To show that the conditional statement [(p → q) ∧ (q → r)] → (p → r) is a tautology without using truth tables, we can use a logical proof known as the Law of Implication.
To show that the conditional statement [(p → q) ∧ (q → r)] → (p → r) is a tautology without using truth tables, we can employ a logical proof known as a direct proof.
First, let's assume that the antecedent, [(p → q) ∧ (q → r)], is true.
This means that both (p → q) and (q → r) are true simultaneously.
Using the definition of implication, (p → q) can be written as (~p ∨ q) and (q → r) can be written as (~q ∨ r).
So we have (~p ∨ q) ∧ (~q ∨ r) as the conjunction of the two implications.
Now, we need to prove that (p → r) is also true.
Using the definition of implication, (p → r) can be written as (~p ∨ r).
To show that (p → r) is true, we need to prove that ~p ∨ r is true.
We can do this by considering the two cases:
If ~p is true, then ~p ∨ r is true regardless of the truth value of r.
If ~p is false, then p is true, and since (p → q) and (q → r) are both true, q and r must also be true.
Thus, ~p ∨ r is true.
In both cases, ~p ∨ r is true, which means (p → r) is true.
Since both the antecedent [(p → q) ∧ (q → r)] and the consequent (p → r) are true, we can conclude that the conditional statement [(p → q) ∧ (q → r)] → (p → r) is a tautology.
Therefore, using a direct proof, we have shown that the given conditional statement is always true and satisfies the definition of a tautology.
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Andre is making a quilt with blue and green fabrics. He has 15 options for the blue portion of the quilt and wants to use 4 different fabrics. How many different ways are there for Andre to select the blue fabrics?
There are 1365 different ways for Andre to select the blue fabrics for his quilt.
To determine the number of different ways Andre can select the blue fabrics for his quilt, we can use the concept of combinations.
Andre wants to select 4 different fabrics from a pool of 15 options for the blue portion of the quilt. This can be calculated using the formula for combinations:
C(n, r) = n! / (r! × (n-r)!)
where n is the total number of options and r is the number of selections.
In this case, we have:
n = 15 (number of options for blue fabrics)
r = 4 (number of fabrics to be selected)
Plugging in these values into the combination formula, we get:
C(15, 4) = 15! / (4! × (15-4)!)
Simplifying the equation, we have:
C(15, 4) = 15! / (4! × 11!)
Now, let's calculate the factorial terms:
15! = 15 × 14 × 13 × 12 × 11!
4! = 4 × 3 × 2 × 1
11! = 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
Substituting these factorial terms into the equation:
C(15, 4) = (15 × 14 × 13× 12 × 11!) / (4 × 3 × 2 × 1 × 11!)
Simplifying further:
C(15, 4) = (15 × 14 × 13 × 12) / (4 × 3 × 2 × 1)
C(15, 4) = 1365
Therefore, there are 1365 different ways for Andre to select the blue fabrics for his quilt.
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2)
A high school basketball team won exactly 65 percent
of the games it played during last season. Which of
the following could be the total number of games the
team played last season?
A) 22
B) 20
C) 18
D) 14
Answer:
To find the answer, we can use the formula:
number of won games / total number of games played = percentage won
Let x be the total number of games played. We know that the percentage won is 65%, or 0.65 as a decimal. So we can set up the equation:
number of won games / x = 0.65
To solve for x, we can cross-multiply:
number of won games = 0.65x
We want to find a whole number value for x that makes sense. One way to do this is to try each answer choice and see if it gives a whole number value for the number of won games. Let's start with choice A:
If the team played 22 games, then the number of won games is:
number of won games = 0.65 * 22 = 14.3
This is not a whole number value, so we can rule out choice A.
We can repeat this process for each answer choice. When we try choice C, we get:
number of won games = 0.65 * 18 = 11.7
This is also not a whole number value, so we can rule out choice C.
When we try choice D, we get:
number of won games = 0.65 * 14 = 9.1
This is also not a whole number value, so we can rule out choice D.
Therefore, the only remaining answer choice is B, which gives us:
number of won games = 0.65 * 20 = 13
This is a whole number value, so the team could have played 20 games in total last season.
find the surface area
The Surface Area of Cone is 706.5 cm².
We have dimension of Cone
height = 12 cm
Radius = 9 cm
Now, l = √9² + 12² = √81 + 144 = 25 cm
So, the Surface Area of Cone
= πrl
= 3.14 x 9 x 25
= 706.5 cm²
Thus, the Surface Area of Cone is 706.5 cm².
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Pls help me in math!!!!!!!!!!
In the given triangle value of b is,
⇒ b = 16
We have to given that,
A triangle with three angles (b + 20), (b + 32) and 6b.
Now, WE know that;
Sum of all interior angles of triangle is 180 degree.
Hence., We can formulate;
⇒ (b + 20) + (b + 32) + 6b = 180
Solve for b,
⇒ 8b + 52 = 180
⇒ 8b = 180 - 52
⇒ 8b = 128
⇒ b = 128/8
⇒ b = 16
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A health store sells two different sized square granola bars. the side length of the smaller granola bar, C(x), is modeled by the function C(x)=1/4 square root x +2, where x is the area of the larger granola bar in square inches. which graph shows C(x)
The graph that best represents these characteristics is the one that shows an increasing curve with a flattened slope and is shifted upward. Option C
The function C(x) = (1/4)√x + 2 represents the side length of the smaller granola bar, where x is the area of the larger granola bar in square inches. To determine which graph shows C(x), we need to analyze the characteristics of the function.
First, let's consider the behavior of the function C(x) = (1/4)√x + 2:
Square Root Function: The term √x indicates that the function involves the square root of x. This means that as x increases, C(x) will also increase, but at a decreasing rate.
Scaling Factor: The coefficient (1/4) scales the square root of x. It affects the steepness of the function and causes C(x) to increase more slowly compared to a simple square root function.
Vertical Shift: The constant term (+2) shifts the entire function vertically upward by 2 units. This means that the graph will be raised by 2 units compared to a standard square root function.
Based on these characteristics, we can conclude that the graph of C(x) will be an increasing curve with a flattened slope compared to a simple square root function, and it will be shifted upward by 2 units.
Among the answer choices, the graph that best represents these characteristics is the one that shows an increasing curve with a flattened slope and is shifted upward. Without the actual graphs provided, it is difficult to specify the exact choice, but it should exhibit these features described above.
It is essential to refer to the available graphs or visually analyze the functions to determine the correct graph that shows C(x). Option C is correct.
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The circle below has center D. Suppose that m BC = 42°. Find the following.
The measure of angle BDC is 42° and the measure of angle BAC is 21°.
Given that, the measure of arc BC of circle = 42°.
An arc of a circle is a section of the circumference of the circle between two radii. A central angle of a circle is an angle between two radii with the vertex at the centre. The central angle of an arc is the central angle subtended by the arc. The measure of an arc is the measure of its central angle.
From the given circle,
The measure of arc BC = Angle BDC = 42°
Here, Angle BDC = 2×Angle BAC
42° = 2×Angle BAC
Angle BAC = 21°
Therefore, the measure of angle BDC is 42° and the measure of angle BAC is 21°.
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Which expression is equivalent to 3 x 5 x 5 x5 x 5 x 5 x5 ?
Answer:
3(5)^6
Step-by-step explanation:
There is 6, 5's in the expression meaning our answer has to be 3(5)^6.
:)
Answer:
the answer for the question given above is equal to 46875
what is the condensed form of the following logarithm: 2log w + 5log v
Answer: log w²v⁵
Step-by-step explanation:
2log w + 5log v >Use exponent log rule
log w² + log v⁵ >There is addition between, use
multiplication rule
log w²v⁵
The average national utility price is $270.48. Over a 6-month period, what is the average utility price in Orlando? How
does this compare with the national average?
The average utility price in Orlando for the 6 month period, and the way it compares to the national average is d. $ 308. 83 ; higher than the national average .
How to find the average ?The information for Orlando is April, the cost was 288 dollars; May, 310 dollars; June, 325 dollars; July, 294 dollars; August, 293 dollars; September, 343 dollars.
The average is:
= ( 288 + 310 + 325 + 294 + 293 + 343 ) / 6
= 1, 853 / 6
= $ 308. 83
This average is higher than the national average of $ 308. 83.
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Full question is:
The average national utility price is $270.48. Over a 6-month period, what is the average utility price in Orlando? How does this compare with the national average?
A graph titled Orlando, Florida, has month on the x-axis and utility price on the y-axis. In April, the cost was 288 dollars; May, 310 dollars; June, 325 dollars; July, 294 dollars; August, 293 dollars; September, 343 dollars.
a. $370.60; higher than the national average
b. $292.17; higher than the national average
c. $38.35; lower than the national average
d. $308.83; higher than the national average
100 Points! Geometry question. Photo attached. Determine whether each pair or figures is similar. If so, write the similarity statement and scale factor. If not, explain your reasoning. Please show as much work as possible. Thank you!
The figures are similar because the ratio of segment BC to segment EF is equal to the of segment AB to segment DE.
The scale factor is equal to 3/2.
What are the properties of similar triangles?In Mathematics and Geometry, two (2) triangles are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.
Additionally, the lengths of corresponding sides or corresponding side lengths are proportional to the lengths of corresponding altitudes when two (2) triangles are similar.
Based on the side, angle, side (SAS) similarity theorem, we can logically deduce the following:
Scale factor = BC/EF = AB/DE
Scale factor = 9/6 = 6/4
Scale factor = 3/2 = 3/2 (similar)
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100 Points! Geometry question. Photo attached. Please show as much work as possible. Thank you!
The length of the arc LM is 8.72 cm.
We have,
The length of an arc is the distance that runs through the curved line of the circle making up the arc.
The length of an arc is expressed as;
l = tetha/360 × 2πr
tetha = R
R = 100°
and, radius = 5 units
so, we get,
l = 100/360 × 2 × 3.14 × 5
l = 8.72 cm (1.dp)
therefore the length of the arc LM is 8.72 cm
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Don't forget to show your work. Thank you!
The probability that a point chosen randomly inside the rectangle and is inside the Square or Trapezoid is 2/15.
We know that, probability of an event
= Number of favourable outcomes/Total number of outcomes.
Here, area of a rectangle = Length × Breadth
= 15×8
= 120 square units
Area of a triangle = 1/2 × Base × Height
= 1/2 ×(√10²-6²)×6
= 0.5×8×6
= 24 square units
Area of a trapezium = 1/2 (Sum of parallel sides)×Height
= 1/2 ×(5+7)×2
= 12 square units
Area of a square = side² = 2²
= 4 square units
(a) Probability of landing in trapezium or Square = 12/120 + 4/120
= 16/120
= 2/15
(b) Probability of landing inside the rectangle but outside the triangle = 16/120
= 2/15
Therefore, the probability that a point chosen randomly inside the rectangle and is inside the Square or Trapezoid is 2/15.
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Using a calculator or statistical software, find the linear regression line for the data in the table below.
Using the regression with the values rounded to the nearest hundredth, find the value of y at x=7. Round your answer to the nearest hundredth if necessary.
x y
0 2.83
1 3.33
2 6.99
3 8.01
4 7.62
5 7.66
Rounded to the nearest hundredth, the value of y at [tex]x = 7[/tex] is approximately [tex]12.78[/tex], according to the concept of linear regression line.
To find the linear regression line for the given data, we can use statistical software or a calculator. The equation of the linear regression line is of the form [tex]y = mx + b[/tex], where m represents the slope and b represents the y-intercept.
Using the provided data, the linear regression line equation is:
[tex]\[ y = 1.3642x + 3.2324 \][/tex]
To find the value of [tex]y[/tex] at [tex]x = 7[/tex], we substitute [tex]x = 7[/tex] into the equation:
[tex]\[ y = 1.3642(7) + 3.2324 \][/tex]
Simplifying the equation, we get:
[tex]\[ y = 9.5494 + 3.2324 \]\[ y = 12.7818 \][/tex]
In conclusion, the linear regression analysis allows us to determine the relationship between the variables x and y in the given data set. By finding the equation of the linear regression line, we can estimate the value of y for any given x. In this case, the linear regression line equation is [tex]y = 1.3642x + 3.2324[/tex]. By substituting [tex]x = 7[/tex] into the equation, we find that the estimated value of y is approximately [tex]12.78[/tex]. This analysis provides a useful tool for predicting and understanding the relationship between variables, allowing us to make informed decisions and interpretations based on the data.
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Simplify 6 5/8 + 6 5/6
The required sum of mixed fraction answer is 12 35/24.
Given that 6 5/8 + 6 5/6.
To find the sum of two or more mixed fractions, add the whole part of two fractions, add the fraction part of two fractions, then add these sums.
The sum (s1) of whole parts of two fractions is
s1= 6 + 6 = 12.
The sum(s2) of fraction parts of two fractions is
s2 = 5/8 +5/6.
L.C.M of 8 and 6 is 24.
Multiply each fraction by 12/12 gives,
s2 = 5/8 × 24/24 + 5/6 × 24/24.
s2 = 15/24 + 20/24.
s2 = 35/24.
The sum of mixed fractions is s1 + s2
Sum = 12 + 35/24
Thus, the required sum of mixed fraction is 12 35/24.
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Select the three inequalities that include 3 in the solution set.
x > 1.4
x < 2.6
x > 4.2
x < 5.1
x < 8.2
The solution set which include 3 are x > 1.4, x < 5.1 and x < 8.2.
Given the inequalities that include 3 in the following inequalities
x > 1.4, x < 2.6, x > 4.2, x < 5.1 and x < 8.2.
To find the solution set which include 3, write the solution set which consists of integer.
The solution set of x > 1.4 is { 2, 3, 4, 5, 6, ........}
The solution set of x < 2.4 is { 2, 1. 0, -1, ...............}
The solution set of x > 4.2 is { 5, 6. 7, 8, ...............}
The solution set of x < 5.1 is { 5, 4, 3, 2, 1, ...............}
The solution set of x < 8.2 is { 8, 7, 6, 5, 4, 3, ...............}
Hence, the solution set which include 3 are x > 1.4, x < 5.1 and x < 8.2.
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f(x)=x-4 Function Family: Parent Function: D: ; R: End Behavior: As x→∞, f(x)→ As x→→∞, f(x)- Increasing Interval: Decreasing Interval:
May I please get a little help with this question? Thank you so much.
The y-intercept of the function is (0, c)
The coefficients b determine the horizontal shift of the parabola compared to the parent function
If a is negative, the parabola opens downward
The y-intercept of the function is (0, c).
This means that when x = 0, the y-value is equal to c.
The constant term c represents the y-coordinate of the point where the parabola intersects the y-axis.
The coefficient b determines the horizontal shift of the parabola compared to the parent function.
The value of b affects the position of the vertex and determines if the parabola is shifted to the left or right.
A positive value of b shifts the parabola to the left, while a negative value of b shifts it to the right.
If a is negative, the parabola opens downward.
The coefficient a determines the shape of the parabola.
If a is positive, the parabola opens upward, and if a is negative, the parabola opens downward. The sign of a determines the direction in which the parabola faces.
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determine the volume of the following spheres with the given dimensions. to break the code find the mean of all your answers. round your answers to the hundreth place. use 3.14 for an approximation for pi.
The calculated volume of the sphere is 44602.24 ft³
How to determine the volume of the sphereFrom the question, we have the following parameters that can be used in our computation:
Radius = 22 ft
The volume of a sphere can be expressed as;
V = 4/3πr³
Where
r = 22
substitute the known values in the above equation, so, we have the following representation
V = 4/3π * 22³
Evaluate
V = 44602.24
Therefore the volume of the sphere is 44602.24 ft³
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Question
Determine the volume of the following spheres with the given dimensions. to break the code find the mean of all your answers. round your answers to the hundreth place. use 3.14 for an approximation for pi.
Radius = 22 ft
NEED HELP ASAP WILL GIVE BRAINLIEST HELP!
Consecutive Interior Angles.
They are on the same side of a line that cuts through two other lines.
Use the function f(x) to answer the questions:
f(x) = 2x2 − 5x + 3
Part A: What are the x-intercepts of the graph of f(x)? Show your work.
Part B: Is the vertex of the graph of f(x) going to be a maximum or a minimum? What are the coordinates of the vertex? Justify your answers and show your work.
Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph.
The x-intercepts of the graph of f(x) are x = 3/2 and x = 1,the Vertex of the graph of f(x) is (5/4, 3/8), and it is a minimum point, The vertex is at (5/4, 3/8). This is the minimum point of the graph.
Part A: To find the x-intercepts of the graph of f(x), we set f(x) equal to zero and solve for x.
2x^2 - 5x + 3 = 0
To factor this quadratic equation, we look for two numbers that multiply to give 3 (the coefficient of the constant term) and add up to -5 (the coefficient of the linear term). These numbers are -3 and -1.
2x^2 - 3x - 2x + 3 = 0
x(2x - 3) - 1(2x - 3) = 0
(2x - 3)(x - 1) = 0
Setting each factor equal to zero, we get:
2x - 3 = 0 --> x = 3/2
x - 1 = 0 --> x = 1
Therefore, the x-intercepts of the graph of f(x) are x = 3/2 and x = 1.
Part B: To determine whether the vertex of the graph of f(x) is a maximum or a minimum, we look at the coefficient of the x^2 term, which is positive (2 in this case). A positive coefficient indicates that the parabola opens upwards, so the vertex will be a minimum.
To find the coordinates of the vertex, we can use the formula x = -b/2a. In the equation f(x) = 2x^2 - 5x + 3, the coefficient of the x term is -5, and the coefficient of the x^2 term is 2.
x = -(-5) / (2*2) = 5/4
Substituting this value of x back into the equation, we can find the y-coordinate:
f(5/4) = 2(5/4)^2 - 5(5/4) + 3 = 25/8 - 25/4 + 3 = 3/8
Therefore, the vertex of the graph of f(x) is (5/4, 3/8), and it is a minimum point.
Part C: To graph f(x), we can use the information obtained in Part A and Part B.
- The x-intercepts are x = 3/2 and x = 1. These are the points where the graph intersects the x-axis.
- The vertex is at (5/4, 3/8). This is the minimum point of the graph.
We can plot these points on a coordinate plane and draw a smooth curve passing through the x-intercepts and the vertex. Since the coefficient of the x^2 term is positive, the parabola opens upwards, and the graph will be concave up.
Additionally, we can consider the symmetry of the graph. Since the coefficient of the linear term is -5, the line of symmetry is given by x = -(-5) / (2*2) = 5/4, which is the x-coordinate of the vertex. The graph will be symmetric with respect to this line.
By connecting the plotted points and sketching the curve smoothly, we can accurately graph the function f(x).
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30 POINTS!! PLEASE
A surveyor wants to determine the width of a river. She surveys the area and finds the following measures below.
She uses a pair of similar triangles, to help her find the answer.
Answer:
D = 90 FT
Step-by-step explanation:
40 / d = 20 / 45
cross multiply
45 × 40 = 20d
1800 = 20d
divide both sides by 20
1800 / 20 = 20d / 20
d = 90 ft
The following number is a BASE-7 NUMBER. Convert this to a base 10 number and enter your answer in the box.
463421
(Remember, the above number is in Base 7!)
Number in base 10:
The base 10 equivalent of the base 7 number 463421 is 83174.
To convert a number from base 7 to base 10, we multiply each digit of the base 7 number by the corresponding power of 7 and sum up the results.
For the number 463421 in base 7:
= 4 x 7⁵ + 6 x 7⁴ + 3 x 7³ + 4 x 7² + 2 x 7¹ + 1 x 7⁰
= 4 x 16807 + 6 x 2401 + 3 x 343 + 4 x 49 + 2 x 7 + 1 x 1
= 67228 + 14406 + 1029 + 196 + 14 + 1
= 83174
Therefore, the base 10 equivalent of the base 7 number 463421 is 83174.
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A surveyor is standing 30 feet from the base of a tall building. The surveyor measures the angle of elevation from the ground to the top of the building to be 65 degrees. Find the height of the building to the nearest foot
Answer:
Height = 64 feet
Step-by-step explanation:
(***It's very important to model these kinds of problems. I attached a picture modeling the problem so show how modeling makes it easier to solve the problem)
We know that the building is vertical, and the ground is horizontal. Thus, the ground and the building make a right angle.
Therefore, the top of the building, the base of the building, and the surveyor form a right triangle, where
The distance between the surveyor and the building is a leg of the triangle,The height, h of the building (aka distance between the top of the building and the base) is another leg of the triangle,and the distance between the top of the building and the surveyor is the hypotenuse.Because we're working with a right triangle, we can now find h, the heigh of the building using one of the trigonometric ratios.
We know that the angle of elevation is made by the ground and the distance between the surveyor and top of the building.
When the 65° (angle of elevation) is our reference angle, the height of the building is the opposite side and the distance between the base of the building and the surveyor is the adjacent side.
Thus, we can use the tangent ratio, which is: tan (θ) = opposite / adjacent:
We can now plug in 65 for θ and 30 for the adjacent side. This will allow us to find the opposite side, which is the heigh of the building:
tan (65) = h / 30
30 * tan(65) = h
64.33520762
64 = h
Thus, the height of the building to the nearest foot is 64 feet.
50 Points! Multiple choice geometry question. Photo attached. Thank you!
The equation of the circle is (x - 2)² + (y - 3)² = 36
Determining the equation of the circleFrom the question, we have the following parameters that can be used in our computation:
The circle
Where, we have
Center = (a, b) = (2, 3)
Radius, r = 6 units
The equation of the circle is represented as
(x - a)² + (y - b)² = r²
So, we have
(x - 2)² + (y - 3)² = 6²
Evaluate
(x - 2)² + (y - 3)² = 36
Hence, the equation is (x - 2)² + (y - 3)² = 36
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1
2
3
4
5
I
Statement
+
ZHKI ZGKH
HJ I GI
H
m2GKH+mZHKI = 180°
m2GKH + m2GKH = 180°
m2GKH = 90°
Reason
Given
Angles forming a linear pair sum to 180°
Definition of congruence
Answer:
3. Substitution because it said angle HKI = angle GKH, so we substitutioned that angle for the other one. I'm not sure about 4. If you provide us with the answer choices for that one, then I could help
subtract 8.263 from 13.48.
Answer:
13.48 - 8.263 = 5.217
Answer:5.217
Step-by-step explanation: caculate it
The slop of the graphed line is 2/3
The formulas that represent the linear function in this problem are given as follows:
y - 2 = 2/3(x - 1).y - 4 = 2/3(x - 4).f(x) = 2x/3 + 4/3.How to define a linear function?The slope-intercept equation for a linear function is presented as follows:
y = mx + b
The line has a slope of 2/3, hence:
y = 2x/3 + b.
When x = 1, y = 2, hence the intercept b is obtained as follows:
2/3 + b = 2
b = 6/3 - 2/3
b = 4/3.
Hence the slope-intercept equation of the line is given as follows:
f(x) = 2x/3 + 4/3.
The line goes through points (1,2) and (4,4), hence the point-slope equations to the line are given as follows:
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What is the value of x
Enter your answer in the box
Answer:
x = 8 units
Step-by-step explanation:
Because this is a right triangle, we can find x using the Pythagorean theorem, which is given by:
a^2 + b^2 = c^2, where
a and b are the shorter sides called legs,and c is the longest side called the hypotenuse (always opposite the right angle).In the triangle, the 6 unit side and the x unit sides are the legs while the 10 unit side is the hypotenuse.
Thus, we can solve for x by plugging in 6 for a and 10 for c in the theorem and solving for a (aka x, simply a in the theorem):
a^2 + 6^2 = 10^2
a^2 + 36 = 100
a^2 = 64
a = 8
x = 8
Thus, x is 8 units.
Which statement is true? 3x-4y=4 or 3x-4y=8
Answer:3x-4y=8
Step-by-step explanation:
write an explicit formula for an the nth term of the sequence 40, 33, 26
Answer:
[tex]a_{n}[/tex] = - 7n + 47
Step-by-step explanation:
there is a common difference between consecutive terms , that is
33 - 40 = 26 - 33 = - 7
this indicates the sequence is arithmetic with nth term
[tex]a_{n}[/tex] = a₁ + d(n - 1)
where a₁ is the first term and d the common difference
here a₁ = 40 and d = - 7 , then
[tex]a_{n}[/tex] = 40 - 7(n - 1) = 40 - 7n + 7 = - 7n + 47