Adam says that the number 4 has four factors. The above statement is incorrect as 4 only three factors 1,2, and 4.
What are the factors?A factor is a number that completely divides another number. To put it another way, if adding two whole numbers results in a product, then the numbers we are adding are factors of the product because the product is divisible by them.Factors can be found using either division or multiplication. Rules of divisibility can also be applied. There are a finite number of factors for all integers.The factor of a number is never more than the number; it is always less than or equal to the number.Every integer, excluding 0 and 1, has a minimum of two factors: 1 and the actual number.Multiplication and division are used to find factors.To learn more about Factors with the given link
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Given the following coordinates complete the reflection transformation.
A(1,−5)
B(2,−2)
C(5,−2)
Transformation: Complete the double reflection over the lines y=−1 followed by y=1.
The double reflection generates the following three points: A''(x, y) = (1, - 1), B''(x, y) = (2, 2) and C''(x, y) = (5, 2).
How to generate a set of point by rigid transformations
In this problem we must apply two rigid transformations to find three points. The formula for reflection over an axis parallel to the y-axis is defined below:
P'(x, y) = (x', k) - [P(x, y) - (x', k)] (1)
Where:
x' - x-coordinate of the point P(x, y).P(x, y) - Original pointP'(x, y) - Resulting pointIf we know that A(x, y) = (1, - 5), k = - 1 and k' = 1, then the resulting points are:
Point A
A'(x, y) = (1, - 1) - [(1, - 5) - (1, - 1)]
A'(x, y) = (1, - 1) - (0, - 4)
A'(x, y) = (1, 3)
A''(x, y) = (1, 1) - [(1, 3) - (1, 1)]
A''(x, y) = (1, 1) - (0, 2)
A''(x, y) = (1, - 1)
Point B
B'(x, y) = (2, - 1) - [(2, - 2) - (2, - 1)]
B'(x, y) = (2, - 1) - (0, - 1)
B'(x, y) = (2, 0)
B''(x, y) = (2, 1) - [(2, 0) - (2, 1)]
B''(x, y) = (2, 1) - (0, - 1)
B''(x, y) = (2, 2)
Point C
C'(x, y) = (5, - 1) - [(5, - 2) - (5, - 1)]
C'(x, y) = (5, - 1) - (0, - 1)
C'(x, y) = (5, 0)
C''(x, y) = (5, 1) - [(5, 0) - (5, 1)]
C''(x, y) = (5, 1) - (0, - 1)
C''(x, y) = (5, 2)
The double reflection generates the following three points: A''(x, y) = (1, - 1), B''(x, y) = (2, 2) and C''(x, y) = (5, 2).
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A rope that is 245cm long is cut into three pieces. The ratio of the lengths of the first piece to the second piece is 2/3, and the ratio of the lenghts of the second piece to the third piece is 4/5. What is the length of the longest of the the three pieces?
The length of the longest piece of the cut rope is; 105 cm
How to work with ratios?We are given that;
Length of rope = 245 cm
We need to make the ratio of second piece equal in both the ratio to find the ratio of all three pieces.
2:3
4:5
Multiply 1st ratio by 4 and 2nd ratio by 3:
Now, the ratio becomes: 8:12 and 12:15
And the ratio of three pieces can be represented as:
8: 12: 15, this ratio is the first piece: second piece: third piece
Thus;
8x + 12x + 15x = 245
35x = 245
x = 245/35
So, the pieces lengths will be;
First piece = 8 * 7 = 56 cm
Second piece = 12 * 7 = 84 cm
Third piece = 15 * 7 = 105 cm
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Kelley writes the expression n 2 to model the phrase "xander studied two more hours than nandini." which best explains the accuracy of kelley’s expression? it is accurate. in the phrase "two more hours than nandini," "two" is "2," "more" is " ," and nandini’s study time is unknown or "n," so 2 n or n 2 are correct translations. it is inaccurate. in the phrase "two more hours than nandini," "two" is "2," "more" is " ," and nandini’s study time is unknown or "n," so 2 n is the correct translation. it is inaccurate. in the phrase "two more hours than nandini," "two" is "2," "more than" is ">," and nandini’s study time is unknown or "n," so 2 greater-than n is the correct translation. it is inaccurate. in the phrase "two more hours than nandini," "two" is "2," "more than" is "<," and nandini’s study time is unknown or "n," so 2 less-than n is the correct translation.
It is accurate. In the phrase “two more hours than Nandini,” n + 2 are correct translations. Then the correct option is A.
What is Algebra?Algebra is used to analyze mathematical symbols, and logic is used to manipulate those symbols.
As an example, Kelley models the sentence "Xander studied two more hours than Nandini" using the equation n + 2.
Consequently, the following will describe Kelley's expression's accuracy the best:
It is precise. Since "two" is translated as "2," "more" as "+," and "Nandini's study time is unknown or "n," 2 + n or n + 2 are the appropriate translations for the sentence "two more hours than Nandini."
So, option A is the best choice.
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I understand that the question you are looking for is :
Kelley writes the expression n + 2 to model the phrase “Xander studied two more hours than Nandini.” Which best explains the accuracy of Kelley’s expression?
It is accurate. In the phrase “two more hours than Nandini,” “two” is “2,” “more” is “+,” and Nandini’s study time is unknown or “n,” so 2 + n or n + 2 are correct translations.
It is inaccurate. In the phrase “two more hours than Nandini,” “two” is “2,” “more” is “+,” and Nandini’s study time is unknown or “n,” so 2 + n is the correct translation.
It is inaccurate. In the phrase “two more hours than Nandini,” “two” is “2,” “more than” is “>,” and Nandini’s study time is unknown or “n,” so 2 greater-than n is the correct translation.
It is inaccurate. In the phrase “two more hours than Nandini,” “two” is “2,” “more than” is “<,” and Nandini’s study time is unknown or “n,” so 2 less-than n is the correct translation.
Please help me on this one I got it wrong
Answer: E. 10,000,000 cubic millimeters
Step-by-step explanation: To find out any result from cubic centimeters to cubic millimeters, you multiply the cubic centimeters' value by 1000 to get the cubic millimeters' value.
10,000 * 1,000 = 10,000,000 cubic millimeters.
Hope this helped!
Step-by-step explanation:
The correct answer is E 10,000,00
25. Jen ran x miles last week and 18 miles
this week. Write an expression to represent
the total number of miles Jen ran over the
last two weeks.
Solve: If she ran 34 miles over the last two
weeks, how many did she run last week?
Answer:
Step-by-step explanation:
If Jen ran 34 miles over the last 2 weeks and only 18 miles the second week, she would've had to have run 16 miles the first week.
34= 18+x
I hope this helps!
find unknown angles, value of 'x' and unknown sides:
At time t, the position of a body moving along the s-axis is s=t^3 -18t^2+60t m. Find the displacement of the body from t=0 to t=3.
a. 56 m
b. 67 m
c. 105 m
d. 45 m
The displacement between t = 0 and t = 3 is 45 meters, so the correct option is d.
How to find the displacement?The displacement is defined as the difference between the final position and the initial position.
The initial position is what we get when we evaluate in t = 0.
s = 0^3 -18*0^2+60*0 = 0
The initial position is 0 meters.
The initial position is what we get when we evaluate in t = 3.
s = 3^3 -18*3^2+60*3 = 45
The final position is 45 meters.
Then the displacement is:
D = 45m - 0m = 45m
The correct option is d.
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The functions f(x) and g(x) are graphed.
f(x)
324
2
-6-5-4-3-2-1₁.
297 49
-2+
-3
-4
g(x)
1 2 3 4 5 6 x
Which represents where f(x) = g(x)?
Of(0) = g(0) and f(2)= g(2)
Of(2)= g(0) and f(0) = g(4)
Of(2)= g(0) and f(4) = g(2)
Of(2)= g(4) and f(1) = g(1)
Answer: Option 1
Step-by-step explanation:
The graphs intersect at x=0 and x=2.
The answer that represents the point where f(x) = g(x) is; A: f(2) = g(2) and f(0) = g(0)
What are functions?A function in math is visualized as a rule, which gives a unique output for every input x. Mapping or transformation is used to denote a function in math.
Given is a graph we need to find the point of intersection where f(x) = g(x)
How to find the coordinates of Intersection :-
From the attached graph, we see that is represents the relationship between the functions f(x) and g(x).
Now, the coordinates of the points where f(x) = g(x) will be the coordinates of the points where both graphs intersect.
The first point of intersection, we see that;
f(0) = g(0) = 4
Second point shows;
f(2) = g(2) = 0
Thus, we can conclude that;
f(2) = g(2) and f(0) = g(0)
Hence, the answer that represents the point where f(x) = g(x) is; A: f(2) = g(2) and f(0) = g(0)
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Show that the curve x = 7 cos(t), y = 4 sin(t) cos(t) has two tangents at (0, 0) and find their equations
The equations with tangents at (0,0) are [tex]y = \frac{4}{7} x[/tex] and
[tex]y = -\frac{4}{7} x[/tex].
In this question,
The curves are x = 7 cos(t), y = 4 sin(t) cos(t)
Two tangents at (0, 0)
In this case, the parametric derivative of x and y are expressed in terms of t.
The first derivative dy/dx can be expressed as
[tex]\frac{dy}{dx}=\frac{\frac{dy}{dt} }{\frac{dx}{dt} }[/tex]
Now, dy/dt is obtained by differentiate y with respect to t,
[tex]\frac{dy}{dt}= 4[cos(t)(cos(t))+sin(t)(-sin(t))][/tex]
⇒ [tex]\frac{dy}{dt}= 4[cos^{2} (t)-sin^{2} (t)][/tex]
Now, dx/dt is obtained by differentiate x with respect to t,
[tex]\frac{dx}{dt} =7(-sin(t))[/tex]
⇒ [tex]\frac{dx}{dt} =-7sin(t)[/tex]
Thus, [tex]\frac{dy}{dx}=\frac{4[cos^{2}(t)-sin^{2}(t ) ]}{-7sin(t)}[/tex]
At (0,0) x = 0 and y = 0, Then
0 = 7 cos(t)
0 = 4 sin(t) cos(t)
and
cos(t) = 0
sin(t) cos(t) = 0
There are two values between -π and π which satisfy these equations simultaneously are
t = π/2
t = -π/2
The equation of a straight line given a point and its slope is
y-y₀ = m(x-x₀)
The two tangents lies at (0,0), so the equation becomes
y = mx
Then the two straight lines will be
y = m₁x and
y = m₂x
For t = π/2,
[tex]m_1=\frac{dy}{dx}=\frac{4[cos^{2}(\frac{\pi }{2} )-sin^{2}(\frac{\pi }{2} ) ]}{-7sin(\frac{\pi }{2} )}[/tex]
⇒ [tex]m_1=-\frac{4[0-1]}{7(1)}[/tex]
⇒ [tex]m_1=\frac{4}{7}[/tex]
For t = -π/2,
[tex]m_2=\frac{dy}{dx}=\frac{4[cos^{2}(-\frac{\pi }{2} )-sin^{2}(-\frac{\pi }{2} ) ]}{-7sin(-\frac{\pi }{2} )}[/tex]
⇒ [tex]m_2=-\frac{4[0-1]}{-7(1)}[/tex]
⇒ [tex]m_2=-\frac{4}{7}[/tex]
Thus the equations with tangents at (0,0) are [tex]y = \frac{4}{7} x[/tex] and
[tex]y = -\frac{4}{7} x[/tex].
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After a rotation of 90° about the origin, the coordinates of the vertices of the image of a triangle are A'(5, 3), B'(–2, 1), and C'(1, 7). What are the coordinates of the vertices of the pre-image?
A -
B -
C -
The coordinates of the vertices of the preimages are A ( 3, 5) , B ( 1, 2) and C ( 7, -1)
How to determine the coordinatesIt is important to note the following about the 90 degrees rotation about the origin
A( x ,y ) becomes A' ( -y, x ) switch x and y and make y negativeNote that
A ( x, y) is the coordinates of the preimage
A' ( -y , x) is the coordinates after the said rotation about the origin
Converting the transformed coordinates to the preimage coordinates
For vertex A
A' ( 5, 3)
Compare with A' ( -y, x)
x = 3
y = -5
Substitute the values
A ( 3, 5 )
For vertex B
B' ( -2, 1)
Compare with B' ( -y, x)
x = 1
y = 2
Substitute the values
B ( 1, 2)
For vertex C
C' ( 1, 7)
Compare with C' ( -y, x)
x = 7
y = -1
Substitute the values
C ( 7, -1)
Thus, the coordinates of the vertices of the preimages are A ( 3, 5) , B ( 1, 2) and C ( 7, -1)
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A team plays 93 games in a season. The team won 15
more than twice as many games as they lost.
How many wins and losses did the team have?
PLS HELPPPP THE WORDING IS CONFUSING
Answer:67 wins, 26 losses
Step-by-step explanation: They played 93 games. Let L = their losses and w = their wins. They won 15 more than twice as many games they lost so w = 2L + 15. So, 93 = 3L + 15. 93 - 15 = 78. 78 divided by 3 is 26 so they lost 26 games and won the rest of them or 93-26 = 67 wins and 26 losses.
X
-3
-1
1
3
f(x) = -x + 7
f(x)
8
2323
20
6
Determine the input that would give an output value of
W/N!
2
X=
=-3x+7
Answer: 19
Step-by-step explanation:
Multiplying both sides of the equation by [tex]-3[/tex] gives that [tex]x=19[/tex].
what book has the solution in it?
Answer:
Book 3
Step-by-step explanation:
So we have four statements:
1: Book 1 or Book 2
2: Book 3 or Book 4
3: Book 4
4: Not Book 3
We also have the condition that only 1 of the statements are correct, and more important, the other 3 are incorrect.
So it may be a bit difficult to see at first, but the solution, has to be book 3. The reason for this is if we assume statement 1, statement 2, or statement 3 is true, that would imply that the statement 4 is false, meaning the solution is in book 3.
If we assume that statement 2 is true, then statement 4 is false, meaning that book 3 has the solution, which aligns statement 2. It also aligns with statement 1 being false, because it's not book 1 or 2, and it also aligns with statement 3 being false, because the solution is not in book 4.
Let's look at why assuming any other statement to be true, leads to contradictions.
So let's assume the first statement is true, this means the solution is book 1 or 2. But this means the statement 4 is false, meaning book 3 has the solution. This contradicts the first statement which we assume is true, thus the first statement is not true.
Let's now assume the third statement is true, this means statement four is false meaning it's book 3, but this contradicts the third statement which says it's book 4. Thus this statement cannot be true
Let's assume the last option is true, this means book 3 has no solution, and since all the other are false, book 4 has no solution, book 3 or 4 has no solution, book 1 or 2 has no solution either. This means that none of the books have solutions, and assuming one of the books does have a solution, this statement cannot be true.
y=4x−5 y=2x+3 Is (4,11)(4,11)left parenthesis, 4, comma, 11, right parenthesis a solution of the system?
The solution to the systems if equation y = 4x - 5, y = 2x + 3 is (x, y) = (4, 11)
Equationy = 4x - 5
y = 2x + 3
Equation both equations4x - 5 = 2x + 3
collect like terms4x - 2x = 3 + 5
2x = 8
divide both sides by 2x = 8/2
x = 4
Substitute x = 4 intoy = 4x - 5
= 4(4) - 5
= 16 - 5
y = 11
Therefore, the solution to the system of be equation is x = 4 and y = 11
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Find the midpoint of the line segment joining (–4, –2) and (2, 8).
HELP!!! MATH!!! 100 PTS !!!
Suppose f is a one-to-one, differentiable function and its inverse function f−1 is also differentiable. One can show, using implicit differentiation (do it!), that
(f^−1)′(x)=1/(f′(f^−1(x)))
Find (f^−1)′(6) if f(−1)=6 and f′(−1)=3/7.
Answer:
(f^−1)′(6)=1/(f'(f^-1(6)))
(f^−1)′(6)=1/(f'(f^-6)))
I hope this helps.
Arlana graphed the system of equations that can be used to solve x cubed minus 5 x squared 2 = negative x cubed 17 x.
The roots of the resulting polynomial equation are [tex](x_{1} ,y_{1} ) = (-2 , -26), (x_{2} ,y_{2} ) = ( 0.11 , 1.94 )[/tex] and [tex]( x_{3} , y_{3} ) = ( 4.39, - -9.81)[/tex] , respectively.
What is a polynomial easy definition?
A mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a nonnegative integral power (such as a + bx + cx2).
Ariana must graph two polynomic expressions:
[tex]f(x) = x^{3} -5 . x^{2} + 2[/tex] and [tex]g(x) = -x^{3} + 17 . x[/tex]
The roots of the resulting polynomial equation are found by means of this formula-
f(x) - g(x) = 0.........................1
Then, we must look for every point so that f(x) = g(x)
The roots of the resulting polynomial equation are [tex](x_{1} , y_{1} ) = ( -2,-26) , (x_{2} ,y_{2} ) = ( 0.114 , 1.937 )[/tex] and [tex](x_{3},y_{3} ) = ( 4. 386 , -9. 812)[/tex] respectively.
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Answer:
D
Step-by-step explanation:
.
y = x2 + 3
y = x + 5
A vector in the xy plane has a magnitude of 25 and an x component of 12. The angle it makes with the positive x axis is?
The angle it makes with the positive x axis in the xy plane is 61.3145 degrees.
In this question,
A normal vector to the plane is any vector that starts at a point in the plane and has a direction that is orthogonal (perpendicular) to the surface of the xy plane .
The magnitude of the xy plane, hypotenuse = 25 units
The x component of the xy plane, base = 12 units
Let θ be an angle, then the angle it makes with the positive x axis is
[tex]\theta = cos^{-1}(\frac{base}{hypotenuse} )[/tex]
⇒ [tex]\theta = cos^{-1}(\frac{12}{25} )[/tex]
⇒ [tex]\theta = cos^{-1}(0.48)[/tex]
⇒ [tex]\theta = 61.3145[/tex]
Hence we can conclude that the angle it makes with the positive x axis in the xy plane is 61.3145 degrees.
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Show your work and hurry please
Answer:
15
Step-by-step explanation:
Note: Take note , absolute value will give the magnitude of the value. (Which means ignoring the + / - signs.)
Eg. | - 19 | = 19
Therefore,
[tex] \frac{12(30 - (9 + {4}^{2})) }{10 - | - 6| } \\ = \frac{12(30 - (9 + 16))}{10 - 6} \\ = \frac{12(30 - 25)}{4} \\ = \frac{12(5)}{4} \\ = \frac{60}{4} \\ = 15[/tex]
Select the correct answer from each drop-down menu.
-60
8-
6-
4
2-
-4-2 O
-2-
-4-
-6-
-8-
-N
4
2
-6
Quadrilateral 1 and quadrilateral 2 are polygons that can be mapped onto each other using similarity transformations. The
transformation that maps quadrilateral 1 onto quadrilateral 2 is a
followed by a dilation with a scale factor of
Reset
Nexts
The transformation that maps quadrilateral 1 onto quadrilateral 2 is a translation followed by a dilation with a scale factor of 2.
What is a transformation?A transformation can be defined as the movement of a point on a cartesian coordinate from its original (initial) position to a new location.
The types of transformation.In Geometry, there are different types of transformation and these include the following:
DilationReflectionRotationTranslationBased on similarity transformation of both quadrilateral 1 and quadrilateral 2, we can infer and logically deduce that the transformation which directly maps quadrilateral 1 onto quadrilateral 2 is a translation followed by a dilation with a scale factor of 2, as shown in the image attached below.
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Find the coordinates of the vertices of the figure after the given transformation: T<−5,−2>
The vertices of the figure after the transformation: T<−5, −2>; 5 units left and 2 units down is X'(-3, -3), V'(-4, 0), E'(-1, -1), K'(0, -5)
What is transformation?Transformation is the movement of a point from its initial location to a new location. Types of transformation are reflection, translation, rotation and dilation.
Rigid transformations are transformation that preserve the shape and size of a figure such are reflection, rotation and translation.
The vertices of the figure after the transformation: T<−5, −2>; 5 units left and 2 units down is X'(-3, -3), V'(-4, 0), E'(-1, -1), K'(0, -5)
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need help finding the measurement of s
Answer:
∠S = 66°
Step-by-step explanation:
A parallelogram's 4 angles always add up to 360°, and opposite angles are the same. (∠S = ∠U; ∠T = ∠V)
So, ∠S + ∠T = 180°.
180° = (2x + 4x + 12 + 6)°
180° = (6x + 18)°
162° = (6x)°
27° = x°
(2x + 12)° = ∠S
(2(27) + 12)° = ∠S
(54 + 12)° = ∠S
66° = ∠S
Which of the following fractions has the largest value 11/20 10/21 9/19 8/17 or 6/13
Answer:
11/20
Step-by-step explanation:
You could turn them all into equivalent fractions, but the common denominator would be 1,763,580. Yuck.
Look at the relationship for the numerators to the denominators. The only numerator that is more than half of its denominator is 11/20. Half of 20 would be 10 and 11 has a larger value than 10, so as a decimal, it would be more than .5. All of the other numerators are less than half, so that they have to less than .5.
If you changed all of the numbers to decimals, you would see that 11/20 has the largest value.
Solve the equation by first using a sum-to-product formula. (enter your answers as a comma-separated list. let k be any integer. round terms to three
Solutions of the equation are 22.5°, 30°.
The given equation is sin(5θ) - sin(3θ) = cos(4θ)
We take left side of the equation
sin(5θ) - sin(3θ) = 2cos ((5θ+3θ)/2) (sin(5θ-3θ)/2)
=2cos4θsinθ [From sum-product identity]
Now we can write the equation as
2cos(4θ)sin(θ) = cos(4θ)
2cos(4θ)sinθ - cos(4θ) = 0
cos(4θ)[2sinθ - 1] = 0
cos(4θ) = 0
4θ = 90°
θ = 90/4
θ = 22.5°
and (2sinθ - 1) = 0
sinθ = 1/2
θ = 30°
Therefore, solutions of the equation are 22.5°, 30°
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x =
30°
xo
90°
Report a pro
Answer:
here is the answer
hope it helps u
happy to help
stay safe and healthy
step by step explanation :x=90(being adjacent angle)
now ,
x+30=180(being alternate angle)
x=180-30
x=150#
i think my answer ia wrong sorry -_-
Answer:
x=90
Step-by-step explanation:
x is on a straight line with 90 degrees (given angle measure angle)
you should do 180-90=90 to find that angle x is 90 degrees.
Geometry: fill in the blanks, ASAP! It’s urgent
Answer:
3. 125
4. 115
5. m<1 = 70, m<2 = 55, m<3 = 55
6. m<2 = 50, m<3 = 50, m<4 = 80, m<6 = 130
can you guys help me please or you can just explain how to solve it thanks
The composition of transformation that maps ABCD to EHGF is "(x, y) → (x', -y') → (x' + 6, y' + 3)".
What are the transformation rules?The basic transformation rules are:
Translation: (x, y) → (x + a, y + b);Reflection: (x, y) → (x, -y) over x-axis; (x, y) → (-x, y) over y-axis;Dilation: (x, y) → (kx, ky)Rotation 90° counter-clockwise: (x, y) → (-y, x)Rotation 180°: (x, y) → (-x, -y)Calculation:The given quadrilateral ABCD has vertices,
A(-5, 2), B(-3, 4), C(-2, 4), and D(-1, 2)
The transformed quadrilateral EHGF has vertices,
E(1, 1), H(3, -1), G(4, -1), and F(5, 1)
In the map ABCD to EHGF, the transformations that took place are:
1) Reflection over x-axis
2) Translation by (x + a, y + b)
Step 1: Reflection over x-axis; (x, y) → (x, -y)
A(-5, 2) → A'(-5, (-2)) = A'(-5, -2)
B(-3, 4) → B'(-3, (-4)) = B'(-3, -4)
C(-2, 4) → C'(-2, (-4)) = C'(-2, -4)
D(-1, 2) → D'(-1, (-2)) = D'(-1, -2)
So, the reflected quadrilateral is A'B'C'D'
Step 2: Translation by (x, y) → (x + a, y + b);
The reflected quadrilateral A'B'C'D' is now translated by
A'B'C'D' → EHGF
So, (for A'(-5, -2) and E(1, 1))
x + a = 1
⇒ -5 + a = 1
⇒ a = 1 + 5
∴ a = 6
and
y + b = 1
⇒ -2 + b = 1
⇒ b = 1 + 2
∴ b = 3
Thus, the translation is (x + 6, y + 3).
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Can someone please help me with these problems and show work, if the answer isn’t with work I will remove your comment and you won’t get the points, please help
m∠B = 57.52°, m∠B = 70.8°, AB = 46.03 km and AC = 39.08 ft. This can be obtained using the Laws of cosine formula and Laws of sine formula.
Find the required angles and sides:Laws of cosine formula,In a triangle ABC,
⇒ a² = b² + c² - 2bc cos A
⇒ b² = a² + c² - 2ac cos B
⇒ c² = a² + b² - 2ab cos C
where a, b and c are sides of a triangle and A, B and C are the angles of a triangle.
Laws of sine formula,In a triangle ABC,
⇒ [tex]\frac{sinA}{a} =\frac{sinB}{b} =\frac{sinC}{c}[/tex]
where a, b and c are sides of a triangle and A, B and C are the angles of a triangle.
In the question we can use the laws of cosine formula and laws of sine formula,
5) Given that,
AB = c = 13 km
AC = b = 21 km
m∠A = 91°
By using Laws of cosine formula,
⇒ a² = b² + c² - 2bc cos A
a² = 21² + 13² - 2(21)(13) cos 91°
a² = 441 + 169 - 546 (-0.0174524064)
a² = 610 + 9.52901391 = 619.529014
⇒ a = 24.89 km
By using Laws of sine formula,
⇒ [tex]\frac{sinA}{a} =\frac{sinB}{b} =\frac{sinC}{c}[/tex]
sin 91°/24.89 = sin B/21
sin B = 0.999847695×21/24.89 = 0.843583833
⇒ m∠B = 57.52°
6) Given that,
AB = c = 11 cm
BC = a = 13 cm
AC = b = 14 cm
By using Laws of cosine formula,
⇒ b² = a² + c² - 2ac cos B
14² = 13² + 11² - 2(13)(11) cos B
196 = 169 + 121 - 286 cos B
196 = 290 - 286 cos B
cos B = 94/286
cos B = 0.328671329
⇒ m∠B = 70.8°
7) Given that,
AC = b = 24 km
BC = a = 26 km
m∠C = 134°
By using Laws of cosine formula,
⇒ c² = a² + b² - 2ab cos C
c² = 26² + 24² - 2(26)(24) cos 134°
c² = 676 + 576 - 1248 (-0.69465837)
c² = 1252 + 866.933646 = 2118.93365
⇒ AB = c = 46.03 km
8) Given that,
AB = c = 26 ft
BC = a = 21 ft
m∠B = 112°
By using Laws of cosine formula,
⇒ b² = a² + c² - 2ac cos B
b² = 21² + 26² - 2(21)(26) cos 112°
b² = 441 + 676 - 1096 (-0.374606593)
b² = 1117 + 410.568826 = 1527.56883
⇒ AC = b = 39.08 ft
Hence m∠B = 57.52°, m∠B = 70.8°, AB = 46.03 km and AC = 39.08 ft.
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How many ways are there to put five beads on a necklace if there are eight distinct beads to choose from, and rotations and reflections of the necklace are considered the same
Answer:
3,360
Step-by-step explanation:
so, to start with, there are 8 over 5 permutations (the sequence of the picked beads matters, but no repetitions of beads are possible) to pick 5 items out of 8 available ones.
that is
8! / ((8-5)!) = 8! / 3! = 8×7×6×5×4 = 6,720
rotations and reflections are the same thing here for a 1- dimensional sequence.
1 2 3 4 5 rotated around the middle (3) is
5 4 3 2 1
a reflection is again
5 4 3 2 1
so, we need to consider that instead of 8 beads for the first choice we have only 4 beads to choose from to leave the other 4 beads as choices for the last position.
once we have this established, it is sure that the first and the last position cannot have mirrored beads in any permutation. and therefore rotational or reflectional permutations are impossible.
in other words, half of the possible permutations would be rotations/reflections, and we need to eliminate them.
so, the calculation is
4×7×6×5×4 = 8×7×6×5×4/2 = 6,720/2 = 3,360