The expected length of the arc containing the point (1,0) is; π
How to solve Unit Circle Problems?First of all, it is pertinent to note that the lengths of the four arcs in which the circle is divided by the three points and (1, 0) have identical distributions.
The problem is same as dividing a circle of length 2π at four points chosen at random, and labelling one of them (1, 0). Now, let us make L₁, L₂, L₃, L₄ to be the lengths of the four arcs determined by those four points.
Thus, we have;
L₁ + L₂ + L₃ + L₄ = 2π
Whereas, the expected value of several random variables is additive as;
E[ L₁ + L₂ + L₃ + L₄] = [EL₁] + [EL₂] + [EL₃] + [EL₄]
By symmetry the lengths have the same probability distribution and as such; [EL₁] = [EL₂] = [EL₃] = [EL₄] and the sum must be 2π. Thus, we can say that each expected value is 2π/4 = π/4.
Finally, the expected value of the arc containing the point (1, 0) is the sum of the expected values of the two arcs that are adjacent to (1, 0), i.e.,
π/2 + π/2 = π
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The total marks obtained in a mathematics test was 384 if the mean was 24 how many children took the test
Use the distance formula to find the perimeter of the triangle below. Round you answer to the nearest hundredth.
jAlright, let's take this problem step-by-step:
What is the perimeter:
⇒ all the side-lengths added up
⇒ need to find side-length using distance formula
[tex]Distance_._. Formula=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
- (x₁,y₁): first point
- (x₂,y₂): second point
Let's solve:
Find length of AC⇒ first point: (5,7)
⇒ second point: (3,3)
[tex]Distance =\sqrt{(3-5)^2+(3-7)^2} =\sqrt{4+16} =\sqrt{20} =4.472[/tex]
Find length of AB⇒ first point: (5,7)
⇒ second point:(8,6)
[tex]Distance=\sqrt{(8-5)^2+(6-7)^2} =\sqrt{9+1} =\sqrt{10}=3.162[/tex]
Find length of BC⇒ first point: (8,6)
⇒ second point: (3,3)
[tex]Distance = \sqrt{(3-8)^2+(3-6)^2} =\sqrt{25+9} =\sqrt{34}=5.831[/tex]
Now let's solve for the perimeter:
[tex]Perimeter = 4.472+ 3.126+ 5.831=13.429[/tex]
Answer: 13.43 (rounded to the nearest hundredth)
Hope that helps!
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The diagram below shows a square inside a regular octagon. The apothem of the octagon is 15.69 units. To the nearest square unit, what is the area of the shaded region? 13 U 13 U Apothem length: 15.69 O A. 1463 square units B. 816 square units C. 647 square units OD. 764 square units
perimeter of octagon
8(13)104unitsNow area
perimeter×apothem /2104(15.69)/252(15.69)815.88units²Area of square
13²169units²Area of shaded region
815.88-169646.88units²647units²Answer:
C. 647 square units
Step-by-step explanation:
To find the shaded area, subtract the area of the unshaded square from the area of the octagon.
Area of the octagon
[tex]\textsf{Area of a regular polygon}=\dfrac{n\:l\:a}{2}[/tex]
where:
n = number of sidesl = length of one sidea = apothemGiven:
n = 8l = 13a = 15.69Substitute the given values into the formula and solve for A:
[tex]\implies \textsf{Area}=\sf \dfrac{8 \cdot 13 \cdot 15.69}{2}[/tex]
[tex]\implies \textsf{Area}=\sf \dfrac{1631.76}{2}[/tex]
[tex]\implies \textsf{Area}=\sf 815.88\:\:square \:units[/tex]
Area of the square
[tex]\implies \textsf{Area}=\sf 13^2=169 \:\:square \:units[/tex]
Area of the shaded region
= area of the octagon - area of the square
= 815.88 - 169
= 646.88
= 647 square units (nearest square unit)
Josh has a big backyard. He loves Red Pine trees because they can grow as tall as 50m (200 feet).
Josh is 31.2 metres away from the Red Pine tree. The angle of elevation from Josh to the top of tree is 48º. His friend Jake also has a Red Pine tree in his backyard. It is half as tall as Josh's tree. Calculate the height of Jake's tree, rounded to two decimal places
The tangent or tanθ in a right-angle triangle is the ratio of its perpendicular to its base. The height of Jake's tree is 17.325 trees.
What is a Tangent (Tanθ)?The tangent or tanθ in a right-angle triangle is the ratio of its perpendicular to its base. it is given as,
[tex]\rm Tangent(\theta) = \dfrac{Perpendicular}{Base}[/tex]
where,
θ is the angle,
Perpendicular is the side of the triangle opposite to the angle θ,
The base is the adjacent smaller side of the angle θ.
Given Josh is 31.2 meters away from the Red Pine tree. The angle of elevation from Josh to the top of the tree is 48º. Therefore, the height of the Josh tree will be,
Tan(48°) = Height of the tree/ Distance between Josh and tree
Height of the tree = Tan(48°) × 31.2 meter = 34.65 meters
Now, Jake's tree is half as tall as Josh's tree. Therefore, the height of Jake's tree is,
Height of Jake's tree = 34.65 meters = 17.325 meters
Hence, the height of Jake's tree is 17.325 trees.
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What is the range of f(x) = 3x + 9?
Answer:
The range is all real numbers.
Step-by-step explanation:
The function shown is a line (and not a horizontal line).
The range is all the y's on the graph or all the y's that can be generated by the equation of the function.
f(x) = 3x + 9
You could put ANY, literally any, number in place of the f(x) and be able to calculate an x for it. Looking at the graph of this line, you could go up and down the y-axis to infinity in either direction, and the graph of the line would be there.
The range is All Real Numbers.
what is the measure of easy interior angle of a regular octagon?
The sum of the interior angle is 1080 and each angle measure is 135 degrees.
What is a regular polygon?A polygon is a geometric figure with a finite number of sides in two dimensions. On the sides or edges of a polygon, straight-line segments are joined end to end to form a closed shape. The vertices, also known as corners, are the points where two line segments meet and form an angle.
We have an octagon
In an octagon the number of sides = 8
The sum of the angles in the octagon is given by:
= (n - 2)×180
n = 8
= (8 - 2)180
= 6×180
The sum of the interior angle = 1080
Each angle = 1080/8 = 135 degree
Thus, the sum of the interior angle is 1080 and each angle measure is 135 degrees.
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what does ([tex](x^{2}+4x)(x)[/tex] equal
the first definition of an element was made by
All else being equal, if you cut the sample size in half, how does this affect the margin of error when using the sample to make a statistical inference about the mean of the normally distributed population from which it was drawn? m e = startfraction z times s over startroot n endroot endfraction.
how to figure out pythagoras theorem?
Any triangle with one angle equal to 90⁰ produces a Pythagoras triangle and the Pythagoras equation can be applied in the triangle.
What is the Pythagoras Theorem?The Pythagoras theorem states that if a triangle is right-angled (90 degrees), then the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In △ABD and △ACB,
∠A = ∠A (common)
∠ADB = ∠ABC (both are right angles)
Thus, △ABD ∼ △ACB (by AA similarity criterion)
Similarly, we can prove △BCD ∼ △ACB.
Thus △ABD ∼ △ACB,
Therefore, AD/AB = AB/AC.
We can say that AD × AC = AB².
Similarly, △BCD ∼ △ACB.
Therefore,
CD/BC = BC/AC.
We can also say that
CD × AC = BC².
Adding these 2 equations, we get
AB² + BC² = (AD × AC) + (CD × AC)
AB² + BC² =AC(AD +DC)
AB² + BC² = AC²
Hence proved
Thus, any triangle with one angle equal to 90⁰ produces a Pythagoras triangle and the Pythagoras equation can be applied in the triangle.
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Solve for x..........
[tex]5x - x - 8 = 15 + 3x[/tex]
[tex]\quad \huge \quad \quad \boxed{ \tt \:Answer }[/tex]
[tex]\qquad \tt \rightarrow \:x = 23 [/tex]
____________________________________
[tex] \large \tt Solution \: : [/tex]
[tex]\qquad \tt \rightarrow \: 5x - x - 8 = 15 + 3x[/tex]
[tex]\qquad \tt \rightarrow \: 4x - 8 = 15 + 3x[/tex]
[tex]\qquad \tt \rightarrow \: 4x - 3x = 15 + 8[/tex]
[tex]\qquad \tt \rightarrow \: x = 23[/tex]
Answered by : ❝ AǫᴜᴀWɪᴢ ❞
Answer:
x = 23
Explanation:
⇒ 5x - x - 8 = 15 + 3x
collect like terms
⇒ 5x - x - 3x = 15 + 8
add or subtract like terms
⇒ x = 23
Checking:
5(23) - 23 - 8 = 15 + 3(23)
115 - 23 - 8 = 15 + 69
84 = 84
Please help me with my math
The value of z is 119° and the value of y is 28°.
What is corresponding angles?When two parallel lines meet any other line, comparable angles are generated in matching corners or corresponding corners with the transversal, these angles are called corresponding angles.
If the given diagram is modified a bit it will look as shown below, the two angles will be 61° because the two angles are corresponding angles.
Now, since 61° and x lie on the same side of the line, therefore, the sum of the two angles will be,
61° + x = 180°
x = 119°
Also, the angles measured at 61° and (3y-23)° lie on the same side of the transversal, therefore, these two angles are corresponding angles. Thus,
61° = (3y-23)°
61 = 3y - 23
y = 28
Hence, the value of z is 119° and the value of y is 28°.
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9. Classify the shapes shown below. Write the letter of each shape in the category that
describes it. Shapes may fall into more than one category, and not all shapes will be
used.
At least 1 right angle = A, C and E
At least 2 sides of equal length = B and E
At least 1 pair of parallel sides= B and E
What is a Parallel line?Parallel lines are those lines that are apart and they never meet till infinity.
Example of shapes that have sides with parallel line are the rectangle and parallelograms.
The shape B and E are rectangles and parallelograms respectively.
The shape that has angle 90° are right angle triangles and trapezium.
Therefore, the shapes that has right angles are A which is a trapezoid and E which is a rectangle.
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Chris opened up a new credit card and credit line with U.S bank. He is paying a 15% APR on all new purchases
A. if he has a balance of 2,800 on his credit card, how much does he owe in interest to the bank next month?
B. if he sends in a payment to U.S bank for $140, how much of that will go towards paying off his principle balance
Answer:
Which one does not belong to the group
Which are two possible reasons for the change within the frog population?
–3.2, 4.8, –7.2, 10.8, …
The given sequence;–3.2, 4.8, –7.2, 10.8, is a geometric progression common ratio of -1.5.
What is a geometric series?When all the terms of a geometric sequence are added, then that expression is called geometric series.
The given sequence ;
–3.2, 4.8, –7.2, 10.8, …
The common ratio
= -4.8/ 3.2
= - 1.5
The terms having a common ratio of -1.5 best describe the relationship between the successive terms in the sequence of the terms.
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Can someone please teach me how to do this? You can please do like 2 questions and please explain exactly how you got the answers so I can do the rest myself. please
All of these questions require one thing: trigonometric functions.
There are 3 main trigonometric functions, which can only be used on right triangles: sine, cosine, and tangent.
Sine = opposite / hypotenuse
Cosine = adjacent / hypotenuse
Tangent = opposite / adjacent
When trying to figure out what function to use, we always start by looking from the angle. Take problem a, for example. Looking from angle E, of which the value is not given, we have the side opposite and the side adjacent. Therefore, we should use the tangent function.
---The hypotenuse is always the longest side of the triangle. It is never considered the opposite or adjacent side.
Let's set up our function with the given information from problem a.
tan(x) = 9.7 / 5.2
---The tangent of an unknown angle is equal to the quotient of the opposite side and the adjacent side.
Now, solving for the value of x will require a calculator. We'll need to use what's called an inverse trigonometric function. Most calculators have these directly above the regular trigonometric functions, and the inverse functions are accessed using a "second" key.
---Ensure that your calculator is in degrees, not radians!
x = tan^-1(9.7 / 5.2)
x = 61.805 = 62 degrees
Next, let's take a look at problem b. This time, we're solving for a side length instead of an angle. But, we're still going to start by looking from our angle.
Looking from the 38 degree angle, we are given the adjacent side and an unknown hypotenuse. Therefore, we should use the cosine function.
cosine(38) = 53.1 / r
---The cosine of a 38 degree angle is equal to the quotient of 53.1 and an unknown hypotenuse, r.
Use your algebra skills to isolate the variable r.
r * cosine(38) = 53.1
r = 53.1 / cosine(38)
---From here, all you need to do is plug this into your calculator. Since we are solving for a side length (and given an angle), we are just using the regular trigonometric function buttons on the calculator.
r = 67.385 = 67.4 units
Hope this helps!
Select the correct answer.
Which value of x makes this equation true?
-12x2(x + 9) = 5(x + 4)
O A.
OB.
O C.
O D.
Answer:
B. -2
Step-by-step explanation:
-12x -2(x+9) = 5(x+4)
*ALWAYS REMEMBER YOUR BODMAS*
-12x -2x - 18 = 5x + 20
*group the like terms*
-14x - 5x = 20 + 18
-19x = 38
x = -2
The value of x that makes the equation true is: x = -2.
B. -2.
Here, we have to find the value of x that makes the equation true, we need to solve for x.
Let's solve the equation step by step:
-12x - 2(x + 9) = 5(x + 4)
Distribute the -2 and 5 on the right side of the equation:
-12x - 2x - 18 = 5x + 20
Combine like terms on both sides:
-14x - 18 = 5x + 20
Move the variable terms (5x) to one side of the equation and the constant terms to the other side.
Let's subtract 5x from both sides:
-14x - 5x - 18 = 5x - 5x + 20
-19x - 18 = 20
Move the constant term (-18) to the other side of the equation.
Let's add 18 to both sides:
-19x - 18 + 18 = 20 + 18
-19x = 38
Finally, divide both sides by -19 to isolate x:
x = 38 / -19
x = -2
So, the value of x that makes the equation true is: x = -2.
B. -2.
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Find the value of z
[tex]\textsf {Finding y :}[/tex]
[tex]\implies \mathsf {1^{2} + y^{2} = 3^{2}}[/tex]
[tex]\implies \mathsf {y^{2} = 8}[/tex]
[tex]\implies \mathsf {y = 2\sqrt{2}}[/tex]
[tex]\textsf {Finding z :}[/tex]
[tex]\implies \mathsf {(2\sqrt{2})^{2}+8^{2}=z^{2}}[/tex]
[tex]\implies \mathsf {z^{2} = 72}[/tex]
[tex]\implies \mathsf {z = 6\sqrt{2}}[/tex]
Answer:
z =8.49 or 6[tex]\sqrt{2}[/tex]
Step-by-step explanation:
Do the Pythagorean theorem = [tex]a^{2} +b^{2} =c^{2}[/tex]
Length of hypotenuse = 8+1=9
length of one side = 3
so, a²+3²=9²
a²+3²=9²
a²=81-9
a²=72
a=√72=8.49 or 6[tex]\sqrt{2}[/tex]
Pleaseee hurry and answer
what are the zero of f(x)=x(x-8)
The zeroes of the function f(x) = x (x – 8) will be 0 and 8.
What is a factorization?It is a method for dividing a polynomial into pieces that will be multiplied together. At this moment, the polynomial's value will be zero.
The function is given below.
f(x) = x (x – 8)
The zeroes of the function f(x) is given by the factor equals to zero.
Then we have
f(x) = 0
x(x – 8) = 0
x = 0, 8
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In the year 2008, a person bought a new car for $23500. For each consecutive year after that, the value of the car depreciated by 10%. How much would the car be worth in the year 2011, to the nearest hundred dollars?
Answer:
$17131.50
Step-by-step explanation:
23500 decrease 10% =
23500 × (1 - 10%) = 23500 × (1 - 0.1) = 21150
21150 decrease 10% =
21150 × (1 - 10%) = 21150 × (1 - 0.1) = 19035
19035 decrease 10% =
19035 × (1 - 10%) = 19035 × (1 - 0.1) = 17131.5
The vertices of an isosceles triangle are A(-4, 0), B(-10, -2) and C(-6, -6).
What is the equation of the triangle's line of symmetry?
y = - x - 4
y = x - 4
y = x + 4
y = - x + 4
express 3256 as the product of power of 10
Answer:
3.256*10³2
Step-by-step explanation:3.256 * 1000 = 3256 1000 =10³
Find an explicit formula for the arithmetic sequence -11, -3, 5, 13, ....
Note: the first term should be b(1).
b(n) =
Answer:
b(n) = -11 + (n-1)*8
Step-by-step explanation:
Let n be the sequence number, with n=1 the first number b(1) -11
The sequence changes by +8 each step.
b(1) -11 + 8 = -3
b(2) -3 + 8 = 5
b(3) 5 + 8 = 13
b(4) 13
b(1) = -11
b(2) = -3. or b(1) + 1*8
b(3) = 5, or b(1) + 2*8
b(4) = 13, or b(1) + 3*8
We note that each step adds a multiple of 8 to the initial value of -11. This can be stated as (n-1)*8
The formula for this sequence would be b(n) = b(1) + (n-1)*8
b(n) = -11 + (n-1)*8
Check: Does n=3 return the value of 5?
b(n) = -11 + (n-1)*8
b(3) = -11 + ((3)-1)*8
b(3) = -11 + (2)*8
b(3) = -11 + 16
b(3) = 5 YES
What is the solution to the equation below? (Round your answer to two
decimal places.)
7•Inx=2.4
A. x=99.48
B. x = 18.48
C. x=1.41
D. x= 1.9
Answer:
C
Step-by-step explanation:
7 × ln(x) = 2.4
ln(x) = 2.4/7 = 0.342857143...
x = e^0.342857143... = 1.408967466... ≈ 1.41
If one pair of opposite sides of a quadrilateral is both congruent and____ then the quadrilateral is a parallelogram then the
Answer:
parallel
a quadrilateral can be a parallelogram only and only if pair of opposite sides are equal and parallel.
Girl runs across Across a rectangle field diagonal covering a distance of 120m if the length of the field is 100m calculate the width of the field to the nearest metre
Answer:
66m
Step-by-step explanation:
Answer:
66 m
Step-by-step explanation:
ok so if you think about it, when she runs diagonal she's creating a triangle where she is running along the hypotenuse. we can now use the Pythagorean Theorem a^2 + b^2 = c^2 to solve for the other side with two of them being known.
plug values in
100^2 + w^2 = 120^2
square known values
10,000 = w^2 = 14,400
subtract 10,000 from both sides
w^2 = 4400
take the square root
w = 66 (rounded to the nearest whole number)
Solve the equation 7(w +13) = 35.
) What is the value of w?
Answer: -8
Step-by-step explanation:
7(w+13)=35 [given]w+13=5 [divide both sides by 7]w = -8 [subtract 13 from both sides]Answer:
7(w+13)=35
7w + 91 =35
7w=35-91
7w= -56
7w/7 = -56/7
w=-8
3 erasers cost $4.41. Write a proportion that would solve for the cost of 4 erasers.
Answer:
$5.88
Step-by-step explanation:
3 erasers ⇒ $4,41
4 erasers ⇒ $A
A = 4 erasers * $4.41 / 3 erasers
A = $ 5.88
Answer:
see below
Step-by-step explanation:
Proportion:
4.41 is to 3 erasers as 'x' is to 4 erasers:
4.41 /3 = x / 4 <======this is your proportion
or rearrange
4 * 4.41/3 = x
(this results in x = $ 5.88 for 4 erasers )
What is the vertex of the graph of the function below?
y=x² + 4x-6
A. (2,0)
B. (-2,6)
C. (-2,-10)
D. (2,6)
what is 5xy-x^2t+2xy+3x^2t
Answer:
x • (2xt + 7y)
Step-by-step explanation:
1. ((5xy-((x2)•t))+2xy)+(3x2•t)
2. Pull out like factors :
2x2t + 7xy = x • (2xt + 7y)
Final result
x • (2xt + 7y)
Answer:
[tex]7xy + {x}^{2} t[/tex]
Step-by-step explanation:
[tex](5xy + 2xy) + ( - 2 {x}^{2} t + 3 {x}^{2}t) [/tex]
