Answers
6 1/4, √ 40, 13/2
Related Questions
You need to rent a moving truck for a day. You have identified two companies that rent trucks. Company A charges $45 per day plus $0.10 per mile. Company B charges $25 per day plus $0.30 per mile. For how many miles will the cost of renting be the same?
Answers
Answer:
the cost of renting a truck from both companies will be the same for 100 miles.
Step-by-step explanation:
Let's assume that the cost of renting a truck from Company A for x miles is C_A(x) and the cost of renting a truck from Company B for x miles is C_B(x).
From the given information, we know that:
C_A(x) = 45 + 0.10x
C_B(x) = 25 + 0.30x
We need to find the value of x for which C_A(x) = C_B(x). So we can set the two equations equal to each other and solve for x:
45 + 0.10x = 25 + 0.30x
20 = 0.20x
x = 100
Therefore, the cost of renting a truck from both companies will be the same for 100 miles.
Tara used a credit card to pay for a jacket that costs
$150. She did not purchase any other items using the
credit card. The credit card company charges 25%
interest each month on the remaining balance. Tara
pays $40 per month after the monthly interest has
been applies. How long will it take Tara to pay off the
balance? How much does she pay in all?
Answers
It takes Tara 5 months to pay off the balance.
What is the rate of interest?An interest rate tells you how high the cost of borrowing is, or high the rewards are for saving. So, if you're a borrower, the interest rate is the amount you are charged for borrowing money, shown as a percentage of the total amount of the loan.
Given that, Tara used a credit card to pay for a jacket that costs $150.
The credit card company charges 25% interest each month on the remaining balance.
Now, 25% of 150
25/100 ×150
0.25×150= 37.5
So, total money is 150+37.5
= 187.5
Number of months = 187.5/40
= 4.6875
≈ 5
Therefore, it takes Tara 5 months to pay off the balance.
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Recall the equation for simple interest, A = P(1 + rt). Compare the amount of interest earned in 10 years for an investment of $3700 with a 7% annual simple interest rate and a 7% annual interest rate compounded monthly.
simple interest: $______
compound interest: $______
The compound interest investment earned $______more than the simple interest investment.
Answers
For the compound interest investment, the formula is A = P(1 + r/n)^(nt), where n is the number of times the interest is compounded in a year (12 in this case).
Using this formula, the amount of interest earned after 10 years is $3700 x (1 + 0.07/12)^(12 x 10) = $3700 x (1.005833333)^120 = $6388.99.
Therefore, the compound interest investment earned $98.99 more than the simple interest investment.
3(6-4)=-3
How do I show work for this
Answers
The expression is not true.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The expression is,
⇒ 3 (6 - 4) = - 3
Now, We can check as;
⇒ 3 (6 - 4) = - 3
LHS;
⇒ 3 (6 - 4)
⇒ 3 × 2
⇒ 6
RHS;
⇒ - 3
Hence, The expression is not true.
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Use tape diagram to show Noah saved x dollars and Elena saved 1/10 less than that
Answers
If the mean of 4,6,9,y,16 and 19 is 13. What is the value of y.
Answers
The missing term y in the given data having mean is 33.
What is mean?The Arithmetic Mean is the average of the numbers: a calculated "central" value of a set of numbers.
Given that, the mean = 13,
we need to find the value of y.
4+6+9+y+16+19 / 6 = 13
y = 33
Hence, the missing term y in the given data having mean is 33.
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Limosine Company P charges a rental fee plus an
additional charge per hour. The equation y = 50 +
30x shows the total cost, y of renting the limosine,
and the x represents the hours. The cost to rent a
limosine from Company R is shown below in the
table. Which statement is true?
Time (hours)
Total Cost
O
A
B
C
Company R
1
2
3
4
$100 $125 $150 $175
5
$200
Company P has a lower rate of
change than Company R.
Company P charges $25 more
for its rental fee compared to
Company R.
Company R charges less money
for 1 hour of service compared
to Company P.
Answers
The correct statement is -
company {R} charges less money for 1 hour of service compared to company {P}.
What is division?Division is used to split a number into equal parts. It is one of the four arithmetic operations.
Given is that Limousine Company {P} charges a rental fee plus an additional charge per hour. The equation {y} = 50 + 30x shows the total cost. The cost to rent a limousine from Company {R} is shown below in the table as -
Time : 1 2 3 4 5
Cost : $100 $125 $150 $175 $200
We can write the unit charge for company {P} as -
{r₁} = $30
We can write the unit charge for company {R} can be calculated as -
{r₂} = (125 - 100) ÷ (2 - 1)
{r₂} = $25
We can conclude -
company {R} charges less money for 1 hour of service compared to company {P}.
Therefore, the correct statement is -
company {R} charges less money for 1 hour of service compared to company {P}.
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What is the center and radius of the circle with equation (x + 5)2 + (y – 3)2 = 4?
Choose one
(5, –3), r = 16
(5, –3), r = 4
(–5, 3), r = 16
(–5, 3), r = 4
none of these
Answers
The center and radius of the circle in the equation is given by (-5, 3) and 4 respectively.
What is circle?A circle is a round-shaped figure that has no corners or edges. In geometry, a circle can be defined as a closed, two-dimensional curved shape.
Given that, an equation of a circle, (x+5)²+(y-2)² = 4, we need to identify the center and radius of the circle,
The standard form for the equation of a circle is (x−h)²+(y−k)² = r². The center is (h, k) and the radius measures r units.
Comparing with the standard form, we get,
(h, k) = (-5, 2) and r² = 4
Therefore, the center is (-5, 2) and the radius is r = 4
Hence, the center and radius of the circle in the equation is given by (-5, 3) and 4 respectively.
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Accounting /
The following list of accounts was drawn for Tile, Etc., Inc. on December 31, Year 1, after the closing entries were posted:
Account Title
Cash
Accounts receivable
Allowance for doubtful accounts.
Inventory
Accounts payable
Common stock
Retained earnings
$110,000
125,000
18,000
425,000
95,000
450,000
97,000
Tile, Etc. had the following transactions in Year 2:
1. Purchased merchandise on account for $580,000.
2. Sold merchandise that cost $420,000 for $890,000 on account.
3. Sold for $245,000 cash merchandise that had cost $160,000.
4. Sold merchandise for $190,000 to credit card customers. The merchandise had cost $96,000. The credit card company charges a 4
percent fee.
5. Collected $620,000 cash from accounts receivable.
6. Paid $610,000 cash on accounts payable.
7. Paid $145,000 cash for selling and administrative expenses.
8. Collected cash for the full amount due from the credit card company (see item 4).
9. Loaned $60,000 to J. Parks. The note had an 8 percent interest rate and a one-year term to maturity.
10. Wrote off $7,500 of accounts as uncollectible.
11. Made the following adjusting entries:
(a) Recorded uncollectible accounts expense estimated at 1 percent of sales on account.
(b) Recorded seven months of accrued interest on the note at December 31, Year 2 (see item 9).
Required
a. Organize the transaction data in accounts under an accounting equation.
b. Prepare an income statement, a statement of changes in stockholders' equity, a balance sheet, and a statement of cash flows for
Year 2.
Answers
The Total Stockholder's Equity is $1,266,290.00
How to represent the income statementIncome Statement Amount $ Amount $
Sales Revenue 1,585,000.00
Interest Revenue 2,940.00
Total Revenue 1,587,940.00
Less: Expenses
Cost of merchandise sold 793,000.00
Selling and admin expense 158,000.00
Credit card expense 7,650.00
Bad debt reinstated (5,500.00)
Total Operating expense 953,150.00
Net Income 634,790.00
Statement of changes in Stockholder's Equity Amount $ Amount $
Beginning Common Stock 515,000.00
Add: Common Stock issued -
Ending Common Stock 515,000.00
Beginning Retained Earnings 116,500.00
Add: Net Income 634,790.00
Ending Retained Earnings 751,290.00
Total Stockholder's Equity 1,266,290.00
Balance Sheet Amount $ Amount $
Assets
Cash 586,350.00
Inventory 303,000.00
Accounts Receivable 399,200.00
Allowance for uncollectible accounts (10,200.00) 389,000.00
Notes Receivable 63,000.00
Interest Receivable 2,940.00
Total Assets 1,344,290.00
Liabilities & Stockholder's Equity
Liabilities
Current Liabilities
Accounts Payable 78,000.00
Current Liabilities 78,000.00
Stockholder's Equity
Common Stock 515,000.00
Retained Earnings 751,290.00
Total Stockholder's Equity 1,266,290.00
Total Liabilities & Stockholder's Equity 1,344,290.00
Statement of cash flows Amount $ Amount $
Cash flow from operating activities
Cash sales 310,000.00
Inflow from customers 997,350.00
Payment for inventory (675,000.00)
Payment for expenses (158,000.00)
Net Cash flow from operating activities 474,350.00
Cash flow from investing activities
Notes receivable provided (63,000.00)
Cash flow from financing activities
Cash flow from financing activities -
Net Change in cash 411,350.00
Add: Beginning cash balance 175,000.00
Ending cash balance 586,350.00
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An ecologist wishes to mark off a circular sampling region having radius 13 m. However, the radius of the resulting region is actually a random variable R with pdf given below. s(r) = { 1-(13-r)?] 12
Answers
The ecologist needs to carefully choose the size of the sampling region based on the pdf of R.
Sampling is the process of selecting a representative subset of individuals or objects from a larger population to study or analyze.
To ensure that their data is representative of the entire ecosystem, it is important to choose a sampling region that is large enough and well-defined.
In this scenario, an ecologist wishes to mark off a circular sampling region with a fixed radius of 13 meters. However, due to natural variation, the actual radius of the sampling region is a random variable, denoted by R. The probability density function (pdf) of R is given as
=> s(r) = { 1-(13-r)] ¹²,
which means that the probability of a particular value of R is proportional to the area of a circle with that radius.
Now, when the ecologist takes a sample, it is important to consider the effects of this random variation in the sampling region. If the radius of the sampling region is too small, it may not capture the full range of variability in the ecosystem.
On the other hand, if the radius is too large, it may include areas that are not representative of the ecosystem as a whole.
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Find the radius of a sphere with a volume of 26, 244π cubic inches.
Answers
27 inches is the radius of a sphere with a volume of 26, 244π cubic inches.
What is Three dimensional shape?a three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.
Given that volume of a sphere is 26, 244π cubic inches.
We need to find the radius of sphere.
V=4/3πr³
26, 244π=4/3πr³
26, 244=1.333r³
Divide both sides by 1.333
26,244/1.333=r³
19688=r³
Cube root on both sides
∛19688 = r
27=r
Hence, 27 inches is the radius of a sphere with a volume of 26, 244π cubic inches.
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Help me please.
Very urgently
Answers
The answers to each part is given below.
What is function?A function is a elation between a dependent and independent variable.
Given is the function as -
f(x) = x² + 10x + 12
{ 1 } -
Using the quadratic formula, we can write -
x = - 10 ± √(100 - 48)/2
x = {- 10 ± √52}/2
{ 2 } -
We can write the area as -
1/2 x b x h = 96
1/2 x 2(x + 4) x {x} = 96
x(x + 4) = 96
x² + 4x - 96 = 0
Using the quadratic formula, we can write -
x = - 4 ± √(16 + 384)/2
x = -4 ± 20/2
Therefore, the answers to each part is given above.
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find two numbers whose product is 15 and sum is 16
Answers
Answer:
Step-by-step explanation:
l
Answer:
15+1
Step-by-step explanation:
Calculate the area of this rectangle.
1/2 ft.
3
ft.
Answers
The area of a rectangle with the given length breadth is 1.5 square feet.
What is the area of a rectangle?The area occupied by a rectangle within its boundary is called the area of the rectangle. The formula to find the area of a rectangle is Area = Length × Breadth.
Given that, length of a rectangle= 3 ft and width of a rectangle= 1/2 ft.
Area is defined as the total space taken up by a flat surface or shape of an object.
Here, area of a rectangle =Length × Breadth
= 3×1/2
= 3×0.5
= 1.5 square feet
Therefore, the area of a rectangle is 1.5 square feet.
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A construction crew needs to pave a road that is 204 miles long. The crew paves 7 miles of the road each day. The length, L (in miles) that is left to be paved after d days is given by the following function. L (d) =204-7. If 155miles of road is left to be paved how many days has the crew been paving the road? How many miles of road does the crew have left to pave after 13days?
Answers
The number of days required to pave 155 miles of the road will be 7 days. The length of the road paved in 13 days will be 113 miles.
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 the length, L (in miles) that is left to be paved after d days is given by the following function. L (d) =204-7d.
The number of days will be calculated as:-
L (d) =204-7d
155 = 204 - 7d
7dv = 49
d = 7 days
The length of the road paved in 13 days will be calculated as:-
L (d) =204-7d
L (d) =204-7 x 13
L (d) =113 miles
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Jack tenía 3 bolsas de pelotas de golf con b pelotas en cada bolsa; luego su amigo le dio 6
pelotas de golf más.
¿Cuántas pelotas de golf tiene ahora Jack?
Answers
The total number of golf balls with Jack as -
{n} = 3b + 6.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Given is that Jack had 3 bags of golf balls with {b} balls in each bag; then his friend gave him 6 golf balls more.
We can write the total number of golf balls with Jack as -
{n} = 3b + 6
Therefore, the total number of golf balls with Jack as -
{n} = 3b + 6.
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{Question in english -
Jack had 3 bags of golf balls with b balls in each bag; then his friend gave him 6 golf balls more. How many golf balls does Jack have now?}
Kellan went ice skating for 8 hours this week. If he spent an equal amount of time skating each day for 5 days, how many hours did kellan skate each day?
Answers
Answer:
8/5 = 1.6
Step-by-step explanation:
8/5 = 1.6
show me neat sketch of two graphs on the same set of axes that shows how the total amount repaid (T) depends on the number of months (N) for both of the instalment factors.
Answers
Answer:
Step-by-step explanation:
I can explain how the two graphs would look like.
If we are comparing two installment factors with respect to the total amount repaid (T) over time (N), we can plot two separate graphs on the same set of axes. The x-axis would represent the number of months (N) and the y-axis would represent the total amount repaid (T).
For each installment factor, we would plot a line that represents the total amount repaid over time. The shape and slope of each line would depend on the interest rate, the amount borrowed, and the number of payments per year.
The two lines would likely have a similar shape, but different slopes and intercepts. The slope of each line would depend on the interest rate and the amount borrowed, with steeper slopes indicating higher interest rates or larger amounts borrowed. The intercept of each line would depend on the amount borrowed and the number of payments per year.
To compare the two installment factors, we would look at the relative slopes and intercepts of the two lines. If one line has a steeper slope than the other, it would indicate that the total amount repaid increases more rapidly over time for that installment factor. If one line has a higher intercept than the other, it would indicate that the total amount repaid is higher at the beginning of the repayment period for that installment factor.
Overall, by plotting the two graphs on the same set of axes, we can visually compare how the total amount repaid (T) depends on the number of months (N) for both installment factors.
26°
onotzib loinosho
er
11
Answers
Answer:chicken nuggets
Step-by-step explanation:
chickemn nuggets
What was the first step in solving this equation?
Original equation:
9(2p+ 1) = 6p + 15
Fill in the missing work:
Result of the first step:
>
3(2p + 1) = 2p + 5
Write a few sentences describing the missing work.
Answers
The missing step is 9(2p+ 1)/3 = 6p/3 + 15/3.
What is division?
The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.
Given:
An original equation,
9(2p+ 1) = 6p + 15.
Result of the first step:
3(2p + 1) = 2p + 5
To find the missing step:
Divide the original equation by 3 into both sides,
we get,
9(2p+ 1)/3 = 6p/3 + 15/3.
Simplifying further,
3(2p + 1) = 2p + 5
Therefore, the required expression is 9(2p+ 1)/3 = 6p/3 + 15/3.
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An office manager believes that the percentage of employees arriving late is even greater than the previously claimed 7%. She conducts a hypothesis test on a random 200 employee arrivals and finds 23 punching in late. Is this strong evidence against the 0.07 claim?
answer choices
Yes, p-value is 0.0062
Yes, p-value is 2.5
No, p-value is only 0.0062
No, p-value is over 0.10
There is insufficient information to reach a conclusion.
Answers
We do not have strong evidence against the 0.07 claim so, the correct answer is: No, the p-value is over 0.10.
Let's calculate the p-value and compare it to the level of significance (alpha) to determine if there is strong evidence against the 0.07 claim.
Let p be the true proportion of employees arriving late. The null hypothesis is that p = 0.07 and the alternative hypothesis is that p > 0.07.
The sample proportion of employees arriving late is x/n = 23/200 = 0.115. Under the null hypothesis, the sampling distribution of the sample proportion follows a normal distribution with a mean of 0.07 and standard deviation [tex]\sqrt{((0.07 * 0.93) / 200)}[/tex] = 0.0319.
The z-score corresponding to the sample proportion is (0.115 - 0.07) / 0.0319 = 1.41. The p-value is the probability of observing a z-score greater than or equal to 1.41 under the null hypothesis, which can be calculated as P(Z >= 1.41) = 0.078.
Since the p-value is greater than the level of significance alpha (which is typically 0.05 or 0.01), we fail to reject the null hypothesis that the true proportion of employees arriving late is 0.07.
Therefore, we do not have strong evidence against the 0.07 claim so, the correct answer is: No, the p-value is over 0.10.
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Cost of headphones: $20.00 Tax: 6% What is the new cost?
Answers
Answer:
Step-by-step explanation:
e = 20 * (1+6/100) = 20 * 53 50 = 20 * 1.06 =$ 21.2
3 tickets to the museum cost $12.75. At this rate, what is the cost of:
A. 1 ticket?
B. 5 tickets?
Answers
Answer:
1 ticket - $4.25
5 tickets - $21.25
Step-by-step explanation:
12.75/3=4.25 which is one ticket then take the price of one ticket and multiply it by 5, 4.25*5=21.25.
eleven spoons minus s equals 9 spoons
Answers
Answer:
s=2
Step-by-step explanation:
What is the algebraic relationship between the resonant wavelength, λ, and length of an open-closed tube, L, for the n=1,3,5, and 7 harmonics. The options below refer to the following expressions 1. λ= 2L/52. λ= 2L/73. λ=5L4. λ=3L5. λ = 4L/36. λ=4L7.λ = L8. λ= 4L/79. λ=7L10. λ= 4L/5
Answers
The relationship between the resonant wavelength λ and the length of an open-closed tube L for the n=1,3,5, and 7 harmonics follows the general equation λ=(2n+1)L/2n. This means that the nth harmonic has a wavelength that is (2n+1) times the length of the tube divided by 2n.
For example, the wavelength of the 1st harmonic is λ= (21+1) L/21=3L/2, the wavelength of the 3rd harmonic is λ= (23+1) L/23=7L/4, the wavelength of the 5th harmonic is λ= (25+1) L/25=11L/4, and the wavelength of the 7th harmonic is λ= (27+1) L/27=15L/4. Therefore, the algebraic relationship between the resonant wavelength and length of an open-closed tube follows the general equation λ= (2n+1) L/2n, where n is the harmonic number. This equation can be used to calculate the resonant wavelength for any harmonic number.
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Carmen planted some corn in the back yard and measures its height regularly. The last time she checked, the stalks were 52.6 inches tall. Now they are the same height. What is the percent of increase in the height of the stalks?
Answers
The percent of increase in the height of the stalks is given as follows:
0%.
How to obtain the percent of increase?The percent of increase is obtained applying the proportions in the context of this problem.
A proportion is applied because the percent increase is given by the division of the increase by the initial amount, and multiplied by 100%.
The parameters for this problem are given as follows:
Change of 0, as the plant is of the same height as the initial height.Initial height of 52.6 inches.Hence the percent increase is given as follows:
0/52.6 x 100% = 0%.
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NO LINKS!! URGENT HELP PLEASE!!
For #4-6, find the area of each figure, round your answer to one decimal point if necessary.
Answers
Problem 4
Answer: 2400 square inchesExplanation:
Draw a horizontal divider line to split the figure into two rectangles.
The rectangle up top is 30 by 30, so it has area 30*30 = 900 square inches.
The rectangle on the bottom has area 50*30 = 1500 square inches.
total = 900+1500 = 2400 square inches
========================================================
Problem 5
Answer: 12 square inchesExplanation:
Draw a horizontal divider line to split the figure into two rectangles.
A = area of rectangle on top = 2*2 = 4 square inches
B = area of rectangle on bottom = 4*2 = 8 square inches
C = total = A+B = 4+8 = 12 square inches
========================================================
Problem 6
Answer: 2592 square yardsExplanation:
Draw a horizontal divider line to get two rectangles.
A = area of rectangle on top = 36*36 = 1296 square yards
B = area of rectangle on bottom = 72*18 = 1296 square yards
C = total = A+B = 1296+1296 = 2592 square yards.
You have the correct answer. Nice work.
Answer:
4) 2400 in²
5) 12 in²
6) 2592 yd²
Step-by-step explanation:
To calculate the area of each given composite figure, divide the figure into two rectangles and sum the area of the two rectangles.
[tex]\boxed{\begin{minipage}{5cm}\underline{Area of a rectangle}\\\\$A=w\cdot l$\\\\where:\\ \phantom{ww} $\bullet$ \quad $w$ is the width.\\ \phantom{ww} $\bullet$ \quad $l$ is the length.\\\end{minipage}}[/tex]
Question 4Separate the figure into two rectangles by drawing a horizontal line (see attachment).
[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&= 30\cdot 30+50\cdot 30\\&=900+1500\\&=2400\; \sf in^2\end{aligned}[/tex]
Question 5Separate the figure into two rectangles by drawing a horizontal line (see attachment).
[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&=2 \cdot 2+ 4\cdot2 \\&=4+8\\&=12\; \sf in^2\end{aligned}[/tex]
Question 6Separate the figure into two rectangles by drawing a horizontal line (see attachment).
[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&=36 \cdot36 +72 \cdot18 \\&=1296+1296\\&=2592 \; \sf yd^2\end{aligned}[/tex]
What is yield strength determined by?
Answers
1) Consider the second-order linear DE (1- tcot t) y - ty y 0 for 0 < t < T- Let Y1 (t)-t and y2 (t) sin t. a) Are y1 and y2 both solutions to this DE? b) Are yi and y2 linearly independent? If so, then find the general solution of the DE. If not, then find constants A and B, not both zero, such that Ay? By 0. 2) Suppose that is a constant and {31 , y2} is a set of solutions to the DE Find the Wronskian W(x) of {31 , y, given that W(0) = 1.
Answers
The solution of second order linear equation DE (1- tcot t) y - ty y 0 for 0 < t < T is y2(t) = sin(t). There cannot be exist a constant k such that {x^2, e^x} is a set of solutions to the given differential equation.
To check if y1(t) = t is a solution to the given differential equation, we substitute it into the DE and simplify:
(1 - t cot(t))y - t(dy/dt) = (1 - t cot(t))t - t(1) = t - t^2 cot(t) - t = t(1 - t cot(t))
So y1(t) = t is a solution to the DE.
To check if y2(t) = sin(t) is a solution, we substitute it into the DE and simplify:
(1 - t cot(t))y - t(dy/dt) = (1 - t cot(t))sin(t) - t(cos(t)) = sin(t) - t sin(t) (csc(t)) - t(cos(t)) = sin(t) - t tan(t) sin(t) - t(cos(t))
So y2(t) = sin(t) is also a solution to the DE.
To check if y1(t) and y2(t) are linearly independent, we can check if their Wronskian is nonzero. The Wronskian of two solutions y1(t) and y2(t) of a second-order linear differential equation is given by:
W(y1, y2) = y1(t) (dy2/dt) - y2(t) (dy1/dt)
Substituting y1(t) = t and y2(t) = sin(t), we get:
W(y1, y2) = t cos(t) - sin(t)
Since W(y1, y2) is not identically zero (it is zero only at certain values of t), we can conclude that y1(t) and y2(t) are linearly independent.
The general solution to the given differential equation is given by:
y(t) = c1 t + c2 sin(t)
where c1 and c2 are constants determined by the initial or boundary conditions.
Let W(x) be the Wronskian of the solutions y1(x) and y2(x) of the differential equation given by:
(1 - x^2) y'' - 2x y' + k y = 0
Then, we have:
W(x) = y1(x) y2'(x) - y1'(x) y2(x)
Substituting y1(x) = x^2 and y2(x) = e^x, we get:
W(x) = x^2 e^x - 2x e^x - x^2 e^x = -2x e^x
Since we are given that W(0) = 1, we have:
W(0) = -2(0) e^0 = 1
This gives us a contradiction, so there cannot exist a constant k such that {x^2, e^x} is a set of solutions to the given differential equation.
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Determine where f is continuous. If possible, extend f as in example 4.2 to a new function that is continuous on a larger domain.F(x) = {sin x/x if x ≠ 0; 1 if x= 0Continuous everywhere since limx→0 sinx/x=1,and f(0) = 1.
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This function is continuous everywhere, as the limit of sinx/x as x approaches 0 is 1, and f(0) = 1. It cannot be extended to a larger domain, as it is already continuous everywhere.
For example, when x = 0.1, f(x) = 0.998334166, and when x = 0.001, f(x) = 0.998999667.This function is continuous everywhere, as the limit of sinx/x as x approaches 0 is 1, and f(0) = 1. This means that there are no breaks or discontinuities in the function, and it is continuous everywhere. To demonstrate this, we can look at specific values of x. For example, when x = 0.1, f(x) = 0.998334166, and when x = 0.001, f(x) = 0.998999667. This shows that the function is smoothly transitioning from one value to the next, and is therefore continuous everywhere. The function cannot be extended to a larger domain, as it is already continuous everywhere. This means that the function is defined everywhere on its domain, and there is no need to extend it.
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