Answer:
To find a splitting field for f, we first factor f completely over Q:
f = (2x^2 − 4x + 1)(x^4 + 1) = 2(x - 1/2)^2(x^4 + 1)
The roots of the quadratic factor are x = 1/2 ± 1/2√2, which are real, and the roots of the quartic factor are x = ±i. Therefore, a splitting field for f is Q(i, √2).
(a) Q(√2): The roots of the quadratic factor are real, so they are already in Q(√2). The roots of the quartic factor are ±i, which are not in Q(√2). Therefore, the Galois group of f over Q(√2) is isomorphic to Z/2Z.
(b) Q(2): The roots of the quadratic factor are not in Q(2). However, both i and √2 are in Q(2), so Q(2) contains a splitting field for f. Therefore, the Galois group of f over Q(2) is trivial, since there is only one automorphism of Q(2) that fixes Q.
(c) Q(i): The roots of the quadratic factor are not in Q(i). However, √2 is in Q(i), so Q(i) contains a splitting field for f. The Galois group of f over Q(i) is isomorphic to Z/2Z × Z/2Z, since there are two complex roots and two real roots, and the complex roots are conjugates of each other.
(d) Q(ζ): The roots of the quadratic factor are not in Q(ζ). However, both i and √2 are in Q(ζ), since ζ^4 = i and ζ^8 = √2. Therefore, Q(ζ) contains a splitting field for f. The Galois group of f over Q(ζ) is isomorphic to Z/2Z × Z/2Z, since there are two complex roots and two real roots, and the complex roots are conjugates of each other.
(e) R: The roots of the quadratic factor are real, so they are already in R. The roots of the quartic factor are ±i, which are not in R. Therefore, the Galois group of f over R is isomorphic to Z/2Z.
London invested $3,400 in an account paying an interest rate of 6 7/8% compounded
quarterly. Victoria invested $3,400 in an account paying an interest rate of 6 3/8%
compounded monthly. After 5 years, how much more money would London have in
her account than Victoria, to the nearest dollar?
Answer:
Step-by-step explanation:
Answer: 108
Step-by-step explanation:
Ben has some goldfish in a fish tank. The ratio of orange goldfish to yellow goldfish is 3:5.
He has 6f orange goldfish. Write the number of yellow goldfish he has in terms of f.
Answer:
Step-by-step explanation:
If the ratio of orange goldfish to yellow goldfish is 3:5, then we can write:orange goldfish : yellow goldfish = 3 : 5We know that Ben has 6f orange goldfish. We can use this information to set up a proportion and solve for the number of yellow goldfish in terms of f:3/5 = 6f/yellow goldfishTo solve for yellow goldfish, we can cross-multiply:3 * yellow goldfish = 5 * 6fSimplifying:yellow goldfish = 30f/3 = 10fTherefore, Ben has 10f yellow goldfish in terms of f, given that he has 6f orange goldfish and the ratio of orange to yellow goldfish is 3:5.
What’s the equation and solution?
The equation for the rectangle is adding the interior angles:
x + x - 5 + 71 + y + 21 = 360°
Solving that for y, we will get:
2x + y + 87 = 360
How to find the equation?Remember that the sum of the interior angles of any figure with 4 sides must be 360°.
Then we can write an equation:
x + x - 5 + 71 + y + 21 = 360°
2x + y + 87 = 360
Solving it for y:
y = 273 - 2x
Now there is a clear problem in the diagram because the angle in the bottom left is clearly larger than 90°, but it says that the angle measures 71°, so there we have a problem.
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Given f(x)
, find g(x)
and h(x)
such that f(x)=g(h(x))
and neither g(x)
nor h(x)
is solely x.
f(x)=−square root of x+4 minus 3
The functions g(x) = -√x - 3 and h(x) = x + 4 meet the requirements, and f(x) = g(h(x)) is true for f(x) = -√(x+4) - 3.
We are given f(x) and we need to find functions g(x) and h(x) such that f(x) = g(h(x)). The given function is f(x) = -√(x+4) - 3.
To find g(x) and h(x), we can split the given function into two simpler functions. Let's first look at the inner function, which deals with the input of the square root. We can define h(x) as the function inside the square root, which is:
h(x) = x + 4
Now we need to find g(x) such that when h(x) is plugged into it, we get the original function f(x). Since we have the square root and a subtraction, we can define g(x) as:
g(x) = -√x - 3
Now let's verify if f(x) = g(h(x)):
g(h(x)) = g(x + 4) = -√(x + 4) - 3
As you can see, g(h(x)) = f(x), which confirms that the functions g(x) and h(x) we found satisfy the condition.
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Consider the curve given by the equation x2 − y2 = 2x + y + xy − 4. Find the equation
of the tangent line to the curve at the point (1, 1).
The equation of the tangent line at (1,1) is given as follows:
y - 1 = -0.25(x - 1).
How to obtain the equation of the tangent line?The curve for this problem is given as follows:
x² - y² = 2x + y + xy - 4.
Applying implicit differentiation, we obtain the slope of the tangent line, as follows:
2x - 2y(dy/dx) = 2 + (dy/dx) + x(dy/dx) + y
(dy/dx)(1 + x + 2y) = 2x - 2 - y
m = (2x - 2 - y)/(1 + x + 2y).
At x = 1 and y = 1, the slope is given as follows:
m = (2 - 2 - 1)/(1 + 1 + 2)
m = -0.25.
Hence the point-slope equation is given as follows:
y - 1 = -0.25(x - 1).
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5/3X + 1/3X= 2 2/3+ 8/3X
The solution of the given equation 5X/3 + X/3 = 2 (2/3) + 8X/3 is, X = - 4.
The equation is a mathematical statement involving mathematical variable, numerical values, mathematical operation with a equal sign.
The given equation is,
(5/3)*X + (1/3)*X = 2 (2/3) + (8/3)*X
Solving the equation we get,
5X/3 + X/3 = (2*3 + 2)/3 + 8X/3
5X/3 + X/3 - 8X/3 = (6 + 2)/3
(5X + X - 8X)/3 = 8/3
- 2X/3 = 8/3
- 2X = 8
X = 8/(-2)
X = - 4
Hence the solution is, X = - 4.
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the area of a circle is 2464 cm². Calculate the diameters to 2 significant figure
Answer:
56 cm
Step-by-step explanation:
have a good day God bless :)
The figure below shows part of a circle, with central angle as marked. What part of
the full circle does the figure represent? Express your answer as a fraction in simplest
terms.
60°
11/18
Step-by-step explanation:
sorry if i got it wrong
Christo wants to buy a small wine farm worth R8 500 000. He plans to sell his current home for R3 400 000 which he will use as a deposit for the purchase of the farm. He secures a loan with a bank with a repayment period of 10 years and an interest rate of 9, 5% compounded monthly. Calculate his monthly repayments.
Christo's monthly repayments for a loan of R5,100,000 for 10 years at 9,5% interest compounded monthly is R65,992.75.
How is the monthly repayments determined?The monthly repayments can be determined using an online finance calculator as follows:
The monthly repayments show the periodic payments to settle the loan.
Wine Farm Price = R8,500,000
Down Payment = R3,400,000
Loan Term = 10 years
Interest Rate = 9.5%
Start Date = Apr. 2023
Monthly Pay: R65,992.75
Loan Amount = R5,100,000.00
Total of 120 Mortgage Payments = R7,919,130.52
Total Interest = R2,819,130.52
Mortgage Payoff Date = Apr. 2033
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The sides of a triangle are 30, 64, and 90. Use the Pythagorean Theorem to determine if the triangle is right, acute, or obtuse.
Answer:
The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's check if the triangle with sides of 30, 64, and 90 is a right triangle:
30^2 + 64^2 = 900 + 4096 = 4996
90^2 = 8100
Since 4996 is less than 8100, the triangle is not a right triangle.
In an acute triangle, all angles are less than 90 degrees. In an obtuse triangle, one angle is greater than 90 degrees.
To determine if the triangle is acute or obtuse, we need to find the largest side. The largest side is 90. Let's find the sum of the squares of the other two sides:
30^2 + 64^2 = 900 + 4096 = 4996
Since 4996 is less than 90^2, the triangle is acute.
Therefore, the triangle with sides of 30, 64, and 90 is an acute triangle.
the triangle is obtuse
Step-by-step explanation:
[tex] {90}^{2} = 8100[/tex]
[tex] {30}^{2} + {64}^{2} = 4996[/tex]
[tex] {30}^{2} + {60}^{2} < {90}^{2} [/tex]
triangle is obtuse because the square of the largest is greater than the sum of the square of the other 2 sides
A
16.2
16.2-foot slide has an angle of elevation of
49.4
°
49.4° from the ground to the slide. What is the horizontal distance traveled by someone sliding from the top of the slide to the bottom of the slide? Round your answer to one decimal place. Enter
deg
deg after any degree value.
Answer:
Set your calculator to degree mode.
Please sketch the figure to confirm my answer.
cos(49.4°) = d/16.2
d = 16.2cos(49.4°) = 10.5 feet
Candace is flipping a coin a certain number of times. The theoretical probability of her flipping tails on all flips is 1/32 . How many times is she flipping the coin?
Answer:
Cadence is flipping her coin 32 times
Step-by-step explanation:
So the fraction is 1/32 which means 32 is the number of times the coin was flipped.
what is 9 1/8 - 2 1/4 simplified?
Answer:
To simplify 9 1/8 - 2 1/4, we need to first convert the mixed numbers to improper fractions.
9 1/8 = (8 x 9 + 1)/8 = 73/8
2 1/4 = (4 x 2 + 1)/4 = 9/4
Now we can subtract the two fractions:
73/8 - 9/4
Since the two fractions have different denominators, we need to find a common denominator. The least common multiple of 8 and 4 is 8, so we can convert the second fraction to have a denominator of 8:
73/8 - (9/4)x(2/2) = 73/8 - 18/8 = 55/8
Therefore, 9 1/8 - 2 1/4 = 55/8, which is an improper fraction. If we want to convert it back to a mixed number, we can divide the numerator by the denominator:
55/8 = 6 7/8
So the simplified answer is 6 7/8. Answer: 6 7/8.
Answer:
6 7/8
Step-by-step explanation:
9 1/8 - 2 1/4 =
9 1/8
- 2 1/4
---------------
The common denominator for the fractions is 8.
9 1/8
- 2 2/8
---------------
Since the bottom fraction is greater than ther upper fraction, we borrow 1 from the nine. 1 is the same as 8/8, so we take 1 from 9 and add 8/8 to 1/8.
8 9/8
- 2 2/8
---------------
6 7/8
Answer: 6 7/8
The number 58,391 was rounded to 58,000. What place value was the number rounded to? Hundred Thousand Ten thousand Hundred thousand
Answer:
Thousand
Step-by-step explanation:
Solve:
58,391
Knowing that:
Digit Less than 5 goes downDigit greater than 5 goes upWhen rounding to:
Hundred Thousand - 100,000Ten Thousand - 60,000Hundred - 58,400Thousand - 58,000Because the digit to the right in the hundreds place is 3 which is less than 5- Thus, it goes down to 58,000.
RevyBreeze
Answer:
I think it's Thousand
Tom made some beads and packed all the bead in 14 small boxes and 3 large boxes. She filled each small box with the same number of beads and each large box with the same number of beads. There were 4 more beads in each large box than in each small box. 7/9 of the beads made were packed in the small boxes. How many beads were there in each small box?
Answer:
Each small box has 24/17 beads.
Step-by-step explanation:
Let's call the number of beads in each small box "s" and the number of beads in each large box "l".
We know that there are 14 small boxes, so the total number of beads in the small boxes is 14s. We also know that there are 3 large boxes, so the total number of beads in the large boxes is 3l.
From the problem, we know that there are 4 more beads in each large box than in each small box, so we can write:
l = s + 4
We also know that 7/9 of the total number of beads were packed in the small boxes, so we can write:
14s + 3l = 7/9(total number of beads)
We can simplify this expression by substituting l = s + 4:
14s + 3(s + 4) = 7/9(total number of beads)
Expanding and simplifying:
17s + 12 = 7/9(total number of beads)
Now, we need one more equation to solve for s. We know that the total number of beads is equal to the sum of the beads in the small and large boxes:
total number of beads = 14s + 3(s + 4)
Simplifying:
total number of beads = 17s + 12
Now we have two equations that we can use to solve for s:
17s + 12 = 7/9(total number of beads)
total number of beads = 17s + 12
We don't know the total number of beads, but we do know that it is a multiple of both 14 and 3 (since there are 14 small boxes and 3 large boxes). The smallest multiple of both 14 and 3 is 42, so let's assume that the total number of beads is 42x, where x is some number.
Substituting 42x for "total number of beads" in the two equations:
17s + 12 = 7/9(42x)
42x = 17s + 12
Simplifying:
153s + 108 = 98x
42x = 17s + 12
Now we have two equations and two unknowns (s and x). We can solve for s in terms of x by isolating s in the second equation:
17s = 42x - 12
s = (42x - 12)/17
We can substitute this expression for s into the first equation:
17((42x - 12)/17) + 12 = 7/9(42x)
Simplifying:
42x - 12 + 12 = 14x
28x = 24
x = 24/28 = 0.857
Now we know that the total number of beads is:
42x = 42(0.857) = 36
And we know that each small box has:
s = (42x - 12)/17 = (36 - 12)/17 = 24/17
So each small box has 24/17 beads.
Last week a certain brand of bottled water cost $1.25 for a 16-ounce bottle. This week the water is on sale for $1.00 for a 16-ounce bottle. What is the percent decrease in the price of this bottled water?
The percent decrease in the price of this bottled water is 20%
What is percentage?Percentage can simply defined as the ratio of a number or a variable and 100.
Percentage is represented with the symbol, %.
From the information given, we have that;
Original amount for 16-ounce bottle = $1.25
New amount for 16-ounce bottle = $1. 00
The formula for calculating the percent decrease is represented as;
Percent decrease = original value - new value/original value × 100/1
Substitute the values, we have;
Percent decrease = 1.25 - 1.00/1.25 × 100
Subtract the values
Percent decrease = 20%
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Luke is 5 years younger than 3 times Sydney’s age, s. In this situation, what does 3s represent?
Luke’s age
Sydney’s age
three times Luke’s age
three times Sydney’s age
Based on the given context, 3s represent (d) three times Sydney’s age
Explaining what 3s represent?From the question, we have the following parameters that can be used in our computation:
Sydney's age = s
Luke is 5 years younger than 3 times Sydney’s age
The interpretation of the second statement above is
Luke's age = Sydney's age - 5
Mathematically, this can be expressed as
l = s - 5
Where
Luke's age = l and
Sydney's age = s
Also from the question we have:
3s = 3 times Sydney's age
Hence, 3s represent (d) three times Sydney’s age
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A flower bed is in the shape of a triangle with one side twice the length of the shortest side and the third side is feet more than the length of the shortest side. Find the dimensions if the perimeter is feet.
I got you, buddy!
Let x be the length of the shortest side. Then, the length of the second side is 2x, and the length of the third side is x + y, where y is the additional length.
The perimeter of the triangle is the sum of the three sides, so we have:
x + 2x + (x + y) = 60
Simplifying this equation, we get:
4x + y = 60
We also know that the area of a triangle is given by:
A = (1/2)bh
where b is the length of the base, and h is the height. Since the base of the flower bed is the shortest side of length x, we can choose the height to be y. Therefore, we have:
A = (1/2)xy
Now we need to solve for x and y. We can use the equation for the perimeter to solve for y:
y = 60 - 4x
Substituting this into the equation for the area, we have:
A = (1/2)x(60 - 4x)
Simplifying and expanding, we get:
A = 30x - 2x^2
To maximize the area, we can take the derivative of A with respect to x and set it equal to 0:
dA/dx = 30 - 4x = 0
Solving for x, we get:
x = 7.5
Substituting this back into the equation for y, we get:
y = 60 - 4x = 45
Therefore, the dimensions of the flower bed are:
shortest side = x = 7.5 feet second side = 2x = 15 feet third side = x + y = 52.5 feet
QUICK I NEED HELP! ILL MARK BRAINLIEST IF CORRECT
Suppose you deposit $1000 in a savings account that pays interest at an annual rate of 6%. If no money is added or withdrawn from the account, answer the following questions.
a. How much will be in the account after 4 years?
b. How much will be in the account after 16 years?
c. How many years will it take for the account to contain $1,500?
d. How many years will it take for the account to contain $2,00?
Answer:
a. $1,240
b. $1,960
c. 9 years
d. 17 years
Step-by-step explanation:
For interest you multiply the percent with 1,000 and add the number you get for your answer. So 1,000 times 6% is 60. Each year the account earns $60 worth of interest. You multiply 60 by 4 and your result is 240. You add 240 to 1,000 and get your answer for a $1,240.
You repeat the same process for b. 16 times 60 is 960. You add 960 and 1,000 and your answer is $1,960.
So we know the account already has $1,000 so all that has to be done is to divide 500 by 60 and your result is 8.3. However you it is anual interest so you cannot but in a decimal so you round up to 9. Making it 9 years
Same process dividing 1,000 by 60 which gives you 16.6 rounding up to 17. Making it 17 years.
what is the volume of the cube shown?
The volume of the cube shown can be found to be 132. 65 inch ³ .
How to find the volume of a cube ?To ascertain the volume of a cube , knowledge of one edge's length is mandatory . This side would correspond to all other sides; therefore , multiplying it by three inversely gauges its complete space as a cubic structure.
The formula for the volume of a cube is:
= Side x Side x Side
The side is 5. 1 inch so the volume is:
= 5. 1 x 5 .1 x 5. 1
= 132. 65 inch ³
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12 total, 8 girls, 4 boys, probability boys will be selected
0.333 is the probability of choosing a boy.
The probability of boys being selected can be calculated as the ratio of the number of boys to the total number of children:
Probability of selecting a boy = Number of boys / Total number of children
Probability of selecting a boy = 4 / 12 = 1/3
Therefore, the probability of selecting a boy is 1/3 or approximately 0.333.
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15x^2 + x - 2 = (? -1) (? + 2)
3x, 5x
x, 15x
5x, 3x
15x, x
The correct terms in the expression are,
⇒ 3x and 5x
We have to given that;
The quadratic equation is,
⇒ 15x² + x - 2 = (? - 1) (? + 2)
Now, We can simplify as;
⇒ 15x² + x - 2
⇒ 15x² + 6x - 5x - 2
⇒ 3x (5x + 2) - 1 (5x + 2)
⇒ (3x - 1) (5x + 2)
Thus, The correct terms in the expression are,
⇒ 3x and 5x
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Find the partial sum for the sequence.
{0, −1, −3, −6, −10, ...}; S14
S14=
Answer:
-91
Step-by-step explanation:
To find the partial sum for the sequence
[tex]\large \boxed{\mathrm{ \ 0, \ -1, \ -3, \ -6, \ -10,}}[/tex]
We first notice that the first term is
[tex]\large \boxed{\mathrm{0}}[/tex],
And each subsequent term is the sum of the previous term and the next integer in the sequence starting with [tex]-1.[/tex] Therefore, we can use the formula for the sum of an arithmetic series to find the partial sum S14:
S14 = 14/2 * (2(0) + (14-1)(-1+0)/2) = 14/2 * (13/2 * -1) = -91
Therefore, the partial sum for the sequence up to the [tex]14th[/tex] term is [tex]-91.[/tex]
Answer:
[tex]91[/tex]
Step-by-step explanation:
The partial sum of a sequence is the sum of a certain number of terms in the sequence. For the given sequence
{0, −1, −3, −6, −10, ...},
The nth term can be represented as Tn = n(n-1)/2. This means that the partial sum of the first 14 terms of the sequence, denoted by S14, can be calculated by adding up the first 14 terms using the formula: S14 = 0 + (-1) + (-3) + (-6) + (-10) + ... + T14.
Simplifying this expression using the formula for Tn, We get S14 = 91. Therefore, the partial sum for the given sequence up to the 14th term is equal to 91.
For this problem, we are interested in predicting GNP of a country using the birth rate and life expectancy of females for that country. The estimated regression equation for these data is expressed with the following formula.
$\widehat{GNP_i} = -17684.99 - 132.97(\text{Birth Rate})_i + 307.97(\text{Life Expectancy Female})_i$
1. What is the predicted change in GNP for a one unit increase in birth rate, while holding female life expectancy constant?
2. What is the predicted change in GNP for a one unit increase in female life expectancy, while holding birth rate constant?
3. In a certain country, the birth rate is 23 live births per thousand in the population and the female life expectancy is 62. What is the predicted GNP for this country?
4.Suppose the country in number 3 had an observed GNP of 9832. Calculate the residual for this country.
4) residual =actual -predicted =26708-10376.71 =16331.29
How to solve1) predicted change = -140.74
2) predicted change =317.74
3) predicted GNP =-11093.17-140.74*10+317.74*72 =10376.71
4) residual =actual -predicted =26708-10376.71 =16331.29
A statistical method called regression links a dependent variable to one or more independent (explanatory) variables. A regression model can demonstrate whether changes in one or more of the explanatory variables are related to changes in the dependent variable.
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Find the multiplicative inverse of
[tex] \frac{ - 4}{9} + ( \frac{2}{5} + \frac{3}{8} )[/tex]
Multiplicative inverse of given expression is 9/5.
The given expression,
-4/9 + (2/5 + 3/5)
Simplify it,
⇒ -4/9 + (2+3)/5
⇒ -4/9 + 5/5
⇒ -4/9 + 1
⇒ (-4 + 9)/9
⇒ 5/9
So its multiplicative inverse = 1/(5/9) = 9/5
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find an equivalent equation in rectangular coordinates
r sin theta = 10
The equivalent equation of r sinθ = 10 in rectangular coordinates is y² + y⁴/x² - 100 = 0.
What are the rectangular coordinates?
The rectangular coordinates is calculated from the polar equation as follows;
r sinθ = 10
the conversion from polar to rectangular coordinates;
r² = x² + y²
r = √(x² + y²) ----- (1)
y/x = tanθ ------ (2)
r sinθ = 10
√(x² + y²)(y/x) = 10
(x² + y²)(y²/x²) = 100
y² + y⁴/x² = 100
y² + y⁴/x² - 100 = 0
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An employee is considering two job offers.
First offer: $57,000 yearly salary with an 8% matching 401k
Second offer: $63,000 yearly salary with a 4% matching 401k
The employee plans to stay at either job for at least 4 years, assumes there are no salary increases, and will make 401k contributions at the same rate the company matches. After 4 years, the total value of the first offer, including gross income and total 401k contributions, is $264,480.
Which job has the better overall pay structure, and by how much?
(MULTIPLE CHOICE ONLY)
The second job offer is better by $6,780.
The first job offer is better by $6,780.
The second job offer is better by $7,680.
The first job offer is better by $7,680.
The second job offer is better by $21,840.
We have :
An employee is considering two job offers.
First offer: $57,000 yearly salary with an 8% matching 401k
Second offer: $63,000 yearly salary with a 4% matching 401k
Let's calculate the total value of the second offer after 4 years:
Yearly salary is : $63,000
Total gross income after 4 years: $63,000 x 4 = $252,000
The company matches 5% of the employee's salary.
So, the employee contributes 5% of $63,000 = $3,150 per year
After 4 years,
Company matches: $3,150 x 4 = $12,600
Employee contributions: $3,150 x 4 = $12,600
Total 401k contributions: $12,600 + $12,600 = $25,200
The total value of the second offer after 4 years is:
Total value: $252,000 + $25,200 = $277,200
Now, let's compare this to the total value of the first offer, which is : $255,360.
Therefore, the second job offer is better by:
$277,200 - $255,360 = $21,840
Hence, The second job offer is better by $21,840.
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5. How far does the tip of a minute hand on a clock travel in 45 minutes if the
distance from the center to the tip is 10 in? Leave your answer in terms of .
work:
distance traveled =
15
in.
Answer:
The tip of the minute hand on a clock travels 15π inches in 45 minutes if the distance from the center to the tip is 10 inches.
Which numbers are equivalent to 10 ÷ 5? Choose ALL the correct answers.
10
5
5
10
2
1
2
5
HELPPPP
Which linear equation is represented in the graph?
A. y = x – 1
B.y = 2x – 1
C. y = x + 1
D. y = 3x – 1
