14)
Answer:
-2x² + x - 7
Step-by-step explanation:
(6x² - 9x + 7) + (-8x² + 10x - 14)
6x² - 9x + 7 - 8x² + 10x - 14
6x² - 8x² - 9x + 10x + 7 - 14
-2x² + x - 7