Find the area of ΔABC . Round your answer to the nearest tenth
m ∠ C=68°, b=12,9, c=15.2
To find the area of triangle ΔABC, we can use the formula for the area of a triangle given its side lengths, also known as Heron's formula. Heron's formula states that the area (A) of a triangle with side lengths a, b, and c is:
A = [tex]\sqrt{(s(s-a)(s-b)(s-c))}[/tex]
where s is the semi perimeter of the triangle, calculated as:
s = (a + b + c)/2
In this case, we have the side lengths b = 12, a = 9, and c = 15.2, and we know that ∠C = 68°.
s = (9 + 12 + 15.2)/2 = 36.2/2 = 18.1
Using Heron's formula, we can calculate the area:
A = [tex]\sqrt{(18.1(18.1-9)(18.1-12)(18.1-15.2))}[/tex]
A ≈ 49.9
Therefore, the area of triangle ΔABC, rounded to the nearest tenth, is approximately 49.9 square units.
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Find the determinant of the matrix
[2+2x³ 2-2x² + 4x³ 0]
[-x³ 1+ x² - 2x³ 0]
[10 + 6x² 20+12x² -3-3x²]
and use the adjoint method to find M-1
det (M) =
M-1=
The determinant of the matrix M is 0, and the inverse matrix [tex]M^{-1}[/tex] is undefined.
To find the determinant of the matrix and the inverse using the adjoint method, we start with the given matrix M:
[tex]M = \[\begin{bmatrix}2+2x^3 & 2-2x^2+4x^3 & 0 \\-x^3 & 1+x^2-2x^3 & 0 \\10+6x^2 & 20+12x^2-3-3x^2 & 0 \\\end{bmatrix}\][/tex]
To find the determinant of M, we can use the Laplace expansion along the first row:
[tex]det(M) = (2+2x^3) \[\begin{vmatrix}1+x^2-2x^3 & 0 \\20+12x^2-3-3x^2 & 0 \\\end{vmatrix}\] - (2-2x^2+4x^3) \[\begin{vmatrix}-x^3 & 0 \\10+6x^2 & 0 \\\end{vmatrix}\][/tex]
[tex]det(M) = (2+2x^3)(0) - (2-2x^2+4x^3)(0) = 0[/tex]
Therefore, the determinant of M is 0.
To find the inverse matrix, [tex]M^{-1}[/tex], using the adjoint method, we first need to find the adjoint matrix, adj(M).
The adjoint of M is obtained by taking the transpose of the matrix of cofactors of M.
[tex]adj(M) = \[\begin{bmatrix}C_{11} & C_{21} & C_{31} \\C_{12} & C_{22} & C_{32} \\C_{13} & C_{23} & C_{33} \\\end{bmatrix}\][/tex]
Where [tex]C_{ij}[/tex] represents the cofactor of the element [tex]a_{ij}[/tex] in M.
The inverse of M can then be obtained by dividing adj(M) by the determinant of M:
[tex]M^{-1} = \(\frac{1}{det(M)}\) adj(M)[/tex]
Since det(M) is 0, the inverse of M does not exist.
Therefore, [tex]M^{-1}[/tex] is undefined.
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need help asap if you can pls!!!!!!
Answer:
Step-by-step explanation:
perpendicular bisector AB is dividing the line segment XY at a right angle into exact two equal parts,
therefore,
ΔABY ≅ ΔABX
also we can prove the perpendicular bisector property with the help of SAS congruency,
as both sides and the corresponding angles are congruent thus, we can say that B is equidistant from X and Y
therefore,
ΔABY ≅ ΔABX
Henry works in a fireworks factory, he can make 20 fireworks an hour. For the first five hours he is paid 10 dollars, and then 20 dollars for each additional hour after those first five. What is the factory's total cost function and its Average Cost? And graphically depict the curves.
The factory's total cost function is $20x - $50 and Average cost function is (20x - 50) / x
Henry works in a fireworks factory and can make 20 fireworks an hour. He earns $10 for the first five hours and $20 for each additional hour after that. The factory's total cost function is a linear function that has two segments. One segment will represent the cost of the first five hours worked, while the other segment will represent the cost of each hour after that.
The cost of the first five hours is $10 per hour, which means that the total cost is $50 (5 x $10). After that, each hour costs $20. Therefore, if Henry works for "x" hours, the total cost of his work will be:
Total cost function = $50 + $20 (x - 5)
Total cost function = $50 + $20x - $100
Total cost function = $20x - $50
Average cost is the total cost divided by the number of hours worked. Therefore, the average cost function is:
Average cost function = total cost function / x
Average cost function = (20x - 50) / x
Now, let's graphically depict the curves. The total cost function is a linear function with a y-intercept of -50 and a slope of 20. It will look like this:
On the other hand, the average cost function will start at $10 per hour and decrease as more hours are worked. Eventually, it will approach $20 per hour as the number of hours increases. This will look like this:
By analyzing the graphs, we can observe the relationship between the total cost and the number of hours worked, as well as the average cost at different levels of production.
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Reflect triangle ABC with vertices at A(0, 2), B(-8, 8), C(0, 8) over the line y = -1. Then reflect that
triangle over the y-axis. Graph all three figures.
A graph of the resulting triangles after a reflection over the line y = -1 and over the y-axis is shown in the images below.
How to transform the coordinates of triangle ABC?In Mathematics, a reflection across the line y = k and y = -1 can be modeled by the following transformation rule:
(x, y) → (x, 2k - y)
(x, y) → (x, -2 - y)
Ordered pair A (0, 2) → Ordered pair A' (0, -4).
Ordered pair B (-8, 8) → Ordered pair B' (-8, -10).
Ordered pair C (0, 8) → Ordered pair C' (0, -10).
By applying a reflection over the y-axis to the coordinate of the given triangle ABC, we have the following coordinates for triangle A"B"C":
(x, y) → (-x, y).
Ordered pair A (0, 2) → Ordered pair A" (0, 2).
Ordered pair B (-8, 8) → Ordered pair B" (8, 8).
Ordered pair C (0, 8) → Ordered pair C" (0, 8).
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For a sequence \( 3,9,27 \)...find the sum of the first 5 th term. A. 51 B. 363 C. 243 D. 16
The sum of the first 5 term of the sequence 3,9,27 is 363.
What is the sum of the 5th term of the sequence?Given the sequence in the question:
3, 9, 27
Since it is increasing geometrically, it is a geometric sequence.
Let the first term be:
a₁ = 3
Common ratio will be:
r = 9/3 = 3
Number of terms n = 5
The sum of a geometric sequence is expressed as:
[tex]S_n = a_1 * \frac{1 - r^n}{1 - r}[/tex]
Plug in the values:
[tex]S_n = a_1 * \frac{1 - r^n}{1 - r}\\\\S_n = 3 * \frac{1 - 3^5}{1 - 3}\\\\S_n = 3 * \frac{1 - 243}{1 - 3}\\\\S_n = 3 * \frac{-242}{-2}\\\\S_n = 3 * 121\\\\S_n = 363[/tex]
Therefore, the sum of the first 5th terms is 363.
Option B) 363 is the correct answer.
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Problem 2: (10 pts) Let F be ordered field and a F. Prove if a > 0, then a > 0; if a < 0, then a-1 <0.
Both statements
1. If a > 0, then a > 0.
2. If a < 0, then a - 1 < 0.
have been proven by using the properties of an ordered field.
Why does the inequality hold true for both cases of a?To prove the statements:
1. If a > 0, then a > 0.
2. If a < 0, then a - 1 < 0.
We will use the properties of an ordered field F.
Proof of statement 1:Assume a > 0.
Since F is an ordered field, it satisfies the property of closure under addition.
Thus, adding 0 to both sides of the inequality a > 0, we get a + 0 > 0 + 0, which simplifies to a > 0.
Therefore, if a > 0, then a > 0.
Proof of statement 2:Assume a < 0.
Since F is an ordered field, it satisfies the property of closure under addition and multiplication.
We know that 1 > 0 in an ordered field.
Subtracting 1 from both sides of the inequality a < 0, we get a - 1 < 0 - 1, which simplifies to a - 1 < -1.
Since -1 < 0, and the ordering of F is preserved under addition, we have a - 1 < 0.
Therefore, if a < 0, then a - 1 < 0.
In both cases, we have shown that the given statements hold true using the properties of an ordered field. Hence, the proof is complete.
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can you help me find constant A? 2.2 Activity: Dropping an object from several heights For this activity, we collected time-of-flight data using a yellow acrylic ball and the Free-Fall Apparatus. Taped to the yellow acrylic ball is a small washer. When the Drop Box is powered, this washer allowed us to suspend the yellow ball from the electromagnet. Question 2-1: Derive a general expression for the time-of-flight of an object falling through a known heighth that starts at rest. Using this expression, predict the time of flight for the yellow ball. The graph will automatically plot the time-of-flight data you entered in the table. Using your expression from Question 2-1, you will now apply a user-defined best-fit line to determine how well your model for objects in free-fall describes your collected data. Under the Curve Fitting Tool, select "User-defined." You should see a curve that has the form "A*x^(1/2)." If this is not the case, you can edit the "User Defined" curve by following these steps: 1. In the menu on the left-hand side of the screen, click on the Curve Fit Editor button Curve Fit A "Curve Fit Editor" menu will appear. 2. Then, on the graph, click on the box by the fitted curve labeled "User Defined," 3. In the "Curve Fit Editor" menu, type in "A*x^(1/2)". Screenshot Take a screenshot of your data using the Screenshot Tool, which adds the screenshot to the journal in Capstone. Open the journal by using the Journal Tool Save your screenshot as a jpg or PDF, and include it in your assignment submission. Question 2-2: Determine the constant A from the expression you derived in Question 2-1 and compare it to the value that you obtained in Capstone using the Curve Fitting Tool.
Previous question
The constant A is equal to 4.903. This can be found by fitting a user-defined curve to the time-of-flight data using the Curve Fitting Tool in Capstone.
The time-of-flight of an object falling through a known height h that starts at rest can be calculated using the following expression:
t = √(2h/g)
where g is the acceleration due to gravity (9.8 m/s²).
The Curve Fitting Tool in Capstone can be used to fit a user-defined curve to a set of data points. In this case, the user-defined curve will be of the form A*x^(1/2), where A is the constant that we are trying to find.
To fit a user-defined curve to the time-of-flight data, follow these steps:
Open the Capstone app and select the "Data" tab.Import the time-of-flight data into Capstone.Select the "Curve Fitting" tool.Select "User-defined" from the drop-down menu.In the "Curve Fit Editor" dialog box, type in "A*x^(1/2)".Click on the "Fit" button.Capstone will fit the user-defined curve to the data and display the value of the constant A in the "Curve Fit Editor" dialog box. In this case, the value of A is equal to 4.903.
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A quiz consists of 2 multiple-choice questions with 4 answer choices and 2 true or false questions. What is the probability that you will get all four questions correct? Select one: a. 1/64 b. 1/12 c. 1/8 d. 1/100
The probability of getting all four questions correct is 1/16.
To determine the probability of getting all four questions correct, we need to consider the number of favorable outcomes (getting all answers correct) and the total number of possible outcomes.
For each multiple-choice question, there are 4 answer choices, and only 1 is correct. Thus, the probability of getting both multiple-choice questions correct is (1/4) * (1/4) = 1/16.
For true or false questions, there are 2 possible answers (true or false) for each question. The probability of getting both true or false questions correct is (1/2) * (1/2) = 1/4.
To find the overall probability of getting all four questions correct, we multiply the probabilities of each type of question: (1/16) * (1/4) = 1/64.
Therefore, the probability of getting all four questions correct is 1/64.
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Answer in to comments pls cause I can’t see
Answer:
A - the table represents a nonlinear function because the graph does not show a constant rate of change
Step-by-step explanation:
you can tell this is true, because the y value does not increase by the same amount every time
The 1st and 10th terms of an arithmetic series are −1 and 10,
respectively.
Find the sum of the first 10 terms.
The sum of the first 10 terms of the arithmetic series is 45.
To find the sum of the first 10 terms of an arithmetic series, we can use the formula for the sum of an arithmetic series:
Sn = (n/2) * (a1 + an)
where Sn represents the sum of the first n terms, a1 is the first term, and an is the nth term.
Given that the first term (a1) is -1 and the 10th term (an) is 10, we can substitute these values into the formula to find the sum of the first 10 terms:
S10 = (10/2) * (-1 + 10)
= 5 * 9
= 45
Therefore, the sum of the first 10 terms of the arithmetic series is 45.
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Find the distance between the pair of parallel lines with the given equations. (Lesson 3-6)
y=1/2x+7/2y=1/2x+1
The distance between the pair of parallel lines with the equations y = (1/2)x + 7/2 and y = (1/2)x + 1 is 1.67 units.
To find the distance between two parallel lines, we need to determine the perpendicular distance between them. Since the slopes of the given lines are equal (both lines have a slope of 1/2), they are parallel.
To calculate the distance, we can take any point on one line and find its perpendicular distance to the other line. Let's choose a convenient point on the first line, y = (1/2)x + 7/2. When x = 0, y = 7/2, so we have the point (0, 7/2).
Now, we'll use the formula for the perpendicular distance from a point (x₁, y₁) to a line Ax + By + C = 0:
Distance = |Ax₁ + By₁ + C| / √(A² + B²)
For the line y = (1/2)x + 1, the equation can be rewritten as (1/2)x - y + 1 = 0. Substituting the values from our point (0, 7/2) into the formula, we get:
Distance = |(1/2)(0) - (7/2) + 1| / √((1/2)² + (-1)²)
= |-(7/2) + 1| / √(1/4 + 1)
= |-5/2| / √(5/4 + 1)
= 5/2 / √(9/4)
= 5/2 / (3/2)
= 5/2 * 2/3
= 5/3
= 1 2/3
= 1.67 units (approx.)
Therefore, the distance between the given pair of parallel lines is approximately 1.67 units.
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The measure θ of an angle in standard position is given. 180°
b. Find the exact values of cosθ and sin θ for each angle measure.
An angle in standard position is an angle whose vertex is at the origin and whose initial side is on the positive x-axis. The measure of an angle in standard position is the angle between the initial side and the terminal side.
An angle with a measure of 180° is a straight angle. A straight angle is an angle that measures 180°. Straight angles are formed when two rays intersect at a point and form a straight line.
The terminal side of an angle with a measure of 180° lies on the negative x-axis. This is because the angle goes from the positive x-axis to the negative x-axis as it rotates counterclockwise from the initial side.
The angle measure is 180°, and the angle is a straight angle.
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5. Sketch graphs of the following polar functions. Give the coordinates of intersections with 0 = 0 and 0 = π/2. ady = 0/4c. with 0 < 0 < 4. bir sin(201 dr−1+cost d) r = 1- cos(20) e) r = 1- 2 sin
a) The graph originates at the origin( 0, 0) and spirals in exterior as θ increases. b) The graph have two loops centered at the origin. c) The graph is a cardioid. d) The graph has bigger loop at origin and the innner loop inside it.. e) The graph is helical that starts at the point( 1, 0) and moves in inward direction towards the origin.
a) The function with polar equals is given by dy = θ/( 4π) with 0< θ< 4.
We've to find the crossroad points with θ = 0 and θ = π/ 2,
When θ = 0
dy = 0/( 4π) = 0
therefore, when θ = 0, the function intersects the origin( 0, 0).
Now, θ = π/ 2
dy = ( π/ 2)/( 4π) = 1/( 8)
thus, when θ = π/ 2, the polar function intersects the y- axis at( 0,1/8).
b) The polar function is given by r = sin( 2θ).
We've to find the corners with θ = 0 and θ = π/ 2,
When θ = 0
r = sin( 2 * 0) = sin( 0) = 0
thus, when θ = 0, the polar function intersects the origin( 0, 0).
Now, θ = π/ 2
r = sin( 2 *( π/ 2)) = sin( π) = 0
thus, when θ = π/ 2, the polar function also intersects the origin( 0, 0).
c) The polar function is given by r = 1 cos( θ).
To find the corners with θ = 0 and θ = π/ 2,
At θ = 0
r = 1 cos( 0) = 1 1 = 2
thus, when θ = 0, the polar function intersects thex-axis at( 2, 0).
At θ = π/ 2
r = 1 cos( π/ 2) = 1 0 = 1
thus, when θ = π/ 2, the polar function intersects the circle centered at( 0, 0) with compass 1 at( 1, π/ 2).
d) The polar function is given by r = 1- cos( 2θ).
To find the corners with θ = 0 and θ = π/ 2
At θ = 0
r = 1- cos( 2 * 0) = 1- cos( 0) = 0
thus, when θ = 0, the polar function intersects the origin( 0, 0).
At θ = π/ 2
r = 1- cos( 2 *( π/ 2)) = 1- cos( π) = 2
therefore, when θ = π/ 2, the polar function intersects the loop centered at( 0, 0) with compass 2 at( 2, π/ 2).
e) The polar function is given by r = 1- 2sin( θ).
To find the point of intersection with θ = 0 and θ = π/ 2,
When θ = 0
r = 1- 2sin( 0) = 1- 2( 0) = 1
thus, when θ = 0, the polar function intersects the circle centered at( 0, 0) with compass 1 at( 1, 0).
When θ = π/ 2
r = 1- 2sin( π/ 2) = 1- 2( 1) = -1
thus, when θ = π/ 2, the polar function intersects the negative y-axis at( 0,-1).
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The correct question is given below-
Sketch graphs of the following polar functions. Give the coordinates of intersections with theta = 0 and theta = π/2. a.dy = theta/4pi. with 0 < 0 < 4. b.r =sin(2theta) c.r=1+costheta d) r = 1- cos(2theta) e) r = 1- 2 sin(theta)
Record the following information below. Be sure to clearly notate which number is which parameter. A.) time of five rotations B.) time of one rotation C.) distance from the shoulder to the elbow D.) distance from the shoulder to the middle of the hand. A. What was the average angular speed (degrees/s and rad/s) of the hand? B. What was the average linear speed (m/s) of the hand? C. Are the answers to A and B the same or different? Explain your answer.
The average angular speed of the hand is ω = 1800 / t rad/s and 103140 / t degrees/s and the average linear speed of the hand is 5D / t m/s. The answers to A and B are not the same as they refer to different quantities with different units and different values.
A) To find the average angular speed of the hand, we need to use the formula:
angular speed (ω) = (angular displacement (θ) /time taken(t))
= 5 × 360 / t
Here, t is the time for 5 rotations
So, average angular speed of the hand is ω = 1800 / trad/s
To convert this into degrees/s, we can use the conversion:
1 rad/s = 57.3 degrees/s
Therefore, ω in degrees/s = (ω in rad/s) × 57.3
= (1800 / t) × 57.3
= 103140 / t degrees/s
B) To find the average linear speed of the hand, we need to use the formula:linear speed (v) = distance (d) /time taken(t)
Here, the distance of the hand is the length of the arm.
Distance from shoulder to middle of hand = D
Similarly, the time taken to complete 5 rotations is t
Thus, the total distance covered by the hand in 5 rotations is D × 5
Therefore, average linear speed of the hand = (D × 5) / t
= 5D / t
= 5 × distance of hand / time for 5 rotations
C) No, the answers to A and B are not the same. This is because angular speed and linear speed are different quantities. Angular speed refers to the rate of change of angular displacement with respect to time whereas linear speed refers to the rate of change of linear displacement with respect to time. Therefore, they have different units and different values.
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A 3500 lbs car rests on a hill inclined at 6◦ from the horizontal. Find the magnitude
of the force required (ignoring friction) to prevent the car from rolling down the hill. (Round
your answer to 2 decimal places)
The magnitude of the force required to prevent the car from rolling down the hill is 1578.88 Newton.
How to calculate the magnitude of the force?In accordance with Newton's Second Law of Motion, the force acting on this car is equal to the horizontal component of the force (Fx) that is parallel to the slope:
Fx = mgcosθ
Fx = Fcosθ
Where:
F represents the force.m represents the mass of a physical object.g represents the acceleration due to gravity.Note: 3500 lbs to kg = 3500/2.205 = 1587.573 kg
By substituting the given parameters into the formula for the horizontal component of the force (Fx), we have;
Fx = 1587.573cos(6)
Fx = 1578.88 Newton.
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The magnitude of the force required to prevent the car from rolling down the hill is approximately 367.01 lbs.
To find the magnitude of the force required to prevent the car from rolling down the inclined hill, we can analyze the forces acting on the car.
The weight of the car acts vertically downward with a magnitude of 3500 lbs. We can decompose this weight into two components: one perpendicular to the incline and one parallel to the incline.
The component perpendicular to the incline can be calculated as W_perpendicular = 3500 * cos(6°).
The component parallel to the incline represents the force that tends to make the car roll down the hill. To prevent this, an equal and opposite force is required, which is the force we need to find.
Since we are ignoring friction, the force required to prevent rolling is equal to the parallel component of the weight: F_required = 3500 * sin(6°).
Calculating this value gives:
F_required = 3500 * sin(6°) ≈ 367.01 lbs (rounded to 2 decimal places).
Therefore, the magnitude of the force required to prevent the car from rolling down the hill is approximately 367.01 lbs.
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1 hectare is defined as 1 x 10^4 m^2. 1 acre is 4.356 x 10^4 ft. How many acres are in 2.0 hectares? (Do not include units in your answer).
There are approximately 0.4594 acres in 2.0 hectares.
To solve this problemWe need to use the conversion factor between hectares and acres.
Given:
[tex]1 hectare = 1[/tex] × [tex]10^4 m^2[/tex]
[tex]1 acre = 4.356[/tex] × [tex]10^4 ft[/tex]
To find the number of acres in 2.0 hectares, we can set up the following conversion:
[tex]2.0 hectares * (1[/tex] × [tex]10^4 m^2 / 1 hectare) * (1 acre / 4.356[/tex] × [tex]10^4 ft)[/tex]
Simplifying the units:
[tex]2.0 * (1[/tex] × [tex]10^4 m^2) * (1 acre / 4.356[/tex] ×[tex]10^4 ft)[/tex]
Now, we can perform the calculation:
[tex]2.0 * (1[/tex] × [tex]10^4) * (1 /[/tex][tex]4.356[/tex] ×[tex]10^4)[/tex]
= 2.0 * 1 / 4.356
= 0.4594
Therefore, there are approximately 0.4594 acres in 2.0 hectares.
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Write a two-column proof.
Given: ΔQTS≅ ΔX W Z, TR , WY are angle bisectors.
Prove: TR /WY = QT/XW
Statement | Reason
----------------------------------------------------------
1. ΔQTS ≅ ΔXWZ | Given
2. TR bisects ∠QTS | Given
3. WY bisects ∠XWZ | Given
4. ∠QTS ≅ ∠XWZ | Corresponding parts of congruent triangles are congruent (CPCTC)
5. ∠QTR ≅ ∠XWY | Angle bisectors divide angles into congruent angles
6. ΔQTR ≅ ΔXWY | Angle-Angle (AA) criterion for triangle congruence
7. TR ≅ WY | Corresponding parts of congruent triangles are congruent (CPCTC)
8. TR/WY = QT/XW | Division property of equality
In the given statement, it is stated that triangle QTS is congruent to triangle XWZ (ΔQTS ≅ ΔXWZ).
The given information also states that TR is an angle bisector of angle QTS, and step 3 states that WY is an angle bisector of angle XWZ.
Based on the congruence of triangles QTS and XWZ (ΔQTS ≅ ΔXWZ), we can conclude that the corresponding angles in these triangles are congruent. Therefore, ∠QTS ≅ ∠XWZ.
Because TR is an angle bisector of ∠QTS and WY is an angle bisector of ∠XWZ, they divide the respective angles into congruent angles. Thus, ∠QTR ≅ ∠XWY.
Using the Angle-Angle (AA) criterion for triangle congruence, we can conclude that triangles QTR and XWY are congruent (ΔQTR ≅ ΔXWY).
By the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) property, we know that corresponding sides of congruent triangles are congruent. Therefore, TR ≅ WY.
Finally, using the Division Property of Equality, we can divide both sides of the equation TR ≅ WY by the corresponding sides QT and XW to obtain the desired result, TR/WY = QT/XW.
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Teresa y su prima Gaby planea salir de vacaciones a la playa por lo que fueron a comprar lentes de sol y sandalias por los lentes de sol y un par de sandalias Teresa pago $164 Gaby compro dos lentes de sol y un par de sandalias y pagó $249 cuál es el costo de los lentes de sol y cuánto de las sandalias
El costo de los lentes de sol es de $85 y el costo de las sandalias es de $79.
Para determinar el costo de los lentes de sol y las sandalias, podemos plantear un sistema de ecuaciones basado en la información proporcionada. Sea "x" el costo de un par de lentes de sol y "y" el costo de un par de sandalias.
De acuerdo con los datos, tenemos la siguiente ecuación para Teresa:
x + y = 164.
Y para Gaby, tenemos:
2x + y = 249.
Podemos resolver este sistema de ecuaciones utilizando métodos de eliminación o sustitución. Aquí utilizaremos el método de sustitución para despejar "x".
De la primera ecuación, podemos despejar "y" en términos de "x":
y = 164 - x.
Sustituyendo este valor de "y" en la segunda ecuación, obtenemos:
2x + (164 - x) = 249.
Simplificando la ecuación, tenemos:
2x + 164 - x = 249.
x + 164 = 249.
x = 249 - 164.
x = 85.
Ahora, podemos sustituir el valor de "x" en la primera ecuación para encontrar el valor de "y":
85 + y = 164.
y = 164 - 85.
y = 79.
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The exterior angle of a regular polygon is 5 times the interior angle. Find the exterior angle, the interior angle and the number of sides
Answer:The interior angle of a polygon is given by
The exterior angle of a polygon is given by
where n is the number of sides of the polygon
The statement
The interior of a regular polygon is 5 times the exterior angle is written as
Solve the equation
That's
Since the denominators are the same we can equate the numerators
That's
180n - 360 = 1800
180n = 1800 + 360
180n = 2160
Divide both sides by 180
n = 12
I).
The interior angle of the polygon is
The answer is
150°
II.
Interior angle + exterior angle = 180
From the question
Interior angle = 150°
So the exterior angle is
Exterior angle = 180 - 150
We have the answer as
30°
III.
The polygon has 12 sides
IV.
The name of the polygon is
Dodecagon
Step-by-step explanation:
Does √x³= ³√x² for all, some, or no values of x Explain.
√x³= ³√x² some values of x.
Let's assume that this equation is true for some value of x. Then:√x³= ³√x²
Cubing both sides gives us: x^(3/2) = x^(2/3)
Multiplying both sides by (2/3) gives: x^(3/2) * (2/3) = x^(2/3)
Multiplying both sides by 3/2 gives us: x^(3/2) = (3/2)x^(2/3)
Thus, we have now determined that if the equation is true for a certain value of x, then it is true for all values of x.
However, the converse is not necessarily true. It's because if the equation is not true for some value of x, then it is not true for all values of x.
As a result, we must investigate if the equation is true for some values of x and if it is false for others.Let's test the equation using a value of x= 4:√(4³) = ³√(4²)2^(3/2) = 2^(4/3)3^(2/3) = 2^(4/3)
There we have it! Because the equation does not hold true for all values of x (i.e. x = 4), we can conclude that the answer is "some values of x."
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Look at the image below. Identify the coordinates for point X, so that the ratio of AX : XB = 5 : 4
The coordinates of X that partitions XY in the ratio 5 to 4 include the following: X (-1.6, -7).
How to determine the coordinates of point X?In this scenario, line ratio would be used to determine the coordinates of the point X on the directed line segment AB that partitions the segment into a ratio of 5 to 4.
In Mathematics and Geometry, line ratio can be used to determine the coordinates of X and this is modeled by this mathematical equation:
M(x, y) = [(mx₂ + nx₁)/(m + n)], [(my₂ + ny₁)/(m + n)]
By substituting the given parameters into the formula for line ratio, we have;
M(x, y) = [(5(2) + 4(-6))/(5 + 4)], [(5(-11) + 4(-2))/(5 + 4)]
M(x, y) = [(10 - 24)/(9)], [(-55 - 8)/9]
M(x, y) = [-14/9], [(-63)/9]
M(x, y) = (-1.6, -7)
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
After graduation you receive 2 job offers, both offering to pay you an annual salary of $50,000:
Offer 1: $70,000 salary with a 4% raise after 1 year, 4% raise after 2 years, and a $3700 raise after the 3rd year.
Offer 2: $60,000 salary, with a $3500 dollar raise after 1 year, and a 6% raise after 2 years, and a 3% after the 3rd year.
Note: Assume raises are based on the amount you made the previous year.
a) How much would you make after 3 years working at the first job?
b) How much would you make after working 3 years at the second job?
c) Assume the working conditions are equal, which offer would you take. Explain.
With offer 1, you would make $78,216, while with offer 2, you would make $70,354.04. Therefore, offer 1 provides a higher overall income over the 3-year period.
Compare two job offers: Offer 1 - $70,000 salary with 4% raise after 1 year, 4% raise after 2 years, and $3700 raise after 3rd year. Offer 2 - $60,000 salary with $3500 raise after 1 year, 6% raise after 2 years, and 3% raise after 3rd year.After 3 years working at the first job, you would start with a salary of $70,000.
After the first year, you would receive a 4% raise, which is 4% of $70,000, resulting in an additional $2,800. After the second year, you would again receive a 4% raise based on the previous year's salary of $72,800 (original salary + raise from year 1), which is $2,912. Then, in the third year, you would receive a $3,700 raise, bringing your total earnings to $70,000 + $2,800 + $2,912 + $3,700 = $78,216.After 3 years working at the second job, you would start with a salary of $60,000.
After the first year, you would receive a $3,500 raise, bringing your salary to $63,500. After the second year, you would receive a 6% raise based on the previous year's salary of $63,500, which is $3,810. Finally, in the third year, you would receive a 3% raise based on the previous year's salary of $67,310 (original salary + raise from year 2), which is $2,019. Adding these amounts together, your total earnings would be $60,000 + $3,500 + $3,810 + $2,019 = $70,354.04.Assuming the working conditions are equal, the better offer would be offer 1 because it results in higher total earnings after 3 years.
With offer 1, you would make $78,216, while with offer 2, you would make $70,354.04. Therefore, offer 1 provides a higher overall income over the 3-year period.
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Write 220 : 132 in the form 1 : n
The expression given can be expressed in it's splest term as 5 : 3
Given the expression :
220 : 132To simplify to it's lowest term , divide both values by 44
Hence, we have :
5 : 3At this point, none of the values can be divide further by a common factor.
Hence, the expression would be 5:3
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Consider the vectors x(¹) (t) = ( t (4) (a) Compute the Wronskian of x(¹) and x(²). W = -2 t² D= -[infinity] (b) In what intervals are x(¹) and x(²) linearly independent? 0 U and x ²) (t) = (2) must be discontinuous at to = P(t) = (c) What conclusion can be drawn about coefficients in the system of homogeneous differential equations satisfied by x(¹) and x(²)? One or more ▼ of the coefficients of the ODE in standard form 0 (d) Find the system of equations x': = 9 [infinity] t² 2t P(t)x.
(e) The overall solution is given by the equation x(t) = C1t^3 + C2/t^3,, where C1 and C2 are arbitrary constants.
(a) The Wronskian of x(1) and x(2) is given by:
W = | x1(t) x2(t) |
| x1'(t) x2'(t) |
Let's evaluate the Wronskian of x(1) and x(2) using the given formula:
W = | t 2t^2 | - | 4t t^2 |
| 1 2t | | 2 2t |
Simplifying the determinant:
W = (t)(2t^2) - (4t)(1)
= 2t^3 - 4t
= 2t(t^2 - 2)
(b) For x(1) and x(2) to be linearly independent, the Wronskian W should be non-zero. Since W = 2t(t^2 - 2), the Wronskian is zero when t = 0, t = -√2, and t = √2. For all other values of t, the Wronskian is non-zero. Therefore, x(1) and x(2) are linearly independent in the intervals (-∞, -√2), (-√2, 0), (0, √2), and (√2, +∞).
(c) Since x(1) and x(2) are linearly dependent for the values t = 0, t = -√2, and t = √2, it implies that the coefficients in the system of homogeneous differential equations satisfied by x(1) and x(2) are not all zero. At least one of the coefficients must be non-zero.
(d) The system of equations x': = 9t^2x is already given.
(e) The general solution of the differential equation x' = 9t^2x can be found by solving the characteristic equation. The characteristic equation is r^2 = 9t^2, which has roots r = ±3t. Therefore, the general solution is:
x(t) = C1t^3 + C2/t^3,
where C1 and C2 are arbitrary constants.
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a man finds 1 hundred dollars and he keeps one half of it, gives 1 fourth if it to someone and and gives another 1 fifth of it to some else and he puts the rest in savings. how much did he give everyone
Determine the first three nonzero terms in the Taylor polynomial approximation for the given initial value problem. x ′′
+8tx=0;x(0)=1,x ′
(0)=0 The Taylor approximation to three nonzero terms is x(t)=+⋯.
The first three nonzero terms in the Taylor polynomial approximation for the given initial value problem are: 1 - t^2/8 + t^4/128.
Given the initial value problem: x′′ + 8tx = 0; x(0) = 1, x′(0) = 0. To find the first three nonzero terms in the Taylor polynomial approximation, we follow these steps:
Step 1: Find x(t) and x′(t) using the integrating factor.
We start with the differential equation x′′ + 8tx = 0. Taking the integrating factor as I.F = e^∫8t dt = e^4t, we multiply it on both sides of the equation to get e^4tx′′ + 8te^4tx = 0. This simplifies to e^4tx′′ + d/dt(e^4tx') = 0.
Integrating both sides gives us ∫ e^4tx′′ dt + ∫ d/dt(e^4tx') dt = c1. Now, we have e^4tx' = c2. Differentiating both sides with respect to t, we get 4e^4tx' + e^4tx′′ = 0. Substituting the value of e^4tx′′ in the previous equation, we have -4e^4tx' + d/dt(e^4tx') = 0.
Simplifying further, we get -4x′ + x″ = 0, which leads to x(t) = c3e^(4t) + c4.
Step 2: Determine the values of c3 and c4 using the initial conditions.
Using the initial conditions x(0) = 1 and x′(0) = 0, we can substitute these values into the expression for x(t). This gives us c3 = 1 and c4 = -1/4.
Step 3: Write the Taylor polynomial approximation.
The Taylor approximation to three nonzero terms is x(t) = 1 - t^2/8 + t^4/128 + ...
Therefore, the starting value problem's Taylor polynomial approximation's first three nonzero terms are: 1 - t^2/8 + t^4/128.
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Create an inequality that needs to reverse the symbol to be true and one that does not need to be reversed.
Reverse
Do Not Reverse
Answer:
See below
Step-by-step explanation:
An easy example of an inequality where you need to flip the sign to be true is something like [tex]-2x > 4[/tex]. By dividing both sides by -2 to isolate x and get [tex]x < -2[/tex], you would need to also flip the sign to make the inequality true.
One that wouldn't need to be reversed is [tex]2x > 4[/tex]. You can just divide both sides by 2 to get [tex]x > 2[/tex] and there's no flipping the sign since you are not multiplying or dividing by a negative.
Find the sum of the first 50 terms of the arithmetic sequence
with first term 6 and common difference 1/2
.
Answer:
S₅₀ = 912.5
Step-by-step explanation:
the sum of n terms of an arithmetic sequence is
[tex]S_{n}[/tex] = [tex]\frac{n}{2}[/tex] [ 2a₁ + (n - 1)d ]
where a₁ is the first term and d the common difference
here a₁ = 6 and d = [tex]\frac{1}{2}[/tex] , then
S₅₀ = [tex]\frac{50}{2}[/tex] [ (2 × 6) + (49 × [tex]\frac{1}{2}[/tex]) ]
= 25(12 + 24.5)
= 25 × 36.5
= 912.5
ST and TS have the same eigenvalues. = Problem 24. Suppose T E L(F2) is defined by T(x, y) eigenvalues and eigenvectors of T. [10 marks] (y,x). Find all [10 marks]
Given a linear transformation T in L(F2) such that T(x, y) = (y, x) and it has the same eigenvalues as ST.
We need to find all eigenvalues and eigenvectors of T.
[tex]Solution: Since T is a linear transformation in L(F2) such that T(x, y) = (y, x),[/tex]
let us consider T(1, 0) and T(0, 1) respectively.
[tex]T(1, 0) = (0, 1) and T(0, 1) = (1, 0).For any (x, y) in F2, it can be written as (x, y) = x(1, 0) + y(0, 1).[/tex]
Therefore, T(x, y) = T(x(1, 0) + y(0, 1)) = xT(1, 0) + yT(0, 1) = x(0, 1) + y(1, 0) = (y, x)
[tex]Thus, the matrix of T with respect to the standard ordered basis B of F2 is given by A = [T]B = [T(1, 0) T(0, 1)] = [0 1; 1 0][/tex]
The eigenvalues and eigenvectors of A are calculated as follows: We find the eigenvalues as:|A - λI| = 0⇒ |[0-λ 1;1 0-λ]| = 0⇒ λ2 - 1 = 0⇒ λ1 = 1 and λ2 = -1
Therefore, the eigenvalues of T are 1 and -1.
Now, we find the eigenvectors of T corresponding to each eigenvalue.
[tex]For eigenvalue λ1 = 1, we have(A - λ1I)X = 0⇒ [0 1; 1 0]X = [0;0]⇒ x2 = 0 and x1 = 0or, X1 = [0;0][/tex]is the eigenvector corresponding to λ1 = 1.
For eigenvalue λ2 = -1, we have(A - λ2I)X = 0⇒ [0 1; 1 0]X = [0;0]⇒ x2 = 0 and x1 = 0or, X2 = [0;0] is the eigenvector corresponding to λ2 = -1.
Since T has only two eigenvectors {X1, X2}, therefore the diagonal matrix D = [Dij]2x2 with diagonal entries as the eigenvalues (λ1, λ2) and the eigenvectors as its columns (X1, X2) such that A = PDP^-1where, P = [X1 X2].
[tex]Then, the eigenvalues and eigenvectors of T are given by λ1 = 1, λ2 = -1 and X1 = [1;0], X2 = [0;1] respectively.[/tex]
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