From the following three options for running the appropriate t-test, Both Options (2) and (3) could be applicable depending on the specific research question being investigated.
The appropriate t-test to use in this scenario is paired t-test. This test compares before and after values for one type using excel. In this scenario, you would be comparing either the altimeter error of ACME or brand x before adjustment and calibration and after adjustment and calibration to see if there are any differences in the means (performance).
Based on the scenario described, Option 1 is not appropriate because there is no hypothesized population mean to compare against. Therefore, we can exclude this option.
Option 2, comparing one brand against another, could be appropriate if we want to investigate if there are any significant differences in altimeter error between brands. This would be a two-sample t-test.
Option 3, comparing the altimeter error of one brand before and after calibration, could also be appropriate to see if the adjustment and calibration had a significant impact on the altimeter's performance. This would be a paired t-test.
This test is the appropriate test to use because the dependent variable data in this scenario is the ratio and the type of question requires a paired t-test. In this scenario, you are comparing the performance of the two brands before and after adjustment and calibration, which requires paired t-test to answer the research question.
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refer to exercise 7.11. suppose that in the forest fertilization problem the population standard deviation of basal areas is not known and must be estimated from the sample. if a random sample of n = 9 basal areas is to be measured, find two statistics g1 and g2 such that p (g1 ≤ ( y - u ) ≤ g2 ) = 90
Confidence interval = (y ± t∗s/√n)g1 = y - t*s/√ng2 = y + t*s/√n Substituting the values, g1 = 26.22 - 1.860*(0.11)/√9 = 25.84g2 = 26.22 + 1.860*(0.11)/√9 = 26.59Therefore, the statistics g1 and g2 that will satisfy the required inequality are 25.84 and 26.59 respectively.
The formula for finding the confidence interval is as follows: n − 1, where t is the value of the t-distribution corresponding to the specified confidence level and the sample size minus one.
As per the given exercise 7.11, suppose that in the forest fertilization problem the population standard deviation of basal areas is not known and must be estimated from the sample.
If a random sample of n = 9 basal areas is to be measured, find two statistics g1 and g2 such that p(g1 ≤ (y - u) ≤ g2) = 90
To find the statistics g1 and g2 that will satisfy the required inequality
the following formula can be used: Confidence interval = [tex](y ± t∗s/√n)[/tex]
From the formula, we can see that the confidence interval depends on the values of y, s, t and n.
The value of y is the sample mean
the value of s is the sample standard deviation
And the value of n is the sample size.
The value of t depends on the confidence level desired and the degrees of freedom for the t-distribution. In this case, the confidence level is 90%, which means that we want to find the value of t that will give us a total area of 0.90 under the t-distribution curve with 8 degrees of freedom .Using the t-table, the value of t can be found to be 1.860, where the value for 90% and 8 degrees of freedom is 1.860.t = 1.860Now, we need to calculate the value of s, which is the sample standard deviation.
Since we do not have any information about the population standard deviation, we will use the sample standard deviation as an estimate of the population standard deviations = σ/√nσ = s*√nσ = 0.11*√9σ = 0.33Substituting the values in the confidence interval formula
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Find the expected value of the winnings
from a game that has the following
payout probability distribution:
The expected probability distribution is $4.35
We must multiply each supplied chance by the respective payout value in order to determine the expected value.
As a result, we have the payout probability distribution shown below:
0.35(1) + 2(0.2) + 5(0.1) + 8(0.2) + 10(0.15) (0.15)
= 0.35 + 0.4 + 0.5 + 1.6 + 1.5
= $4.35
When the set of possible outcomes is discrete in nature, the distribution is referred to as a discrete probability distribution. When a dice is rolled, all conceivable results, for instance, are discrete and result in a large number of outcomes. A different name for it is the probability mass function.
statistical distributions that can be used
Uniform discrete distribution The chances of each outcome are equal.
Single-trial distribution with two potential results.
A series of Bernoulli events are called the binomial distribution.
Poisson Distribution: The likelihood that a particular occurrence may or may not occur.
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1. Describe the historical data on Nando’s sales, including a discussion of thegeneral direction of sales and any seasonal tendencies that might beoccurring. 2. Discuss, giving your justifications, which time series forecasting techniquesare appropriate for producing forecasts with this data set. 3. Apply the appropriate forecasting techniques and compare the models basedon ex post forecasts. Choose the best model. 4. Use your chosen forecasting model to generate forecasts for each of themonths in year 2021. 5. Discuss how these forecasts might be integrated into the planning operationsand policy makings in NIH
In Rosettenville, a suburb of Johannesburg, South Africa, Robert Brozin and Fernando Duarte acquired the Chicken Land restaurant in 1987, launching Nando's.
The eatery was renamed Nando's in honor of Fernando. The restaurant incorporated influences from former Mozambican Portuguese colonists, many of whom had relocated to Johannesburg's southeast after their country gained independence in 1975. Expansion was an essential component of their vision from the beginning. Nando's had already grown from one restaurant in 1987 to four by 1990. It became increasingly difficult to implement a common strategy and decision-making became inefficient as new outlets were maintained as separate businesses.
In 1995, Nando's International Holdings (NIH) was established as a new international holding because managing this growingly complex global structure had become extremely challenging. The South African branch of Nando's Group Holdings (NGH) was successfully listed on the Johannesburg Stock Exchange on April 27, 1997. NGH was 54% owned by NIH, with the remaining 26% available to the general public and former joint venture partners. The main goals of the share offer and listing were to broaden the group's capital base and enable group restructuring.
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Which statement is correct? A -5 < 0 because -5 is to the left of 0 on a number line B -5 < 0 because -5 is to the right of 0 on a number line C -5 > 0 because -5 is to the left of 0 on a number line D -5 > 0 because -5 is to the right of 0 on a number line
The correct statement is A -5 < 0 because -5 is to the left of 0 on a number line. Here's a step-by-step explanation:
On a number line, the smaller numbers are to the left and the larger numbers are to the right. Since -5 is to the left of 0, -5 is a smaller number and 0 is a larger number. Therefore, -5 is less than (or <) 0.
So the statement A -5 < 0 because -5 is to the left of 0 on a number line is correct.
Find the measures of one exterior angle of a regular polygon if the sum of the interior angles is 3960 degrees.
The exteriοr angle οf the pοlygοn is 15°.
What is pοlygοn?A pοlygοn has as many pairs οf interiοr and exteriοr as number οf its sides. Further each pair οf exteriοr and interiοr add up tο 180°.
Sum οf all the exteriοr angles οf any pοlygοn is 360°.
As sum οf all the interiοr angles οf the pοlygοn is 3960°.
Sum οf all pairs οf exteriοr and interiοr angles is
=> 3960°+360°=4320°
As each pair is 180° , Number οf sides οf pοlygοn = 4320/180 = 24
Sum οf exteriοr angles is always 360° and if pοlygοn is regular (i.e. all sides and angles are equal), size οf each exteriοr angle is
=> 360°/24 = 15°
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What is the image of (-2, 6) after a dilation by a scale factor of 1/2 centered at the
origin?
The image of (-2, 6) after a dilation by a scale factor of 1/2 centered at the origin is given as follows:
(-1, 3).
What is a dilation?A dilation is a transformation that changes the size of a figure, but not its shape. Specifically, a dilation is a type of similarity transformation that involves multiplying the coordinates of each point in a figure by a scale factor. This causes the figure to either enlarge or reduce in size.
The scale factor in the context of this problem is given as follows:
1/2.
The coordinates of the original point are given as follows:
(-2, 6).
Multiplying the coordinates of the original point by the scale factor, the coordinates of the image are given as follows:
(-1,3).
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Baseball hats are on sale for 12% off the original price of the sale price is $12.50 what was the original price? round the answer to the nearest cent
The original price of the baseball hat before 12% off in the sale was $14.20.
What is a percentage?A number can be expressed as a fraction of 100 using a percentage. It frequently serves to indicate a portion of a total and is represented by the sign %. We may state that 25% of the class is made up of guys, for instance, if there are 100 pupils in the class and 25 of them are male.
The Roman term per centum, which meaning "by the hundred," is where the word "percent" originates.
Let us suppose the original price of the baseball hat = x.
Given that, baseball hats are on sale for 12% off the original price.
That is,
12.50 = x(1 - 12/100)
12.50 = x(100 - 12)/100
12.50 = x(0.88)
x = 14.2
Hence, the original price of the baseball hat was $14.20.
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Drag the term to the corresponding probability. Each term may be used only once. CertainLikelyUnlikelyImpossible Probability Term Probability < 12 Probability = 0 Probability = 1 Probability > 12
Thus, the probability are Zero represents the impossibility.
1/2 equals probable and unlikely
1 = event happens.
Because we don't know how something will turn out, we might talk about the probability of one outcome or the potential for several outcomes.
Typically, the likelihood is stated as the proportion of favourable events to all other potential outcomes in the sample space.
The probability of an event is calculated using the formula P(E) = (Number of favourable outcomes) (Sample space).
Given that probability can only be a number between 0 and 1,
The zeroth value is the impossible.
so there is a 50% chance for both likely and unlikely.
If 1, the event must take place.
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what is the average time gap between the first cyclists time and each of the remaining cyclists' times (second through fifth) in the 1995 volta a catalunya cycle race if we know the result?
The average time gap between the first cyclist's time and each of the remaining cyclists' times (second through fifth) in the 1995 Volta a Catalunya cycle race is approximately 6 minutes and 7 seconds.
To calculate this, we need to subtract the time of the first cyclist from each of the remaining cyclists' times (second through fifth).The time for the first cyclist was 41:38:33.
The times for the remaining cyclists were as follows:
We can calculate the difference for each cyclist by subtracting the first cyclist's time from their own time:
Adding up all of the times and dividing by four, we get an average of 00:06:07.
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Image attached, please help.
Step-by-step explanation:
so, it is a geometric sequence (every new term is created by multiplying the previous term by a fixed constant.
in this case this constant is 2.
a1 = 3
a2 = 3×2 = 6
a3 = a2×2 = a1 × 2×2 = 3×4 = 12
so, we see
an = 3 ×2^(n-1)
or
an+1 = 3 × 2^n
Answer:
[tex]A_n=3 \cdot 2^{n-1}[/tex]
Step-by-step explanation:
A recursive formula for a sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence.
Given recursive rule:
[tex]\begin{cases}A_1=3\\A_{(n+1)}=2A_n\end{cases}[/tex]
According to the recursive formula, each term in the sequence is double the previous term. This means the sequence is geometric with the common ratio, r = 2.
An explicit formula for a sequence allows you to find the nth term of the sequence. The explicit formula for a geometric sequence is:
[tex]\boxed{A_n=A_1 \cdot r^{n-1}}[/tex]
where:
A₁ is the first term.r is the common ratio.Substituting the given value of A₁ and the found value of r into the formula, the explicit formula for the given sequence is:
[tex]A_n=3 \cdot 2^{n-1}[/tex]
This can also be written as:
[tex]A_{(n+1)}=3 \cdot 2^{n}[/tex]
Find the measure of the arc. *
X
42°
V
W
?
O 42 degrees
84 degrees
0 21 degrees
O 81 degrees
According to the given circle, the measure of the arc is option C. 21°.
Let's look at the figure provided. Since arc AC has a measure of 84°, we know that angle AOC, which is the central angle that subtends arc AC, has a measure of 84° as well. We can use this information to find the measure of angle AOB, which is an inscribed angle that includes angle ABC.
Since arc AC is 84°, arc AB must have a measure of 180° - 84° = 96°. This means that angle AOB, which spans arc AB, must have a measure of half of 96°, or 48°.
Now we can use the property of inscribed angles to find the measure of angle ABC. We know that angle AOB is twice the measure of angle ABC, so angle ABC must have a measure of half of 48°, or 24°.
Therefore, the answer is C. 21°.
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Jennifer bought 14.75 gallons of gasoline for her car at a cost of $2.95 a gallon. which is closest to the amount she paid for the gasoline?
The total amount paid by Jennifer for 14.75 gallons of gasoline at the cost of $2.95 per gallons is equal to $43.51.
Total gallons of gasoline bought by Jennifer for her car = 14.75 gallons
Cost of gasoline per gallons= $2.95
The total amount Jennifer paid for the gasoline is equal to, amount paid by Jennifer= number of gallons of gasoline* cost per gallon
substitute the value in the formula we get, = amount paid by Jennifer= 14.75 gallons* $2.95/gallon= $43.51
Therefore, the closest amount Jennifer paid for the gallons of gasoline is equal to $43.51.
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is this A.less then
b.equal too
c.more then
Answer:
A. Less than
Step-by-step explanation:
Please help me with my math!!
Answer:
5
Step-by-step explanation:
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To find the current that supplies the maximum wattage, we need to take the derivative of the power function with respect to I, set it equal to zero, and solve for I. The derivative of the power function is:
dW/dI = 120 - 24I
Setting this equal to zero and solving for I, we get:
120 - 24I = 024I = 120I = 5
Therefore, the current that supplies the maximum wattage is 5 amps. We can verify that this is a maximum by checking the second derivative of the power function with respect to I. The second derivative is:
d^2W/dI^2 = -24Since the second derivative is negative, the function has a maximum at I = 5.
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Answer:
We can find the current that supplies the maximum wattage by finding the value of I that maximizes the power function W(I) = 120I - 12I^2.
To do this, we can take the derivative of W with respect to I and set it equal to zero:
dW/dI = 120 - 24I = 0
Solving for I, we get:
I = 5
Therefore, the current that supplies the maximum wattage is 5 amps.
Answer: b. 5
The perimeter of a rectangle is 3.5 times the length. The width is 2.5 cm less than the length. What is the width?
It is 7.5
the steps are in the photo
Simplify the equation
i= square root -1
The equation [tex](\sqrt(6)-i\sqrt(2))/(i\sqrt(3))[/tex] simplifies to 1 - i where numerator and denominator is multiplied by conjugate.
What is conjugate?Conjugate can refer to a pair of complex numbers that differ only in the sign of their imaginary parts. For example, the conjugate of the complex number a + bi is a - bi.
According to question:To simplify this expression, we need to rationalize the denominator, which means getting rid of the square root in the denominator. To do this, we can multiply the numerator and denominator by the conjugate of the denominator, which is -i√3.
[tex](\sqrt(6)-i\sqrt(2))/(i\sqrt(3)) * (-i\sqrt(3))/(-i\sqrt(3))\\= (\sqrt(6)(-i\sqrt(3)) - i\sqrt(2)(-i\sqrt(3))) / (i\sqrt(3)*(-i\sqrt(3)))\\= (-i\sqrt(18) + 3\sqrt(2)) / 3\\= (-i\sqrt(9*2) + 3\sqrt(2)) / 3\\= (-i\sqrt(9) * \sqrt(2) + 3\sqrt(2)) / 3\\= (-3i * \sqrt(2) + 3\sqrt(2)) / 3\\= (3 - 3i) / 3\\= 1 - i[/tex]
Therefore, [tex](\sqrt(6)-i\sqrt(2))/(i\sqrt(3))[/tex] simplifies to 1 - i.
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If P = 2y² + 4xy + 4
Q = − 3y² + 7 - 3xy
R=- 3xy + 8
Find P+Q=R.
Answer:
P = [tex]2y^{2}[/tex] + 4xy +4
Q = [tex]-3y^{2}[/tex] + 7 -3xy
R = -3xy +8
Step-by-step explanation:
Kate plans to save 5% of her income. She just earned 80.Howmuchmoneyshouldsheputintosavings?
$4
Multiply 80 by 0.05 to get the answer of $4
Answer:
4 (insert the currency needed)
Step-by-step explanation:
To find our answer we have to find 5% of Kate's wages and to do this we have to do 5 divided by 100. Then that answer is multiplied by 80!
5 ÷ 100 = 0.0580 × 0.05 = 4This means she has to put £4 into her savings!
Hope this helps, have a lovely day! :)
A number is decreased by 40% and the result is 54. What is the number?
A. 94
B. 90
C. 75.6
D. 21.6
E. 14
Answer:
B. 90
Step-by-step explanation:
Let x be the original number.
We know that the number is decreased by 40%, which means we need to subtract 40% of the original number from the original number:
x - 0.4x = 54
Simplifying this equation, we get:
0.6x = 54
Dividing both sides by 0.6, we get:
x = 90
Therefore, the original number is 90.
So, the answer is B) 90.
60°
d. Opposite C:
e. Adjacent C:
B
90°
1. Consider the triangle ABC. Which side is being described by the following labels?
a. Hypotenuse:
b. Opposite A:
c. Adjacent A:
30°
Side BC is the hypotenuse of the right triangle ABC. Against A: The side that is opposed to angle A is side BC. Adjacent A: Side AB is the side that faces angle A. The side that is in opposition to angle C is side AB.
What does a math hypotenuse mean?
An "opposite" side in a right triangle is the one that is across from a specific angle, while a "adjacent" side is the one that is next to a specific angle. The hypotenuse is the longest side in a right triangle. The sides of right triangles are described with particular terminology. It goes without saying that the side opposite the right angle is the hypotenuse of a right triangle.
Side BC is the hypotenuse of the right triangle ABC. Against A: The side that is opposed to angle A is side BC. Adjacent A: Side AB is the side that faces angle A. The side that is in opposition to angle C is side AB.
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4x 2 +6x−13=3x 2 to the nearest tenth.
The solutions to the equation are x = -4 and x = 1.
What is quadratic formula?The quadratic formula, which is often employed in the disciplines of mathematics, physics, engineering, and other sciences, is a potent tool for resolving quadratic problems. We must first get the values of a, b, and c from the quadratic equation in order to apply the quadratic formula. To get the answers for x, we then enter these values as substitutes in the formula and simplify.
The given equation is 4x² + 6x - 13 = 3x².
Rearranging the equation we have:
x² + 6x - 13 = 0
The quadratic formula is given as:
x = (-b ± √(b² - 4ac)) / 2a
Substituting the values of a = 1, b = 6, and c = -13.
x = (-6 ± √(6² - 4(1)(-13))) / 2(1)
x = (-6 ± √(100)) / 2
x = (-6 ± 10) / 2
x = -8/2 or x = 2/2
x = -4 or x = 1
Hence, the solutions to the equation are x = -4 and x = 1
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Please help me with my math!
Answer:
-3
Step-by-step explanation:
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To rewrite the quadratic equation 9 = -3x^2 -18x - 25 in the form y = a(x - p)^2 + q, we need to complete the square. First, we factor out the leading coefficient of -3:
-[tex]3(x^2 + 6x + 25/3) = -3(x^2 + 6x + 9 + 16/3)[/tex]
Next, we add and subtract 9 inside the parentheses to complete the square:
[tex]-3(x^2 + 6x + 9 - 9 + 16/3) = -3((x + 3)^2 - 1/3)[/tex]
Simplifying the expression further, we get:
[tex]-3(x + 3)^2 + 3 = y[/tex]
Comparing this to the standard form y = a(x - p)^2 + q, we can see that a = -3, p = -3, and q = 3. Therefore, the value of p is -3.
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Answer:
[tex]p = - 3[/tex]Step-by-step explanation:
To find:-
The value of p .Answer:-
We are here given that a quadratic equation is written in the form of y = a(x-p)² + q and we are interested in finding out the value of "p" .
So , the given quadratic equation to us is ,
[tex]\longrightarrow y = -3x^2-18x-25\\[/tex]
Now complete the square on the RHS side of the equation as ,
Firstly make the coefficient of x² as 1 . This can be done by taking out -3 as common.
[tex]\longrightarrow y = -3\bigg(x^2+6x +\dfrac{25}{3}\bigg)\\[/tex]
we can rewrite it as ,
[tex]\longrightarrow y = -3\bigg( x^2+2(3)(x) +\dfrac{25}{3}\bigg) \\[/tex]
Add and subtract 3² , as ;
[tex]\longrightarrow y = -3\bigg( x^2+2(3)(x) + 3^2-3^2+\dfrac{25}{3}\bigg) \\[/tex]
Rearrange the terms as ,
[tex]\longrightarrow y = -3 \left\{ (x^2+2(3)(x) + 3^2) - 9 +\dfrac{25}{3}\right\} \\[/tex]
Notice the terms inside the small brackets are in the form of [tex]a^2+b^2+2ab[/tex] which is the whole square of [tex] (a+b)[/tex] . Hence , we can write it as ,
[tex]\longrightarrow y =-3\bigg\{ (x+3)^2 + \dfrac{-27+25}{3}\bigg\} \\[/tex]
Simplify,
[tex]\longrightarrow y = -3\bigg\{ (x+3)^2 -\dfrac{2}{3}\bigg\} \\[/tex]
Open the curly brackets by multiplying the terms inside the brackets by -3 as ,
[tex]\longrightarrow y = -3(x+3)^2 - 2 \\[/tex]
Now compare it with [tex] y = a(x-p)^2+q [/tex] . On comparing we get ,
[tex]\longrightarrow \boxed{\boldsymbol{ p =-3}}\\[/tex]
Hence the value of p is -3 .
please i need help rn
Therefore, if the relationship between time and distance is proportional and the turtle swims 25 feet in 40 seconds, the table above shows the corresponding distances for other times.
What is proportion?A proportion is a statement that two ratios or fractions are equal. It is a way to express the relationship between two quantities that are proportional to each other. In other words, if we have two ratios that are equal, we can write them as a proportion. Proportions are used in a wide range of fields, including mathematics, science, finance, and engineering. They are often used to make predictions, estimate values, or analyze data.
Here,
To determine the missing values in the table, we can use the fact that the relationship between time and distance is proportional. We can set up a proportion using the initial values given:
25 feet / 40 seconds = d feet / t seconds
Simplifying the left-hand side, we get:
5/8 = d/t
5/8 * 20 = 12.5
Using this proportion, we can fill in the missing values in the table:
Time Distance
(seconds) (feet)
1 5
12.5 10
30 187.5
40 25
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A student thinks of a number. They square it and then subtract 9. Their answer is 315. What number is the student thinking of?
The number that the student is thinking of is 18.
What is number?
A number is an arithmetic value that is used to calculate and represent a quantity. Numerical symbols, such as "3," are written to represent numbers.
Let x be the number that the student is thinking of.
According to the problem, the student squares the number and subtracts 9:
x² - 9 = 315
To solve for x, we can add 9 to both sides of the equation:
x² = 324
Then, we can take the square root of both sides:
x = ±18
Since x must be a positive number (according to the problem statement), we can discard the negative solution. Therefore, the number that the student is thinking of is 18.
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Here are two employment adds from a newspaper. Use them to solve the problem.
Driver needed for hauling service.
Must have phone & current
driver's license. $14/hr to start.
$18/hr after 28 days. Call Jerry
777-555-7314
What is the hourly wage that the driver will be paid during the first 18 days? Write your answer in dollar amt. ($8.00)
For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac).
BIUS
Paragraph
Arial
10pt
✓E V
Nurse needed for long-term care
facility. RN $21.50/hr. LPN $14/hr.
Paid Bonus: 1 wk. vacation after 6
mos. + medical benefits. Call
Peggy 777-555-3433
AVA
>
I % 0
Based on the information, it can be inferred that the hourly wage that the driver will receive during the first 18 days would be 14/h.
How to calculate the hourly wage that the driver will receive during the first 18 days?To calculate the hourly wage that the driver will receive during the first 18 days, we must carefully read the information in the advertisement. In this case, the hourly wage is posted there.
Based on this information, we can infer that the driver's salary would be $14/hour for the first 28 days. Additionally, then his salary would increase to $18/hour.
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Identify the slope and y intercept for the equation y = 2x + 3
help with number 2 question
Step-by-step explanation:
To make it simple, since the radius is 15, two points on the circle could be (15,0) and (0,-15). Two more straightforward options are (-15,0) or (0,15)
Use a ruler and pair of compasses to make an accurate drawing of line
AB and its perpendicular bisector, as shown. You must show all of your
construction lines.
Mark point C on your drawing.
Measure the length of AC in your drawing to 1 d.p.
Step-by-step explanation:
1. Draw a line of 8cm.
2. Take a compass and keep it in the length of more than 8 cm.
3. Draw an arc from point A and B which will intersect at point between A and B.
4. Draw a straight line from the arc.
5. You will find out that the line will be drawn exactly between A and B at 4cm.
HW7.4. Find the frequency with the largest amplitude Find the frequency w for which the particular solution to the differential equation dạy dy 2- + dt2 + 2y = eiwt dt has the largest amplitude. You can assume a positive frequency w > 0. Probably the easiest way to do this is to find the particular solution in the form Aeiwt and then minimize the modulus of the denominator of A over all frequencies w. W= number (rtol=0.01, atol=1e-08) ?
Answer:
the frequency with the largest amplitude Find the frequency w for which the particular solution to the differential equation dạy dy 2- + dt2 + 2y = eiwt dt ...
Step-by-step explanation: basically bigger is better
Determine whether the set StartSet left bracket Start 3 By 1 Matrix 1st Row 1st Column 1 2nd Row 1st Column 0 3rd Row 1st Column negative 3 EndMatrix right bracket comma left bracket Start 3 By 1 Matrix 1st Row 1st Column negative 3 2nd Row 1st Column 1 3rd Row 1st Column 6 EndMatrix right bracket comma left bracket Start 3 By 1 Matrix 1st Row 1st Column 1 2nd Row 1st Column negative 1 3rd Row 1st Column 0 EndMatrix right bracket EndSet 1 0 −3 , −3 1 6 , 1 −1 0 is a basis for set of real numbers R cubedℝ3. If the set is not a basis, determine whether the set is linearly independent and whether the set spans set of real numbers R cubedℝ3. Which of the following describe the set?
The set [1 0 −3 , −3 1 6 , 1 −1 0] is not a basis for set of real numbers R cubed(ℝ3).
The reason why it is not a basis is because it is not linearly independent. However, the set does span set of real numbers R cubed(ℝ3).To determine if the given set is a basis for set of real numbers R cubed(ℝ3), we need to test for linear independence and for span.Linear independence
The given set is said to be linearly independent if and only if the only solution to the equation a[1 0 −3] + b[−3 1 6] + c[1 −1 0] = [0 0 0] is the trivial solution where a, b and c are constants.If the given set is linearly independent then it is a basis for R3; if it is not linearly independent, it is not a basis for R3.
SpanThe given set is said to span R3 if every vector in R3 can be written as a linear combination of vectors in the given set.
If the given set spans R3, then it can be considered a basis for R3.For us to test if the given set is linearly independent, we can form a matrix by placing the three given vectors into the columns of a 3 x 3 matrix as follows:[1 0 1] [−3 1 −1] [−3 6 0]
By expanding the determinant of the matrix above, we get: det(A) = 0 - 0 - (-3) = 3
Since the determinant is non-zero, we can say that the given set is linearly independent. Since the given set is linearly independent, we can then use it to span R3. Hence the given set does not form a basis for R3 but it is linearly independent and spans R3.
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