Answer: 127
Step-by-step explanation:
Vertical Angles mean that those 2 angles are equal.
Thus,
You set the two equations equal to each other and solve for the value of x.
5x+12 = 6x-11
x = 23
Next
Once you find the value of x you need to plug in 23 back into any equation to get the value of your angle.
5(23) + 12 = 127
I need this quickly. Thanks!
The evaluation of the sinusoidal sine function y = 2·sin(x) + 1 indicates;
Maximum value: 3 at x = π/2
Minimum value: -1 at x = 3·π/2
What is a sinusoidal function?A sinusoidal function is a periodic function based on either the sine or cosine functions, which oscillates, smoothly, between high and low values.
The specified function is; y = 2·sin(x) + 1
The domain of the input value x is; 0 ≤ x < 2·π
The function is a sinusoidal function which has the form; y = a·sin(b·(x - h)) + k
Therefore, we get;
The amplitude, a = 2
The frequency factor, b = 1
The horizontal shift, h = 0
The vertical shift, k = 1
The points in the interval 0 ≤ x < 2·π, where the sine of the angle is a maximum is; x = π/2, such that we get; sin(π/2) = 1
The points in the interval 0 ≤ x < 2·π, where the sine of the angle is a minimum is; x = 3·π/2, such that we get; sin(3·π/2) = -1
Therefore; Maximum value is;2 × sin(π/2) + 1 = 3 at the point x = π/2
Minimum value is; 2 × sin(3·π/2) + 1 = -1, at the point x = 3·π/2
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(b) Two groups of students are travelling together. One group consists of 40 boys and 25 girls, the other group consists of 50 boys and 35 girls. Find the percentage of boys and girls in the combined group. there 20% mreived
The percentage of boys in the combined group is 60% and the percentage of girls in the combined group is 40%.
Given the number of boys and girls in two groups of students travelling together, we have to determine the percentage of boys and girls in the combined group. Let's solve this problem step by step. Solution:Step 1: Let the number of boys in the combined group be B and the number of girls in the combined group be G.B = 40 + 50 = 90 (Adding the number of boys from both groups)G = 25 + 35 = 60 (Adding the number of girls from both groups)Step 2: Find the total number of students in the combined group.Total students in the combined group = B + G = 90 + 60 = 150 studentsStep 3: Calculate the percentage of boys and girls in the combined group.Percentage of boys = (Number of boys in the combined group/Total number of students in the combined group) × 100= (90/150) × 100 = 60%Percentage of girls = (Number of girls in the combined group/Total number of students in the combined group) × 100= (60/150) × 100 = 40%
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Please answer ASAP I will brainlist
(a) The amount for total expenditures in 2010 was about $160.3 billion.
(b) The first full year in which expenditures exceeded $108 billion was 2010.
What is an exponential function?In Mathematics and Geometry, an exponential function can be represented by using this mathematical equation:
f(x) = a(b)^x
Where:
a represents the initial value or y-intercept.x represents x-variable.b represents the rate of change, common ratio, decay rate, or growth rate.Based on the information provided above, the total expenditures on benefits (in billions of dollars) can be modeled by the following exponential function;
[tex]h(x) = 23.4(1.08)^x[/tex]
Part a.
x = 5 + (2015 - 1995)
x = 25 years.
Therefore, the amount for total expenditures in 2015 is given by:
[tex]h(25) = 23.4(1.08)^{25}[/tex]
h(25) = 160.3 billion.
Part b.
For the first full year in which the total expenditures exceeded $110 billion, we have:
[tex]110= 23.4(1.08)^x\\\\\frac{110}{23.4} =1.08^x[/tex]
By taking the natural log (ln) of both sides of the equation, we have:
x = ln(4.7008547008547008547008547008547)/ln(1.08)
x = 1.5477443434287605/0.0769610411361284
x = 20.12 ≈ 20 years
First full year = (20 - 5) + 1995
First full year = 2010.
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Complete Question:
The total expenditures on benefits (in billions of dollars) can be approximated by the function [tex]h(x) = 23.4(1.08)^x[/tex], where x = 5 corresponds to the year 1995
(a) What was the amount for total expenditures in 2015?
(b) What was the first full year in which expenditures exceeded $110 billion?
Please Help!! - Find constants a and b so that (8, −7) is the solution of the system. Please answer in exact numbers.
Step-by-step explanation:
Plug in 8 for x, and -7 for y in both systems.
[tex]8a- 7b = 10[/tex]
[tex]8b - 7a = - 5[/tex]
Now, we use the elimination method to isolate a or b.
Let multiply the top equation by 8 and the bottom one by 7.
[tex]64a - 56b= 80[/tex]
[tex]56b - 49a = - 35[/tex]
Eliminate the b variable by adding the two systems.
[tex]15 a= 45[/tex]
[tex]a = 3[/tex]
Now, plug 3 for a into either the systems to find b.
[tex]24 - 7b = 10[/tex]
[tex] - 7b = - 14[/tex]
[tex]b = 2[/tex]
Which variation is represented by this situation:
If one hose can fill a swimming pool in 7 hours, how long will it take 3 hoses to fill the pool?
please answer this question below correctly
Answer:
a) sum
b) shorter
c) longest
Step-by-step explanation:
a) sum
b) shorter
c) longest
Spanning trees assignments
Spanning trees assignments are a type of network problem that requires finding the minimum path or tree that spans all nodes or vertices in the network.
This is useful for optimizing the design and efficiency of networks, such as computer networks, transportation systems, and power grids.
A spanning tree is a subset of a graph that connects all vertices without any cycles, making it a tree.
Finding the minimum spanning tree involves selecting the minimum weight edge to add to the tree, while ensuring that there are no cycles.
This can be done using algorithms such as Kruskal's algorithm or Prim's algorithm.
Spanning trees assignments are important in network design and optimization and have applications in various fields such as computer science, electrical engineering, and transportation planning.
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Find f(0) for the
piece-wise function.
f(x) =
if x ≤ 0
x+1 if x>0
X
f(0) = [?]
[tex]f(x)=x[/tex] for [tex]x\leq0[/tex], therfore [tex]f(0)=0[/tex].
the first domain is true for f(0) so we substitute value of x in the first function
[tex]f(x) = x \\ f(0) = 0[/tex]
At a sale , a suit is being sold for 26% of the regular price. The sale price is 109.20 what's the regular price?
Answer: 147.57
Step-by-step explanation:
x - .26x = 109.20
0.26 = 26%
subtract 26% from 100% = 74%
109.20 / 0.74 = 147.57
Check work: 147.57 × .26 = 38.37
Then 147.57 - 38.37 = 109.20
Translate this sentence into an equation Dons score decreased by23 is 43. Use the variable d to represent Dons score
The equation that represents the given sentence "Don's score decreased by 23 is 43" is d - 23 = 43, where d represents Don's score.
To translate the given sentence into an equation, let's break it down:
"Don's score" can be represented by the variable d.
"Decreased by 23" indicates a subtraction operation. So we subtract 23 from Don's score.
"Is" signifies the equal sign in the equation.
"43" is the result or value that Don's score decreased by 23 should be equal to.
Combining all the information, we can write the equation:
d - 23 = 43
This equation states that when we subtract 23 from Don's score (represented by d), the result should be equal to 43. This equation accurately represents the given sentence.
To find the value of Don's score (d), we can solve the equation by isolating the variable:
d - 23 = 43
Adding 23 to both sides of the equation:
d - 23 + 23 = 43 + 23
Simplifying:
d = 66
Therefore, Don's score, represented by the variable d, is equal to 66 based on the given equation.
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The area of a square shaped table is 144/a square meters. Find the length of a side of the table.
Answer: If the area of the square is 144 sq. meter, then the side is 12 meters long.
Step-by-step explanation:
:P
Find the number of students in the class if 2/3 of the students are girls and the number of boys is 21.
Geometry
Solve both and show work
Answer:
y = - 5x - 1 and y = [tex]\frac{1}{3}[/tex] x + 8
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
14
y = - 5x + 4 ← is in slope- intercept form
with slope m = - 5
• Parallel lines have equal slopes , then
y = - 5x + c ← is the partial equation
to find c substitute P (- 1, 4 ) into the partial equation
4 = - 5(- 1) + c = 5 + c ( subtract 5 from both sides )
- 1 = c
y = - 5x - 1 ← equation of parallel line
15
y + 3x = 13 ( subtract 3x from both sides )
y = - 3x + 13 ← in slope- intercept form
with slope m = - 3
given a line with slope m then the slope of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{-3}[/tex] = [tex]\frac{1}{3}[/tex] , then
y = [tex]\frac{1}{3}[/tex] x + c ← is the partial equation
to find c substitute P (6, 10 ) into the partial equation
10 = [tex]\frac{1}{3}[/tex] (6) + c = 2 + c ( subtract 2 from both sides )
8 = c
y = [tex]\frac{1}{3}[/tex] x + 8 ← equation of perpendicular line
Answer:
To find the equation of a line perpendicular to another line, we need to determine the slope of the original line and then take the negative reciprocal of that slope.The given line is y + 3x = 13. We can rearrange it to the slope-intercept form (y = mx + b) by isolating y:y = -3x + 13The slope of this line is -3. The negative reciprocal of -3 is 1/3, which will be the slope of the perpendicular line.Now, we have the slope (1/3) and a point (6, 10) that the perpendicular line passes through. We can use the point-slope form (y - y1 = m(x - x1)) to find the equation of the line.Substituting the values into the point-slope form:y - 10 = (1/3)(x - 6)Distributing 1/3 to (x - 6):y - 10 = (1/3)x - 2Adding 10 to both sides to isolate y:y = (1/3)x - 2 + 10Simplifying:y = (1/3)x + 8Therefore, the equation of the line perpendicular to y + 3x = 13 and passing through the point (6, 10) is y = (1/3)x + 8.find the area of an equilateral triangle ABC, whose sides are 7cm each and the height 4.2cm
Step-by-step explanation:
Area of equalaterial triangle is 1/2bh
Here the base is 7.
The height is 4.2
[tex] \frac{1}{2} (7)(4.2) = 14.7[/tex]
Using the above supply/demand graph, what is the price at the point of equilibrium? a. 105 b. 100 c. 95 d. 80 Please select the best answer from the choices provided A B C D
The price at the point of equilibrium is d. 80
How to determine the price at the point of equilibriumFrom the question, we have the following parameters that can be used in our computation:
The supply/demand graph
The price at the point of equilibrium is the point where the demand and the supply lines meet
From the attached graph, we can see that they intersect at P = 80
This means that the price at the point of equilibrium is d. 80
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the sum of 5 and the square of a number
The value of the sum of 5 and the square of a number varies depending on the value of n. When n = 1, the result of the expression is 6, when n = 2, the result is 9, and when n = 3, the result is 14. The same applies when n is negative.
The sum of 5 and the square of a number is an expression that can be represented as (n² + 5), where n represents the number whose square is added to 5. The result of the expression varies depending on the value of n. In this answer, we will examine how to find the sum of 5 and the square of a number using various values of n.
If n = 1, then the square of n is equal to 1, and the sum of 5 and the square of n is equal to 6. Therefore, if n = 1, the result of the expression is 6.
If n = 2, then the square of n is equal to 4, and the sum of 5 and the square of n is equal to 9. Therefore, if n = 2, the result of the expression is 9.
If n = 3, then the square of n is equal to 9, and the sum of 5 and the square of n is equal to 14. Therefore, if n = 3, the result of the expression is 14.
If n = -1, then the square of n is equal to 1, and the sum of 5 and the square of n is equal to 6. Therefore, if n = -1, the result of the expression is 6.
If n = -2, then the square of n is equal to 4, and the sum of 5 and the square of n is equal to 9. Therefore, if n = -2, the result of the expression is 9.
If n = -3, then the square of n is equal to 9, and the sum of 5 and the square of n is equal to 14. Therefore, if n = -3, the result of the expression is 14.
In conclusion, the value of the sum of 5 and the square of a number varies depending on the value of n. When n = 1, the result of the expression is 6, when n = 2, the result is 9, and when n = 3, the result is 14. The same applies when n is negative.
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¿Cuál es el cociente de la división 1,24:0,08?
The Ratio of 1.24 to 0.08 can be expressed as 15.5, 31:2, or 31/2.
To find the ratio of 1.24 to 0.08, we divide 1.24 by 0.08.
1.24 ÷ 0.08 = 15.5
Therefore, the ratio of 1.24 to 0.08 is 15.5.
We can also express this ratio as a fraction. To do this, we write 15.5 as a fraction over 1:
15.5/1
To simplify the fraction, we can multiply both the numerator and the denominator by 10 to get rid of the decimal:
(15.5 × 10)/(1 × 10) = 155/10
Further simplifying the fraction, we divide both the numerator and the denominator by their greatest common divisor, which is 5:
(155 ÷ 5)/(10 ÷ 5) = 31/2
So, another way to express the ratio 1.24:0.08 is 31:2.
In conclusion, the ratio of 1.24 to 0.08 can be expressed as 15.5, 31:2, or 31/2.
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Emily and John each ran at a constant speed for a 100-meter race. Each runner’s distance for the same section of the race is displayed below. Who had a head start, and how big was the head start?
John’s Run
A 2-column table with 4 rows. Column 1 is labeled Time (seconds) with entries 4, 6, 8, 10. Column 2 is labeled Distance (meters) with entries 35, 47.5, 60, 72.5.
had a head start of
meters.
Emily won the race by reaching the finish line at the same time as John or possibly even earlier.
To determine who won the race and by how many seconds, we can analyze the data provided in John's run.
From the table, we can see that John's time intervals are 4 seconds, 6 seconds, 8 seconds, and 10 seconds. The corresponding distances covered by John at these time intervals are 35 meters, 47.5 meters, 60 meters, and 72.5 meters, respectively.
Now, let's consider Emily's run, taking into account that she gave John a 10-meter head start. This means that Emily's distance covered at each time interval will be 10 meters less than John's distance.
Comparing their distances, we have:
At 4 seconds, John's distance is 35 meters. Therefore, Emily's distance is 35 - 10 = 25 meters.
At 6 seconds, John's distance is 47.5 meters. Therefore, Emily's distance is 47.5 - 10 = 37.5 meters.
At 8 seconds, John's distance is 60 meters. Therefore, Emily's distance is 60 - 10 = 50 meters.
At 10 seconds, John's distance is 72.5 meters. Therefore, Emily's distance is 72.5 - 10 = 62.5 meters.
Now, let's compare their times. Emily's time intervals are the same as John's since they both ran the same race:
At 4 seconds, John covered 35 meters while Emily covered 25 meters.
At 6 seconds, John covered 47.5 meters while Emily covered 37.5 meters.
At 8 seconds, John covered 60 meters while Emily covered 50 meters.
At 10 seconds, John covered 72.5 meters while Emily covered 62.5 meters.
By analyzing their distances and times, we can conclude that Emily won the race. To determine the time difference, we can subtract Emily's time from John's time at each interval:
At 4 seconds, John's time is 4 seconds and Emily's time is also 4 seconds. The time difference is 4 - 4 = 0 seconds.
At 6 seconds, John's time is 6 seconds and Emily's time is 6 seconds. The time difference is 6 - 6 = 0 seconds.
At 8 seconds, John's time is 8 seconds and Emily's time is 8 seconds. The time difference is 8 - 8 = 0 seconds.
At 10 seconds, John's time is 10 seconds and Emily's time is 10 seconds. The time difference is 10 - 10 = 0 seconds.
Therefore, Emily won the race by reaching the finish line at the same time as John or possibly even earlier.
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Question
Emily and John each ran at a constant speed for a 100-meter race. Emily gave John a 10-meter head start. Use the given data below to determine who won the race and by how many seconds
John’s Run
A 2-column table with 4 rows. Column 1 is labeled Time (seconds) with entries 4, 6, 8, 10. Column 2 is labeled Distance (meters) with entries 35, 47.5, 60, 72.5.
Select two points to define a line where the polygon will
break into two pieces.
28 cm
10 cm
10 cm
18 cm.
What are the different ways to combine the possible
areas to find the area of the composite figure?
O Multiply the area of the rectangle by 2.
O Add the area of the rectangle with the area of the
triangle.
O Multiply the area of the triangle by 2.
O Divide the area of the rectangle by 2.
The two points are the tips of the vertical dotted line which represents the measure of the rectangles width and the base of the triangle. To obtain the area of the composite figure, calculate and add the area of each shape.
The figure given is a composite figure which consists of a rectangle and triangle. The polygon could be broken into two pieces by drawing a vertical line from the endpoint of the 28cm line segment of the rectangle.
2.)
To obtain the area of the composite figure, we need to calculate the area of the rectangle and triangle separately and then add them together.
Hence, adding the area of each shape gives the area of the composite figure.
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Answer: correct answer is B
Step-by-step explanation:
I did the assignment
Pls help it’s for my homework
Answer: 150cm1 I think
the product of 4 and g score is 72
The expression "the product of 4 and g score is 72" in algebraic notation is 4g = 72
Writing the algebraic expression in algebraic notationFrom the question, we have the following parameters that can be used in our computation:
The product of 4 and g score is 72
Represent the number with g
So the statement can be rewritten as follows:
the product of 4 and g is 72
So, we have the following
4g is 72
"is" means =
So, we have
4g = 72
Hence, the expression "the product of 4 and g score is 72" in algebraic notation is 4g = 72
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Chris conducted a experiment in his 1st period class which has 30 total students Chris had everyone flip a coin and record the result the result was 18 of them came up heads based on this situation what is the experimental probability of members of his class ending up with heads
The experimental probability of members of Chris's class ending up with heads is about 3/5 or 0.6 (60%).
What is the experimentThe experimental probability is one that can be calculated by:
dividing the number of favorable outcomes (the number of heads) by the total number of outcomes (the total number of students).
So, the number of heads recorded was 18, and the total number of students was 30.
Experimental probability = Number of heads / Total number of students
= 18 / 30
So one can simplify the fraction by:
Experimental probability = 3 / 5
Therefore, the experimental probability of members of Chris's class ending up with heads is 3/5 .
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2.
7
X-1
What is the center of f(x)?
Let f(x) =
O (-5, -1)
O (-1,5) =-5.
none of the answer choices
O (-1,-5)
O (-5, 1)
None of the answer choices provided (O (-5, -1), O (-1, 5), O (-1, -5), O (-5, 1)) are correct, as the concept of a center does not apply to Linear functions.
To determine the center of the function f(x), we need more information about the function. The expression given, 2x-1, represents a linear function, which is in the form y = mx + b, where m is the slope and b is the y-intercept.
The center of a function is typically associated with quadratic or circular functions and is not applicable to linear functions. Therefore, we cannot determine the center of the function f(x) = 2x - 1 based on the given information.
Thus, none of the answer choices provided (O (-5, -1), O (-1, 5), O (-1, -5), O (-5, 1)) are correct, as the concept of a center does not apply to linear functions.
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My friend eats 5 apples I eat 7 apples. How many apples eat together?
Answer:
12 apples
Step-by-step explanation:
Apples eaten by friend = 5
Apples eaten by me = 7
Toral number of apples eaten together
= 5 + 7
= 12
________
hope this helps!
Answer:
12 apples
Step-by-step explanation:
If 1 person eats 5 apples, and another person eats 7, we should add the amount of apples together to get the total.
5+7
=12
So, 12 apples were ate altogether.
Hope this helps! :)
● Blondies are squares with 3 inch sides. ● Brownies are squares with 6 inch sides. ● The tray that displays the blondies and brownies has an area of 648 square inches and is completely full. If she has 4 rows of blondies and 4 rows of brownies, what fraction of the area of the tray, in square inches, is blondies? Show your work.
The fraction of the area of the tray occupied by the blondies is 1/9.
To find the fraction of the area of the tray occupied by blondies, we need to determine the area occupied by the blondies and compare it to the total area of the tray.
Let's calculate the area of each individual blondie:
The blondies are squares with 3-inch sides, so the area of each blondie is 3 inches × 3 inches = 9 square inches.
Now, let's calculate the area occupied by the blondies in each row:
Since there are 4 rows of blondies and each row contains 4 blondies, the total number of blondies is 4 rows × 4 blondies per row = 16 blondies.
So, the total area occupied by the blondies is 16 blondies × 9 square inches per blondie = 144 square inches.
Next, let's determine the area occupied by the brownies:
The brownies are squares with 6-inch sides, so the area of each brownie is 6 inches × 6 inches = 36 square inches.
Since there are also 4 rows of brownies and each row contains 4 brownies, the total number of brownies is 4 rows × 4 brownies per row = 16 brownies.
Therefore, the total area occupied by the brownies is 16 brownies × 36 square inches per brownie = 576 square inches.
Now, let's calculate the total area of the tray:
Given that the tray is completely full and has an area of 648 square inches, we can subtract the area occupied by the brownies from the total area to find the remaining area occupied by the blondies:
Total area of the tray - Area occupied by the brownies = Area occupied by the blondies
648 square inches - 576 square inches = 72 square inches.
So, the fraction of the area of the tray occupied by the blondies is:
Area occupied by the blondies / Total area of the tray = 72 square inches / 648 square inches.
To simplify this fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 72 in this case:
72 square inches / 648 square inches = 1/9.
Therefore, the blondies' percentage of the tray's surface area is 1/9.
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Which sentence contains correct punctuation? A. Tammy is a good worker; she gets everything done on time. B. We had the following choices of colors; red, white, or blue. C. I am going home; and I intend to stay there. D. I have packed everything; including a bathing suit, a pair of sunglasses, and a book.
The sentence that contains correct punctuation is "Tammy is a good worker; she gets everything done on time."
Option A contains correct punctuation. It has a semicolon which separates two independent clauses and also makes a connection between them. This is the correct use of punctuation. It is also known as a semicolon conjunction.
Option B is incorrect as there is no semicolon between "colors" and "red." There should be a semicolon instead of a comma to create a connection between the two clauses.
Option C is incorrect as there is no need for a semicolon after "going home" because there is no second independent clause following it.
Option D is incorrect as there should be a semicolon between "everything" and "including" since there are two distinct groups of items being listed.
In conclusion, option A is the correct answer because it follows the rules of proper punctuation by using a semicolon correctly to join two independent clauses.
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please awnser ASAP i will brainlist
a) The graph of the function represents an exponential decay.
b) The numeric values are given as follows:
f(0) = 1.f(1) = 0.3.How to define an exponential function?An exponential function has the definition presented according to the equation as follows:
[tex]y = ab^x[/tex]
In which the parameters are given as follows:
a is the value of y when x = 0.b is the rate of change.The function for this problem is given as follows:
[tex]f(x) = 0.3^x[/tex]
The parameters for this problem are given as follows:
a = 1, b = 0.3.
The numeric values are given as follows:
[tex]f(0) = (0.3)^0 = 1[/tex][tex]f(1) = (0.3)^1 = 0.3[/tex]More can be learned about exponential functions at brainly.com/question/2456547
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(8m + 4) + (8m + 4) combine the like terms to simplify
The expression simplified can be written as:
16m + 8
How to combine the terms and simplify?Here we want to simplify the following expression:
(8m + 4) + (8m + 4)
To simplify, we start by removing the parenthesis:
8m + 4 + 8m + 4
Now we can change the order of the terms so we get:
8m + 8m + 4 + 4
Now we can group the like terms to get:
(8m + 8m) + (4 + 4)
And simplify this to get:
16m + 8
That is the simplified expression.
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Triangle JKL has the following measures :m∠J=97degree, m∠L=31degree, and j=44. What is the length of side k?
46.3
52.0
22.8
34.9
The correct length of side k in triangle JKL is approximately 35.15 units.
The correct answer is option E.
The length of side k in triangle JKL using the correct values for sin(52°) and sin(97°).
Given:
m∠J = 97 degrees
m∠L = 31 degrees
J = 44
According to the Law of Sines, we have:
sin(∠J) / J = sin(∠K) / k
Substituting the given values:
sin(97°) / 44 = sin(∠K) / k
Now, let's solve for k:
k = (44 * sin(∠K)) / sin(97°)
Using a calculator:
k = (44 * sin(52°)) / sin(97°)
k ≈ (44 * 0.79) / 0.99
k ≈ 35.1
Therefore, the correct length of side k in triangle JKL is approximately 35.15 units making option E the correct answer.
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The question probable may be:
Triangle JKL has the following measures :m∠J=97degree, m∠L=31degree, and j=44. What is the length of side k?
A. 46.3
B. 52.0
C. 22.8
D. 34.9
E. 35.1
Given: m ∥ n Prove: ∠4 and ∠6 are supplementary Two parallel lines m and n intersect another line. The first line forms 4 angles numbered 1, 2, 4, and 3 in clockwise direction and the second line forms 4 angles numbered from 5, 6, 8, and 7 in clockwise direction. Proof: Statements Reasons 1. m ∥ n Given 2. m∠6 = m∠7 Vertical angles theorem 3. ? Same-side interior angles theorem 4. m∠4 + m∠7 = 180° Definition of supplementary angles 5. m∠4 + m∠6 = 180° Substitution property of equality 6. ∠4 and ∠6 are supplementary Definition of supplementary angles Select the statement that completes the proof. A. ∠2 and ∠4 are supplementary B. ∠2 and ∠7 are supplementary C. ∠4 and ∠7 are supplementary D. ∠4 and ∠5 are supplementary
This statement follows from the fact that m∠4 + m∠6 = 180°. The statement that completes the proof is C. ∠4 and ∠7 are supplementary.
To complete the proof, let's analyze the given statements and reasons:
m ∥ n (Given) - This statement establishes that lines m and n are parallel.
m∠6 = m∠7 (Vertical angles theorem) - When two lines intersect, the vertical angles formed are congruent.
? (Same-side interior angles theorem) - This step is missing in the given proof. To prove that ∠4 and ∠6 are supplementary, we need to use the same-side interior angles theorem.
The same-side interior angles theorem states that when two parallel lines are intersected by a transversal, the interior angles on the same side of the transversal are supplementary. In this case, lines m and n are parallel, and the transversal is the line that intersects them.
To complete the proof, we can include the missing statement:
∠4 and ∠6 are interior angles on the same side of the transversal (Same-side interior angles theorem).
Now, we can continue with the remaining steps of the proof:
m∠4 + m∠7 = 180° (Definition of supplementary angles) - Supplementary angles add up to 180 degrees.
m∠4 + m∠6 = 180° (Substitution property of equality) - We substitute ∠7 with ∠6 since they are congruent.
∠4 and ∠6 are supplementary (Definition of supplementary angles) - This statement follows from the fact that m∠4 + m∠6 = 180°.
Therefore, the statement that completes the proof is:
C. ∠4 and ∠7 are supplementary.
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