The Galois group of f over R is isomorphic to the symmetric group S4. The given statement "Each intermediate field of F/Q is a Galois extension of Q." is true because f is separable.
The roots of f are given by
x = ±(√(5) + √(2)), ±(√(5) - √(2))
Let φ be the automorphism defined by
φ(√(5) + √(2)) = √(5) + i√(2)
φ(√(5) - √(2)) = √(5) - i√(2)
φ(-√(5) + √(2)) = -√(5) - i√(2)
φ(-√(5) - √(2)) = -√(5) + i√(2)
where i is the imaginary unit. Then φ has order four since φ⁴ is the identity automorphism. Let σ be the automorphism defined by
σ(√(5) + √(2)) = -√(5) + √(2)
σ(√(5) - √(2)) = √(5) - √(2)
σ(-√(5) + √(2)) = -√(5) - √(2)
σ(-√(5) - √(2)) = √(5) + √(2)
Then σ has order two since σ² is the identity automorphism. It can be shown that the Galois group of f over Q is generated by φ and σ. Since φ has order four and σ has order two, the Galois group is a non-abelian group of order eight.
To prove or disprove that each intermediate field of F/Q is a Galois extension of Q, we need to show that each intermediate field is a splitting field of a separable polynomial over Q. Since F is the splitting field of f over Q, any intermediate field of F/Q is also a splitting field of f over Q. Since f is separable (its roots are distinct), every intermediate field of F/Q is a Galois extension of Q, and hence a splitting field of a separable polynomial over Q. Therefore, the statement is true.
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a. Express each of the following as a function of 15°
6. tan165°
7. sin285°
Expressing each as a function of 15°, we have 6. tan165° = tan(150 + 15)° and 7. sin285° = sin(270 + 15)°
Expressing each as a functionFrom the question, we have the following trigonometry function that can be used in our computation:
6. tan165°
7. sin285°
The above expressions are tangent and sine functions of 165 and 285 degrees
By mathematical expression, we have
165° = 150 + 15
285° = 270 + 15
This means that we can replace the expressions with 150 + 15 and 270 + 15
So in terms of 15 degrees, we have
6. tan165° = tan(150 + 15)° and 7. sin285° = sin(270 + 15)°
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IVE SPENT TWO DARN DAYS ON THIS QUESTION AND I DON'T GET IM SUPPOSED TO KNOW THIS OMG `
Answer:
A rhombus has four congruent sides.
The diagonals of a rhombus bisect each other and form right angles.
From these:
[tex] {(2y})^{2} + {2}^{2} = {(3y - 2)}^{2} [/tex]
[tex]4 {y}^{2} + 4 = 9 {y}^{2} - 12y + 4[/tex]
[tex]5 {y}^{2} - 12y = 0[/tex]
[tex]5y - 12 = 0[/tex]
[tex]5y = 12[/tex]
[tex]y = 2.4[/tex]
Answer:
y=2.4
Step-by-step explanation:
We know that a rhombus has 4 equal sides, and 2 diagonals which bisect eachother at an angle of 90 degrees.
Simply saying, a rhombus is made up of 4 right angled triangles.
If QR is 3y-2, that means that each other side must be 3y-2 aswell, which means that SQ is also 3y-2
Now we can use the pythagorean theorem to solve for y, given that the base is 2 (length of QU), the height is 2y (length of SU), and the hypotenuse is 3y-2 (length of SQ)
(3y-2)²=2²+2y² (Expand using binomial formula)
9y²-12y+4=4+4y² (Collect similar terms)
5y²-12y=0 (Factor the expression)
y(5y-12)=0
We know that if a product of 2 factors is 0, atleast one must also be 0
So we have to make both factors equal 0
y=0
5y-12=0
5y=12
y=12/5, or 2.4
We see that 2.4 is the only possible solution.
Hope this helps!
Write an explicit formula for an, the nth term of the sequence 25,30,35,....
Answer:
[tex]a_{n}[/tex] = 5n + 20
Step-by-step explanation:
there is a common difference between consecutive terms in the sequence
30 - 25 = 35 - 30 = 5
this indicates the sequence is arithmetic with explicit formula
[tex]a_{n}[/tex] = a₁ + d(n - 1)
where a₁ is the first term and d the common difference
here a₁ = 25 and d = 5 , then
[tex]a_{n}[/tex] = 25 + 5(n - 1) = 25 + 5n - 5 = 5n + 20
A young couple purchases their first new home in 2011 for $95,000. They sell it to move into a bigger home in 2018 for $105,000. First, we will develop an exponential model for for the value of the home. The model will have the form V(t) = V0e^kt. Let t be years since 2011 and V (t) be the value of the home.
a. The home's value is increasing at a rate of approximately 1.55% per year.
b. The exponential model is V(t) = [tex]95000e^{(0.0155t)}[/tex]
c. The predicted value of the home in 2022 is approximately $123,240.
d. The home's value reached $130,000 approximately 20 years after 2011, which is in the year 2031.
To find the growth rate k, we can use the formula [tex]V(t) = V0e^{(kt)}[/tex]. We have two data points: V(0) = 95000 and V(7) = 105000 (since 2018 is 7 years after 2011). Inputting these values into the formula yields:
V(0) =[tex]V0e^{(0k)}[/tex] = V0
V(7) = [tex]V0e^{(7k)}[/tex]= 105000
When the second equation is divided by the first, the result is:
V(7)/V(0) = [tex]e^{(7k)}[/tex] = 105000/95000 = 1.1053
Taking the natural logarithm of both sides and calculating k yields:
ln(1.1053) = 7k
k ≈ 0.0155
This means that the home's value is increasing at a rate of approximately 1.55% per year.
b. The exponential model is V(t) = [tex]95000e^{(0.0155t)}[/tex].
c. The predicted value of the home in 2022 is approximately $123,240.
To predict the home's value in 2022, we need to find the value of V(t) when t = 11 (since 2022 is 11 years after 2011). Substituting into the exponential model gives:
V(11) = [tex]95000e^{(0.0155*11)}[/tex] ≈ $123,239.62
So the predicted value of the home in 2022 is approximately $123,240.
d. The home's value reached $130,000 approximately 20 years after 2011, which is in the year 2031.
We need to solve for t in equation V(t) = 130,000.
130,000 = [tex]95,000e^{(0.0155t)}[/tex]
Dividing both sides by 95,000 and taking the natural logarithm of both sides, we get:
ln(130,000/95,000) = 0.0155t
Simplifying, we get:
t = (ln(130,000/95,000))/0.0155
t = (ln(1.368)/0.0155
t = 0.313 / 0.0155
t ≈ 20.23
Therefore, the home's value reached $130,000 approximately 20 years after 2011, which is in the year 2031.
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Complete question:
A young couple purchases their first new home in 2011 for $95,000. They sell it to move into a bigger home in 2018 for $105,000. First, we will develop an exponential model for the value of the home. The model will have the form V(t) = V0e^(kt) . Let t be years since 2011 and v (t) be the value of the home.
a. What is the growth rate k for the model? What does that number mean?
b. What is the exponential model?
c. Predict the value of the home in 2022.
d. During what year did the value of the home reach $130,000?
complete the table AB=1?, BC=3?
Answer:
angle B = 60 degree
AB = 2 cm
BC = 2 cm
Step-by-step explanation:
an equilateral triangle is a triangle with the following properties:
1. all sides equal
2. all angels equal to 60 degree
As a result:
angle B = 60 degree
AB = 2 cm
BC = 2 cm
Find the measure of each angle indicated.
P
U
160°
W
95° ?
V
Answer:
? = 65°
Step-by-step explanation:
the exterior angle of a triangle is equal to the sum of the 2 opposite interior angles.
∠ PUV is an exterior angle of the triangle , then
? + 95° = 160° ( subtract 95° from both sides )
? = 65°
Place value of 4 in 265 473
Answer:
The place value of 4 in 265 473 is 40,000
Answer:
Step-by-step explanation:
Order:
3 is in the ones place and has a value of 3
7 is in the tens place and has a value of 70
4 is in the hundreds place and has a value of 400
5 is in the thousands place and has a value of 5000
6 is in the ten thousands place and has a value of 60,000
2 is in the hundred thousands place and has a value of 200,000
Pls help me with this problem pls right answer only pls, quick!
Answer:
This is a quadratic function.
Which set of data could be reasonably modeled by a quadratic function?
The quadratic function is represented by option A. both graph.
From the set of data representing in the graph ,
A quadratic function is a second-degree polynomial of the form f(x) = ax² + bx + c, where a, b, and c are constants.
It represents a parabolic curve when plotted on a graph.
Generally, a quadratic function is suitable for modeling data that shows a U-shaped or inverted U-shaped pattern.
Representation of x-axis and y-axis is not clearly specified in the graph .
We assume different conditions.
Projectile motion,
The trajectory of a ball thrown into the air or a projectile fired from a cannon can be modeled by a quadratic function.
Falling objects,
The distance a dropped object falls over time can be modeled by a quadratic function.
Profit vs. production,
The relationship between a company's profit and the level of production can be modeled by a quadratic function.
The profit generally increases with production but eventually reaches a maximum point before declining.
Trajectory of a car,
The distance traveled by a car accelerating from a stationary position can be modeled by a quadratic function.
Height vs. time,
The height of an object thrown upwards, such as a bouncing ball, can be modeled by a quadratic function.
The height increases, reaches a maximum point, and then decreases.
Here both the graphs represents the quadratic function.
Therefore, the set of the data clearly modelled the quadratic function is present by option A. Both graph.
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The circumference of a circle is 20 � π cm. Find its radius, in centimeters.
The required radius of the circle is 10 cm.
The formula for the circumference C of a circle in terms of its radius r is given by:
C = 2πr
We are given that the circumference of the circle is 20π cm. So we can set this expression equal to 20π and solve for r:
20π = 2πr
Dividing both sides by 2π, we get:
r = 10
Therefore, the radius of the circle is 10 cm.
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3.) Given the regular polygon. Find the measure of each numbered angle.
m<1 = 30
m<2 = 15
m<3 =
75
Work/expanation:
The b. m<angle1 = 45°,
mangle2 = 67.5°
How to solveAn octagon has 8 sides.
Sum of interior angles = 180° x (n-2) = 180° x (8-2) = 180° x 6 = 1,080°
1,080° ÷ 8 sides = 135° interior angle.
Angle 2 = 135° / 2 = 67.50°
Angle 1 = 180° - 135° = 45°
If these were the choices given:
a. mangle1 = 45°, mangle2 = 135°
b. mangle1 = 45°, mangle2 = 67.5° ⇔ THIS IS THE CORRECT ANSWER.
c. mangle1 = mangle2 = 60°
d. mangle1 = 22.5°, mangle2 = 78.75°
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ABC~ DEF. What sequence of transformations will move ABC onto DEF?
A. A dilation by a scale factor of 2, centered at the origin, followed by a reflection over the y-axis.
B. A dilation by a scale factor of 1/2, centered at the origin, followed by the translation (x,y)-> (x+7,y)
C. The translation (x,y)->(x+7,y), followed by a dilation by a scale factor of 2 centered at the origin.
D. A dilation by a scale factor of 2, centered at the origin, followed by the translation (x,y)-> (x+7,y)
Note that the sequence of transformations that will move ABC onto DEF is; "The translation (x,y)->(x+7,y), followed by a dilation by a scale factor of 2 centered at the origin." *Option C)
What is transformation?A transformation is a mathematical function that repositions points in one-, two-, three-, or any other n-dimensional space.
A dilation is a function f from a metric space M into itself that fulfills the identity d=rd for all locations x, y in M, where d is the distance between x and y and r is some positive real integer. Such a dilatation is a resemblance of space in Euclidean space.
Note that translation takes B(0,0) to B'(7, 0),
Takes A(0, 4) to A'(7, 0) and C (3, 0) to C' (10, 0)
when you multiply the new points by the scale factor of 2, you'd get the new figure when the coordinates are graphed.
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Answer:
D. Dilation by 2, translation right by 7.
Step-by-step explanation:
You want to know the sequence of transformations that maps ∆ABC to ∆DEF.
Scale factorSegment DE is 8 units long, and located at x=7. Segment AB is 4 units long and located at x=0.
The scale factor is the ratio of segment lengths:
DE/AB = 8/4 = 2
Triangle DEF is dilated by a factor of 2 (eliminates B).
TranslationTriangle DEF is oriented the same way as triangle ABC, so there is no reflection involved (eliminates A). As we noted, segment DE is 7 units to the right of segment AB, so a translation of x ⇒ x+7 is involved.
SequenceThe translation must occur after the dilation about the origin (eliminates C). If it were to occur first, the translation would be multiplied by the scale factor, so that DE would end up at x=14.
The required sequence of transformations is ...
D. A dilation by a scale factor of 2, centered at the origin, followed by the translation (x, y) ⇒ (x+7, y).
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What is the slope of the line passing through points (4, 6) and (2, 4)? A. 3 B. 6 C. 4 D. 2 E. 1
Answer: E. 1
Explanation: The slope (m) of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
We can use this formula to find the slope of the line passing through points (4, 6) and (2, 4):
m = (y2 - y1) / (x2 - x1)
m = (4 - 6) / (2 - 4)
m = -2 / -2
m = 1
The slope of the line passing through points (4, 6) and (2, 4) is 1.
The answer is E) 1.
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. In ΔABC, m < B = 14°, m < C = 42° and a = 34. Find the length of b to the nearest tenth.
i Got u Bro!
Obtuse scalene triangle.
Sides: a = 34 b = 9.922 c = 27.442
Area: T = 112.86
Perimeter: p = 71.364
Semiperimeter: s = 35.682
Angle ∠ A = α = 124° = 2.164 rad
Angle ∠ B = β = 14° = 0.244 rad
Angle ∠ C = γ = 42° = 0.733 rad
Height: ha = 6.639
Height: hb = 22.75
Height: hc = 8.225
Median: ma = 11.694
Median: mb = 30.495
Median: mc = 20.951
Inradius: r = 3.163
Circumradius: R = 20.506
Vertex coordinates: A[27.442; 0] B[0; 0] C[32.99; 8.225]
Centroid: CG[20.144; 2.742]
Coordinates of the circumscribed circle: U[13.721; 15.239]
Coordinates of the inscribed circle: I[25.76; 3.163]
Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 56° = 2.164 rad
∠ B' = β' = 166° = 0.244 rad
∠ C' = γ' = 138° = 0.733 rad
evaluate.
20
Σ4 (8)n-1
n = 1
S = [?]
Answer:
32.5861
Step-by-step explanation:
Which expression is equivalent to 3w-4-w-93w−4−w−9?
A 2w-13
b 4w - 13
c 2w + 5
d -w - 13
Answer: A
Step-by-step explanation:
To simplify 3w-4-w-9, we need to combine the like terms (terms that have the same variable and exponent).
3w - w = 2w
-4 - (-9) = -4 + 9 = 5
So,
3w - 4 - w - 9 = 2w - 13
Therefore, the expression 3w-4-w-9 is equivalent to 2w-13, which is option A.
Write an explicit formula for an, the n^th term of the sequence
2,−8,32,....
Answer:
36
Step-by-step explanation
6x6
10. A pizza with a diameter of 5 inches is divided into 8 sectors
of equal area.
What is the approximate area of each pizza slice? Round
your answer to the nearest hundredth.
O
O
The approximate area of each pizza slice is 2.45 sq inches
What is the approximate area of each pizza slice?From the question, we have the following parameters that can be used in our computation:
Diameter = 5 inches
This means that
Radius, r = 5/2 inches
Evaluate
Radius, r = 2.5 inches
The approximate area of each pizza slice is calculated as
Area = Circle area/number of sectors
So, we have
Area = 3.14 * 2.5^2/8
Evaluate
Area = 2.45 sq inches
Hence, the approximate area of each pizza slice is 2.45 sq inches
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Ethan's class is handing out balloon arrangements consisting of 4 balloons each. Each balloon has an equally likely chance of being red, blue, or yellow. Ethan created a spinner to simulate this probability. Here is Ethan's data from 30 trials of 4 spins on the spinner (r = red, b = blue, y = yellow):
ybrr bybb rbbb rrry rbbb rryy yybb ryry ryyr rbyr ryyr rryy rrrr yrby bbyy byyr byyr byyb rbby ybry rryy rbrr rybb bbbr rrbr ybry ryrr rryb rrrb rryr
According to this data, what is the experimental probability that an arrangement will consist entirely of balloons that are the same color?
0.012
0.1
0.033
0.33
The requried, experimental probability is 0.1. Option B is correct.
To find the experimental probability that an arrangement will consist entirely of balloons that are the same color, we need to count the number of times that Ethan got an arrangement where all four balloons were the same color and divide that by the total number of trials.
Looking through the data, we can see that the following trials had all four balloons the same color:
rrrr
bbbb
yyyy
So, out of 30 trials, there were 3 trials where all four balloons were the same color. Therefore, the experimental probability is:
3/30 = 0.1
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STAINED GLASS Pablo made the stained glass
window shown. He used an inscribed square and
equilateral triangle for the design.
a. Label the angle measures on the outer edge of the
triangle.
b. Label all of the arcs with their degree measure.
a) The angles outside the side PQ measure 65° and 25°
The angles outside the side QR measure 85° and 95°
The angles outside the side PR measure 55° and 35°
b) The arc PR, PQ and PR measure 120°
a)
Let us assume that the inscribed square be ABCD and equilateral triangle is PQR.
Let side PR of triangle intersects the quadrilateral at points M and N.
So, we get a triangle DMN outside the equilateral triangle.
Here, ∠DMN = 55°, ∠D = 90°
So, ∠MND = 180 - (90 + 55)
= 35°
Similarly, side PQ of triangle PQR intersects the quadrilateral at points X and Y.
So, we get a triangle AXY outside the equilateral triangle.
From triangle PXM we get the measure of angle PXM which is 65 degrees.
As angle PXM and angle AXY are opposite angles, the measure of angle AXY = 65°
∠A = 90°
So, ∠AYX = 180° - (∠A + ∠AXY )
= 180° - (90° + 65°)
= 25°
Let the side QR of triangle PQR intersects the quadrilateral at points L and K.
so we get new quadrilateral LKBC
In this quadrilateral ∠B = 90°, ∠C = 90°
From triangle QYL, we get the measure of angle QLY = 95°
So, the measure of ∠KLB = 95° .......(∠QLY and ∠KLB opposite angles)
So, the remaining angle LKC of quadrilateral LKBC would be,
∠LKC = 360° - (∠B + ∠C + ∠KLB)
∠LKC = 360° - (90 °+ 90° + 95°)
∠LKC = 85°
b)
We know that he angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
so, the measure of arc PR = 120°
the measure of arc PQ = 120°
the measure of arc RQ = 120°
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Random samples of resting heart rates are taken from two groups. Population 1 exercises regularly, and Population 2 does not. The data from these two samples is given below:
Population 1: 69, 68, 62, 68, 68, 62, 69
Population 2: 76, 68, 68, 72, 76, 70, 70, 70
Is there evidence, at an α=0.025
level of significance, to conclude that those who exercise regularly have lower resting heart rates? (Assume that the population variances are equal.) Carry out an appropriate hypothesis test, filling in the information requested.
A. The value of the standardized test statistic:
B. The rejection region for the standardized test statistic:
C. Your decision for the hypothesis test:
A. The value of the standardized test statistic is -2.391.
B. The rejection region for the standardized test statistic is t <-2.160.
C. The decision for the hypothesis test is to reject the null hypothesis.
How to explain the hypothesisEmploying a t-distribution table or calculator along with 13 degrees of freedom (n1 + n2 - 2) enables us to ascertain the rejection region for the t-statistic at an α=0.025 level of significance (two-tailed test):
Rejection Region: t < -2.160 or t > 2.160
Given that -2.391 falls in the rejection region, we repudiate the null hypothesis. At the α=0.025 level of significance, we are provided sufficient evidence to support that those who consistently partake in exercise generally have a lower resting heart rate than those who do not.
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ILL GIVE 20 POINTS !!!! PLSS HELP ASAP
Answer: 10/40 or 1/4
Step-by-step explanation:
Add up all the amount of rolls, so 10+6+4+8+6+6=40
then add the probability of rolling a 3 or 6 = 10
divide that quantity out of the whole
answer is 10/40 and simplified is 1/4
hope it helps
In which quadrant would an angle x lie if sinx is negative and tanx is negative?
Quadrant I
Quadrant II
Quadrant III
Quadrant IV
Step-by-step explanation:
If sin is neg AND tan is neg then cos is positive
(because tan = sin/cos)
this occurs in Q IV
Today, you want to sell a R1 000 par value zero coupon bond you own. The bond matures in five years. How much will you receive for your bond, if the market yield to maturity is currently 5,33%? 10
With a market yield to maturity of 5.33%, the amount that would be received for a R 1,000 par value zero coupon bond that matures in five years, but sold today will be R747.26.
Price calculation for coupon bondsTo calculate the price you will receive for your zero coupon bond, we can use the formula:
P = F / (1 + r)^n
where:
P is the priceF is the par value (face value) of the bondr is the yield to maturity as a decimaln is the number of years to maturity.In this case, the par value is R1,000, the yield to maturity is 5.33%, or 0.0533 as a decimal, and the number of years to maturity is 5.
P = 1000 / (1 + 0.0533)^5P ≈ R747.26Therefore, you can expect to receive approximately R747.26 for your R1,000 par value zero coupon bond when the market yield to maturity is 5.33%.
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Which ones apply to the question
Answer:
1 and 3 are correct.
1 is correct (supplementary angles)
3 is correct (1 and 3 are vertical angles)
4.) What is the area of the sector with a central angle of 60° and a radius of 2m?
Round your answer to the nearest tenth or leave in terms of .
area =
A
2 60
B
2/3
m²
Work:
The "area" of sector which is having "central-angle" of 60° and the radius of 2m is 2.094 m².
The "Sector" of circle is defined as region bounded by "two-radii" of circle and arc intercepted between them. The two radii that bound the sector are called the "sides" of the sector.
The center of the circle is called as vertex of the sector. The area of a sector can be found using the formula : A = (θ/360)πr², where θ is = angle of sector;
The measure of "central-angle" is = 60°, and radius of circle is = 2m,
So, Area is = (60/360)π(2)²,
⇒ Area is = 4π/6 ≈ 2.094 m²,
Therefore, area of sector is 2.094 m².
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The given question is incomplete, the complete question is
What is the area of the sector with a central angle of 60° and a radius of 2m?
Select the correct answer. A company manufactures computers. Function N represents the number of components that a new employee can assemble per day. Function E represents the number of components that an experienced employee can assemble per day. In both functions, t represents the number of hours worked in one day. Which function describes the difference of the number of components assembled per day by the experienced and new employees? A. B. C. D. Reset Next
The function that describes the difference of the number of components assembled per day by the experienced and new employees is:
E(t) - N(t)
This is because E(t) represents the number of components assembled by an experienced employee in t hours, and N(t) represents the number of components assembled by a new employee in t hours. Therefore, the difference between the two functions will give us the difference in the number of components assembled per day by the two types of employees.
Out of the given options, option A represents the correct answer:
E(t) - N(t) = (5t) - (3t) = 2t
Therefore, the correct answer is A.
For the following vectors, (a) find the dot product v•w ; (b) find the angle between v and w , (c) state whether the vectors are parallel, octagonal, or neither. V=-3i-4j, w=6i+8j
A- v•w
B-the angle between v and w is theta ^•?
C- the vectors v and w are?
Need Help!!!! A pre-image has coordinates J(3, -6) and K(-1, -2). The image has coordinates J'(6, 3) and K'(2, -1). Describe the clockwise rotational path of the line segment.
After considering the given data we conclude that the clockwise rotational path of the line segment is a rotation of -59.04 degrees about the point (-6, -1).
We have to evaluate the center and angle of rotation to explain the clockwise rotation of the line segment.
So in the first step, we can evaluate the midpoint of the line segment JK and the midpoint of the line segment J'K'. we can calculate the vector connecting the midpoint of JK to the midpoint of J'K'. This vector is (4-1, 1-(-4) = (3,5)
The center of rotation is the point that is equidistant from the midpoints of JK and J'K'. We can evaluate this point by finding the perpendicular bisector of the line segment connecting the midpoints.
The slope of this line is the negative reciprocal of the slope of the vector we just found, which is -3/5. We can apply the midpoint formula and the point-slope formula to evaluate the equation of the perpendicular bisector:
Midpoint of JK: (1, -4)
Midpoint of J'K': (4, 1)
The slope of the vector: 3/5
(x₁ + x₂)/2, (y₁ + y₂) /2
Point-slope formula: y - y₁ = m(x - x₁)
Perpendicular bisector: y - (-4) = (- 3/5)(x - 1)
Applying simplification , we get: y = (- 3/5)x - 1.2
To evaluate the center of rotation, we need to find the intersection point of the perpendicular bisector and the line passing through the midpoints of JK and J'K'. This line has slope ( 3 - (4)) /(4 - 1) = 7/3 and passes through the point (4, 1). Applying the point-slope formula, we can evaluate its equation:
y - 1 = (7/3)( x - 4)
Apply simplification , we get: y = (7/3)x - 17/3
To evaluate the intersection point, we can solve the system of equations:
y =(- 3/5)x - 1.2 = (7/3)x - 17/3
Evaluating for x and y, we get x = -6 and y = -1.
Therefore, the center of rotation is (-6, -1).
√( 4 - 1)² + ( 1 - ( - 4))²) = 5√(2)
Distance between image points and center of rotation
√( ( 6 - (-6))² + ( 3 - (-1))² = 13
The ratio of these distances gives us the scale factor of the transformation, which is 13/√2).
The angle of rotation is negative as the image moves clockwise direction. We can apply the inverse tangent function to find the angle of the vector connecting the midpoint of JK to the midpoint of J'K':
Angle of vector: arctan(5/3) = 59.04 degrees
Therefore, the clockwise rotational path of the line segment is a rotation of -59.04 degrees about the point (-6, -1).
To learn more about midpoint formula
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Fred brought a shirt and a wallet for a total of 34.26 before tax. The price of the wallet was $15 less than the price of the shirt. What was the price of the shirt?
Answer:
The price of the shirt is $24.63.
Step-by-step explanation:
Let's assume the price of the shirt is x. Then the price of the wallet will be x - 15. We know that the total cost before tax is 34.26, so we can set up the equation:
x + (x - 15) = 34.26
Simplifying, we get:
2x - 15 = 34.26
Adding 15 on both sides:
2x = 49.26
Dividing by 2:
x = 24.63
Therefore, the price of the shirt is $24.63.
