On solving the provided question, we can say that the provided slope is -2/3 equation of line 2x + 3y -7 = 0
what is slope?A line's slope determines how steep it is. A mathematical equation for the gradient is called "gradient overflow" (the change in y divided by the change in x). The slope is the ratio of the vertical change (rise) between two points to the horizontal change (run) between those same two spots. When a straight line's equation is written as y = mx + b, the slope-intercept form of an equation is employed to express it. The y-intercept is located at a location where the line's slope is m, b is b, and (0, b). For instance, the slope of the equation y = 3x - 7 and the y-intercept (0, 7).The line's slope is m. b is b at the point where the y-intercept is, and (0, b). As an illustration, the equation y = 3x - 7 has a slope of 3, and its y-intercept is (0, 7).
the provided slope is -2/3
let us assume x1 = 2 and y1 = 1
equation of line =>
y-1 = -2/3(x-2)
y-1 = -2x/3 + 4/3
3y-3 = -2x + 4
2x + 3y -7 = 0
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Transform the given equation into a system of first order equations. (Let x1 = u, x2 = u', x3 = u'', and x4 = u'''. Enter your answers in terms of x1, x2, x3, and x4.) u(4) − u = 0
a) x1' =
b) x2' =
c) x3' =
d) x4' =
Given the 4th order linear ODE
[tex]u^{(4)} - u = 0[/tex]
we substitute
[tex]x_1 = u[/tex]
[tex]x_2 = {x_1}' = u'[/tex]
[tex]x_3 = {x_2}' = {x_1}'' = u''[/tex]
[tex]x_4 = {x_3}' = {x_2}'' = {x_1}''' = u'''[/tex]
Then the given equation transforms to
[tex]{x_4}' - x_1 = 0[/tex]
but we also need to relate this to the other derivative substitutions. This gives the system of differential equations
[tex]\begin{cases} {x_1}' = x_2 \\ {x_2}' = x_3 \\ {x_3}' = x_4 \\ {x_4}' = x_1 \end{cases}[/tex]
In matrix form,
[tex]\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}' = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}[/tex]
22. What is the area of minor sector DFE?
79.3 square cm
15.2 square cm
37.9 square cm
66.1 square cm
The area of the minor sector DFE is gotten as; D: 66.1 square cm
How to find area of sector?
Formula for area of sector is;
A = (θ/360) * πr²
We are given;
minor angle; θ = 100°
radius; r = 8.7 cm
Thus;
A = (100/360) * π(8.7²)
A = 66.1 square cm
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Larry has 6 shirts and 4 pairs of jeans. How many different combinations of one shirt and one pair of jeans can Larry make
Answer:
24 combinations
Step-by-step explanation:
To get to this answer, I multiplied 6 x 4, since there are 6 shirts and 4 pairs of jeans, there will be a total of 24 combinations. For combinations of two different things, you can just multiply the amount of each item by the other item. hope this helps :)
Select the correct answer.
A container is made by cutting off the bottom of a cone. The container has a diameter of 30 centimeters and a height of 20 centimeters. The
small cone that was removed has a diameter of 10 centimeters and a height of 6 centimeters.
What is the volume of the container, to the nearest cubic centimeter?
A. 6,126 cm³
B. 6,283 cm³
C. 5,812 cm³
D. 5,969 cm³
The volume of the container is 5969 cm³. The correct option is D. 5,969 cm³
Calculating VolumeFrom the question, we are to determine the volume of the container
Volume of the container = Volume of the cone - Volume of the small cone
The volume of a cone is given by the formula,
V = 1/3πr²h
Where V is the volume
r is the radius
and h is the height
From the given information,
For the small cone,
Diameter = 10 cm
∴ Radius, r = 10cm/ 2 = 5 cm
h = 6 cm
For the big cone
diameter = diameter of the container = 30 cm
∴ Radius = 30cm/ 2
Radius = 15cm
h = height of small cone + height of container
h = 6 cm + 20 cm
h = 26 cm
Putting the parameters into
Volume of the container = Volume of the cone - Volume of the small cone
We get,
Volume of the container = 1/3π × 15² ×26 - 1/3π × 5² × 6
Volume of the container = 1/3π (15² ×26 - 5² × 6)
Volume of the container = 1/3π (5850 - 150)
Volume of the container = 1/3π (5700)
Volume of the container = 1/3 × π × 5700
Volume of the container = 5969.026 cm³
Volume of the container ≈ 5969 cm³
Hence, the volume of the container is 5969 cm³. The correct option is D. 5,969 cm³
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9. Construct Arguments Write a two-column
proof for the Angle Bisector Theorem. MP.3
The angle bisector theorems is proved below.
What is the angle bisector theorems?It should be noted that the angle bisector theorem simply states that an angle bisector of a triangle divides the opposite side into two segments which are proportional to the other sides of the triangle.
The way to proof the theorem is illustrated:
Draw a ray CX parallel to AD and then extend BA to intersect this ray at E.
In triangle CBE, DA is parallel to CE.
BD/DC == BA/AE ......... i
Now we want to prove that AE = AC
Since DA is parallel to CE, we have:
DAB = CEA (corresponding angles) ....... ii
DAC = ACE (alternate interior angles) ...... iii
Since AD is the bisector of BAC, we've DAB = DAC.
From the above, ACE makes and isosceles triangle and since the opposite sides are equal, we've AC = CE.
Substitute AC for AE in equation i
BD/DC = BA/AC
Therefore, the angle bisector theorems is proved.
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fit the model, part a, to the data using simple linear regression. give the least squares prediction equation
Answer:
fitting a simple linear regression first select the cell in data set second on the analyse it ribbon tap in statistical analysis group click fit model and then click the simple degradation model 3rd the wild drop down list select the response variable 4th in the X drop down let's select the predicated
pls HELP Which is the correct way to model the equation 5 x + 6 = 4 x + (negative 3) using algebra tiles?
5 positive x-tiles and 6 positive unit tiles on the left side; 4 positive x-tiles and 3 negative unit tiles on the right side
6 positive x-tiles and 5 positive unit tiles on the left side; 3 negative x-tiles and 4 positive unit tiles on the right side
5 positive x-tiles and 6 negative unit tiles on the left side; 4 positive x-tiles and 3 negative unit tiles on the right side
5 positive x-tiles and 6 positive unit tiles on the left side; 4 positive x-tiles and 3 positive unit tiles on the right side
Answer:
The first choice
Step-by-step explanation:
You are setting up as you read it. The left side has 5 x and 6 units, so you would set up 5 positive x times and 6 positive unit tiles. On the the right you are setting up 4 positive x tiles and 3 negative unit tiles. It is not asking you to solve with the algebra tiles, just to set it up.
HELPPPPPPPPPPPPP99 n
Answer:
555 meters
Step-by-step explanation:
toymaker = height/shadow
= 0.9/0.2
CN tower = height/shadow
= x/123.3
solve for x:
;[tex]\frac{0.9}{0.2} = \frac{x}{123.3} \\\\0.2(x) = (0.9)(123.3)\\0.2x = 110.97\\(\frac{10}{2} )\frac{2}{10}x = 110.97(\frac{10}{2})\\ x = 554.85[/tex]
They are all in meters, so we don't need to convert anything so rounded to the nearest meter, the answer is 555 meters
Answer: 555 m
Step-by-step explanation:
We will set up a proportion to solve.
[tex]\displaystyle \frac{\text{vertical side}}{\text{horizontal side}} \;\;\;\;\;\;\;\frac{0.9\;m}{0.2\;m} =\frac{x\;m}{123.3 \;m}[/tex]
Now we will cross multiply.
0.9 * 123.3 = 0.2 * x
110.97 = 0.2x
554.85 = x
x = 554.85
Lastly, we will round the nearest m.
554.85 ➜ 555 m
Complete the following conversions. i) 1 m2 = a cm2 ii) 1 cm2 = b mm2 iii) 1 km2 = c m2
The values that complete the conversions are
a = 10000
b = 100
c = 1000000
Conversion of unitsFrom the question, we are to complete the given conversions
i)
1 m² = a cm²
NOTE: 1 m² = 10000 cm²
∴ a = 10000
ii)
1 cm² = b mm²
NOTE: 1 cm² = 100 mm²
∴ b = 100
iii)
1 km² = c m²
NOTE: 1 km² = 1000000 m²
∴ c = 1000000
Hence, the values that complete the conversions are
a = 10000
b = 100
c = 1000000
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1. Prove the following identities
Answer:
Step-by-step explanation:
[tex]\sf 1.1)\dfrac{Sin \ \theta-Cos \ \theta}{Sin \ \theta + Cos \ \theta}=\dfrac{(Sin \ \theta-Cos \ \theta)}{((Sin \ \theta + Cos \ \theta)}*\dfrac{(Sin \ \theta-Cos \ \theta)}{(Sin \ \theta-Cos \ \theta)}[/tex]
[tex]\sf = \dfrac{(Sin \ \theta-Cos \ \theta)^2}{Sin^2 \ \theta-Cos^2 \ \theta}\\\\=\dfrac{Sin^2 \ \theta+Cos^2 \ \theta-2Sin \ \theta \ Cos\ \theta}{(Sin \ \theta-Cos \ \theta)}\\\\\\\bf Identity: \ (a +b)^2= a^2 + b^2 - 2ab\\\\\\=\dfrac{1-2Sin \ \theta \ Cos \ \theta}{(Sin \ \theta-Cos \ \theta)} = LHS[/tex]
[tex]\sf 1.2) LHS = tan^2 \ x - Sin^2 \ x = \dfrac{Sin^2 \ x}{Cos^2 \ x}-Sin^2 \ x[/tex]
[tex]\sf =\dfrac{Sin^2 \ x}{Cos^2 \ x}-\dfrac{Sin^2 \ x*Cos^2 \ x}{1*Cos^2 \ x}\\\\\\ = \dfrac{Sin^2 \ x - Sin^2 \ x*Cos^2 \ x}{Cos^2 \ x}\\\\\\= \dfrac{Sin^2 \ x *(1 -Cos^2 \ x)}{Cos^2 \ x}\\\\=\dfrac{Sin^2 \ x*Sin^2 \ x}{Cos^2 \ x} \\\\ \bf 1 - Cos^2 \ x = Sin^2 \ x\\\\= \dfrac{Sin^2 \ x}{Cos^2 \ x}*Sin^2 \ x\\\\=tan^2 \ x * Sin^2 \ x = RHS[/tex]
[tex]\sf 1.3) LHS = \dfrac{1-Cos \ x}{Sin \ x}-\dfrac{Sin \ x}{1+Cos \ x} =\dfrac{(1-Cos \ x)(1+Cos \ x)}{Sin \ x*(1+Cos \ x)}-\dfrac{Sin \ x*Sin \ x}{(1+Cos \ x)*Sin \ x}\\[/tex]
[tex]\sf =\dfrac{1 - Cos^2 \ x}{Sin \ x*(1+Cos \ x)}-\dfrac{Sin^2 \ x}{Sin \ x*(1+Cos \ x)}\\\\=\dfrac{Sin^2 \ x}{Sin \ x*(1+Cos \ x)} - \dfrac{Sin^2 \ x}{Sin \ x*(1+Cos \ x)}\\\\=\dfrac{Sin^2 x - Sin^2 \ x}{Sin \ x*(1+Cos \ x)} \\\\= 0 = RHS[/tex]
[tex]\sf 1.4) LHS = Sin x - \dfrac{1}{Sin \ x + Cos \ x}+Cos \ x \\[/tex]
[tex]\sf = \dfrac{Sin \ x *(Sin \ x + Cos \ x) - 1 + Cos \ x * (Sin \ x + Cos \ x)}{Sin \ x + Cos \ x }\\\\\\= \dfrac{Sin \ x * Sin \ x + Sin \ x*Cos \ x -1 + Cos \ x*Sin \ x + Cos \ x*Cos \ x}{Sin \ x + Cos \ x}\\\\\\=\dfrac{Sin^2 \x + Sin \ x \ Cos \ x - 1 + Cos \ x \ Sin \ x + Cos^2 \x}{Sin \ x + Cos \ x}\\\\=\dfrac{Sin^2 \ x + Cos^2 \ x - 1 + Sin \ xCos \x +Sin \ x Cos \ x}{Sin \ x + Cos \ x}\\\\= \dfrac{1 - 1 +2Sin \ x Cos \ x}{Sin \ x + Cos \ x}\\\\= \dfrac{2Sin \ x Cos \ x}{Sin \ x + Cos \ x}[/tex]
What is the simplified form of startroot startfraction 2,160 x superscript 8 baseline over 60 x squared endfraction endroot? assume x ≠ 0.
The simplified form is [tex]6x^{3}[/tex].
What are indices in maths?
An index, or a power, is the small floating number that goes next to a number or letter.The plural of index is indices. Indices show how many times a number or letter has been multiplied by itself.The equation is -
[tex]\sqrt{\frac{2160x^{3} }{60x^{2} } }[/tex]
We can rewrite this using property of multiplication to obtain
[tex]\sqrt{\frac{2160}{60} * \frac{x^{8} }{x^{2} } }[/tex]
We simplify variable expression in the radicand using
[tex]\frac{a^{m} }{a^{n} } = a^{m - n}[/tex]
[tex]\sqrt{36 * x^{8 - 2} }[/tex]
[tex]\sqrt{36 * x^{6} }[/tex]
[tex]\sqrt{36 * (x^{3})^{2} }[/tex]
[tex]\sqrt{36} * \sqrt{(x^{3})^{2} }[/tex]
[tex]6 * x^{3}[/tex]
Therefore, the simplified form is [tex]6x^{3}[/tex]
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The sum of the reciprocals of three consecutive integers is 47/60. What is the sum of the three integers
Answer:
56
Step-by-step explanation:
small brain :(
Find the area of the rhombus.
9 m
9 m
9 m
9 m
The loudness, L, measured in decibels (Db), of a sound intensity, I, measured in watts per square meter, is defined as
1
L-10log where lo-10-12
and is the least intense sound a human ear can hear. What is the approximate loudness
of a dinner conversation with a sound intensity of 10-7?
[tex]\\ \rm\dashrightarrow B=10log\dfrac{I}{I_o}[/tex]
[tex]\\ \rm\dashrightarrow B=10\log\dfrac{10^{-7}}{10^{-12}}[/tex]
[tex]\\ \rm\dashrightarrow B=10log10^5[/tex]
[tex]\\ \rm\dashrightarrow B=10\times 5log10[/tex]
[tex]\\ \rm\dashrightarrow B=10(5)[/tex]
[tex]\\ \rm\dashrightarrow B=50dB[/tex]
Answer:
50 Db
Step-by-step explanation:
Given:
[tex]L=10 \log \dfrac{I}{I_0}[/tex]
[tex]I_0=10^{-12}[/tex]
where:
L is the loudness measured in decibels (Db)I is the sound intensity measured in w/m²To find the approximate loudness of a dinner conversation with a sound intensity of [tex]I=10^{-7}[/tex], substitute the values of I and I₀ into the given equation and solve for L:
[tex]\implies L=10 \log \dfrac{10^{-7}}{10^{-12}}[/tex]
[tex]\textsf{Apply exponent rule} \quad \dfrac{a^b}{a^c}=a^{b-c}:[/tex]
[tex]\implies L=10 \log 10^{-7-(-12)[/tex]
[tex]\implies L=10 \log 10^5[/tex]
[tex]\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax[/tex]
[tex]\implies L=5 \cdot 10 \log 10[/tex]
[tex]\implies L=50 \log 10[/tex]
As the given log does not have a specified base, assume the base is 10.
[tex]\implies L=50 \log_{10} 10[/tex]
[tex]\textsf{Apply log law}: \quad \log_aa=1[/tex]
[tex]\implies L=50 (1)[/tex]
[tex]\implies L=50\:\:\sf Db[/tex]
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A parabola, with its vertex at the origin, has a directrix at y = 3. which statements about the parabola are true? select two options.
The correct options about parabola are:
The focus is located at (0,–3).
The parabola can be represented by the equation x^2 = –12y.
According to the statement
we have given that the vertex is at the origin and directrix at y = 3.
and from these given information and we have to find the all abot the parabola like its focus point etc.
So,
We know that the equation of parabola is
(x-h)^2 = 4p(y-k)
Here The vertex is (h,k). and the focus is at (h,k+p). and the directrix is y(k - p.)
So, From the given information
Vertex at the origin means that h=0 and k=0
Directrix at y = 3 means that p=-3
Directrix at the y-axis means the parabola opens upwards.
Thus, the focus is: (0,-3)
And The p-value becomes
: 4(-3) = -12.
And from all these the equation of the parabola is becomes
(x)^2 = -12y
So, The correct options about parabola are:
The focus is located at (0,–3).
The parabola can be represented by the equation x^2 = –12y.
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Disclaimer: The question was incomplete. Please find the full content Below.
Question:
A parabola, with its vertex at the origin, has a directrix at y = 3. Which statements about the parabola are true? Select two options.
The focus is located at (0,–3).
The parabola opens to the left.
The p value can be determined by computing 4(3).
The parabola can be represented by the equation x2 = –12y.
The parabola can be represented by the equation y2 = 12x.
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A man mows his 100 ft by 200 ft rectangular lawn in a spiral pattern starting from the outside edge. After a bit of hard work he stops for a water break, he is 40.5% done. How wide of a strip has he mowed around the outside edge
Based on the length and width of the rectangular lawn, and the percentage the man has mowed, the strip width would be 40.5 m.
what is the width of the strip mowed already?first, find the area of the lawn:
= 100 x 200
= 20,000 ft²
the area mowed is:
= 40.5% x 20,000
= 8,100 ft²
the width is therefore:
= 8,100 / 200
= 40.5 m
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[tex]8x^3-4x+\frac{2}{3x}-\frac{1}{27x^3}[/tex]
The solution to the expression [tex]8x^3-4x+\frac{2}{3x}-\frac{1}{27x^3}[/tex] is [tex]\frac{216x^6 - 108x^4 + 18x^2 - 1}{27x^3}[/tex]
How to solve the expression?The expression is given as:
[tex]8x^3-4x+\frac{2}{3x}-\frac{1}{27x^3}[/tex]
Take the LCM of the expression
[tex]\frac{8x^3 * 27x^3 - 4x * 27x^3 + 2 * 9x^2 - 1}{27x^3}[/tex]
Evaluate the products
[tex]\frac{216x^6 - 108x^4 + 18x^2 - 1}{27x^3}[/tex]
Hence, the solution to the expression [tex]8x^3-4x+\frac{2}{3x}-\frac{1}{27x^3}[/tex] is [tex]\frac{216x^6 - 108x^4 + 18x^2 - 1}{27x^3}[/tex]
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find the value for the following
[tex]\sqrt{3-2\sqrt{2} }[/tex] = 0.06
(2x)ˣ = 25√5
How to solve an expression?[tex]\sqrt{3-2\sqrt{2} }[/tex] = [tex]\sqrt{3}-\sqrt{2\sqrt{2} }[/tex] = 1.73205080757 - 1.67428223427 = 0.06
4ˣ . 4ˣ⁻¹ = 24
4ˣ . 4ˣ / 4 = 24
4ˣ(1 - 1 / 4) = 24
4ˣ(3 / 4) = 24
multiply both sides by 4 / 3
4ˣ = 32
2²ˣ = 2⁵
2x = 5
x = 5 / 2 = 2.5
Therefore,
(2x)ˣ = 25√5
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Solve the following:
-5² + 10²
(2 × (-5) × 3) + 3³
Give your answer in simplest form.
The population of a bacteria culture are tripling. after 6 days, the population has quadrupled. what is the tripling time? find the answer to the nearest thousandth. b) how long did it take for the population to double in size?
The tripling time of population is 6 days and the population will be doubled in 3.5 days.
Given that the population of a bacteria is tripling in 6 days.
We have to find the tripling time of population and the time when the population will be doubled.
Suppose the population of bacteria in beginning be x.
The population of bacteria seems like arithmetic progression in which a =x, nth term =3x and n=6.
Aritmetic progression is a sequence which is having common difference.
nth term=a+(n-1)d
3x=x+(6-1)*d
3x-x=5d
2x=5d
d=2x/5
d=0.4x
So now we have to find the value of n where population will be 2x.
nth term=a+(n-1)d
put nth term=2x, a=x,d=0.4x
2x=x+(n-1)*0.4x
2x-x=(n-1)*0.4x
x/0.4x=n-1
2.5=n-1
n=2.5+1
n=3.5
Hence the population of bacteria willbe doubled after 3.5 days.
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Drag each tile to the correct box.
Graph the functions as transformations of f(x) = x^(2).
Arrange the parabolas with respect to the position of their vertices from left to right.
Answer:
Step-by-step explanation:
Let a and b be positive integers such that the product of all positive divisors of a equals the product of all positive divisors of b. Prove that a
The property A*B = (A' + A") * ( B'+B") is equal to each other.
According to the statement
We have given that the a and b is the positive integers and we have to prove that the product of all positive divisors of a equals the product of all positive divisors of b.
And in this statement
Now we use Distributive property
Here A and B is the positive integer and
A' and A" are the divisors of A and B' and B" are the divisors of B.
Now,
A = A' + A" and B = B'+B"
Now, Multiply both with each other then
A*B = (A' + A") * ( B'+B")
then
A*B = (A' B'+ A'B") + ( A"B'+A'B")
by this way it is equal to each other
Hence proved.
So, The property A*B = (A' + A") * ( B'+B") is equal to each other.
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please help me it is easy
I just dont understand
Answer:
13%
15%
41%
31%
Step-by-step explanation:
To find the percentage one amount is out of another amount, divide the smaller amount by the bigger amount. You will end up with a decimal (0.xy where x is the 10% place and y is the 1% place).
For example: to find the percentage that 39 is out of 300 students, we divide 39 by 300. This gives us the decimal 0.13. Written as a percentage, that is 13%.
We can do this for all of them.
Here is the math for each one:
39 ÷ 300 = 0.13 → 13%
45 ÷ 300 = 0.15 → 15%
123 ÷ 300 = 0.41 → 41%
93 ÷ 300 = 0.31 → 31%
Once you have all these decimals, you might choose to check your work by adding them up. They should at up to 1.00, or 100%. In this case, they do, so we know these answers are correct.
I hope this helps. Please let me know if you have questions.
Write the algebraic expression for the difference between the squares of two numbers.
The algebraic expression for the difference between the squares of the two numbers, supposing the two numbers to be a and b, is
a² - b² = (a + b)(a - b).
An algebraic expression is a combination of terms, where the terms are separated using mathematical operators like plus (+), minus (-), multiply (*), and divide (/).
The terms are combinations of numerals and variables.
Variables are represented alphanumerically, which can hold any value as per the expression they are used in.
In the question, we are asked to write the algebraic expression for the difference between the squares of two numbers.
We assume the two numbers to be a and b respectively.
Thus, we are asked to write an expression for the difference between the square of a and square of b, that is, we are asked to write an expression for a² - b².
We know that a² - b² is an identity, which can be shown as (a + b)(a - b).
Thus, the algebraic expression for the difference between the squares of the two numbers, supposing the two numbers to be a and b, is a² - b² = (a + b)(a - b).
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calculate 20% of rs 250000
Answer:
50,000
Step-by-step explanation:
20% can be rewritten as 0.2 times 250000: 0.2*250000 = 50,000
c) tan²0 - cot²0 = sec²0 (1- cot²0)
Prove that LHS = RHS So that LHS will be the sec²0 (1- cot²0)
Answer:
Maybe this could help?
Which is the best estimate of negative 14 and startfraction 1 over 9 endfraction (negative 2 and startfraction 9 over 10 endfraction)
The best estimation is 42.
EstimationFinding the most accurate estimate for:
[tex]-14\frac{1}{9}[/tex] x [tex]-2\frac{9}{10}[/tex]
To obtain, we round each integer to the next full number:
-14 x -3
Remember that adding two negatives produces positive results.
So, 14 x 3
Finally, this provides
42
Therefore, the final answer is 42.
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Answer: its 42
Step-by-step explanation:
Which of the following best describes the slope of the line below?
A. Positive
B. Zero
C. Negative
D. Undefined
Answer:
Zero
Step-by-step explanation:
Since the line is a horizontal line, the slope does not change. The slope is zero.
Answer:
Zero
Step-by-step explanation:
A zero slope is when the line is horizontal and when the slope equals to zero
I hope it helps! Have a great day!
bren~
Find the number of 4 digit numbers that contain at least 3 even digits
Answer:
2625 different numbers
Step-by-step explanation:
General outlineCombinatoricsSpecial circumstancesPartitioning the situationConclusionPart 1. CombinatoricsThis type of problem is classified as a "Combinatorics" problem: Counting up the number of ways something can happen.
Many times when attempting to count a large number of items, patterns are recognized and shortcuts to count are developed so that one doesn't need to physically count all of the items. Perhaps the trickiest part of counting with these shortcuts is to ensure that one does not over-count (counting a scenario more than once), although it is equally important that we don't under-count (fail to count a situation that applies).
Part 2. Special circumstancesIt should be noted that for a number to be a 4 digit number, it must have 4 digits, and thus the first digit must not be zero (despite that that would be an even digit, the number itself would not be a 4 digit number).
Part 3. Partitioning the situationThere are 2 main scenarios:
1. All 4 digits are even
2. Exactly 3 digits are even (meaning that exactly one digit is odd).
There are two sub-cases to Scenario 2:
2.1. The first digit is odd, and all three of the other digits are even.
2.2. The first digit is even, and exactly two of the other digits are even (meaning that exactly one of the last three digits is odd).
None of scenarios 1, 2.1, and 2.2 overlap, so we're not over-counting:
If all 4 digits are even, then there can't be exactly 3 even digits. If the first digit is odd, then not all 4 digits are even nor is the first digit even. If exactly two of the last three digits are even, then not all 4 digits are even, nor are all three of the last digits even.Further, this is all of the possibilities for a 4 digit number with least three even digits, so we're not under-counting.
Scenario 1 -- All 4 digits are even.
If all 4 digits are even, then the first digit has fours choices (2,4,6,8), and the next 3 digits each have 5 choices (2,4,6,8,0).
[tex]4*5*5*5=500\text{ choices}[/tex]
Scenario 2.1 -- The first digit is odd, and all three of the other digits are even
If the first digit is odd, there are 5 choices for the first digit (1,3,5,7,9), and the next 3 digits each have 5 choices (2,4,6,8,0).
[tex]5*5*5*5=625\text{ choices}[/tex]
Scenario 2.2 -- The first digit is even, and exactly two of the other digits are even
If the first digit is even, there are 4 choices for the first digit (2,4,6,8), and if exactly two of the next 3 digits are even, then there are 5 choices for each of the two even digits (2,4,6,8,0), and 5 choices for the odd digit (1,3,5,7,9), and there are "3 permuted by 2" ways of ordering those three digits.
[tex]4* \left [(5*5*5) * {}_3 \! P_2 \right ]=\\\\=4*\left [5*5*5 * \dfrac{3!}{2!} \right ]\\\\=4*\left [ 5*5*5 * \dfrac{3*2*1}{2*1} \right ] \\\\=4*\left [5*5*5 * 3 \right ] \\\\=1500\text{ choices}[/tex]
Scenario 2.2 broken down (calculated without using the permutation operation) -- The first digit is even, and exactly two of the other digits are even
Then either the second digit is odd (scenario 2.2.1), the third digit is odd (scenario 2.2.2), or the fourth digit is odd (scenario 2.2.3).
Scenario 2.2.1. The second digit is odd.
The first digit is even: 4 choices for the first digit (2,4,6,8)
The second digit is odd: 5 choices for the odd digit (1,3,5,7,9)
The third and fourth digits are even: 5 choices for each even digit (2,4,6,8,0).
[tex]4*5*5*5=500\text{ choices}[/tex]
Scenario 2.2.2. The third digit is odd.
The first digit is even: 4 choices for the first digit (2,4,6,8)
The second and fourth digits are even: 5 choices for each even digit (2,4,6,8,0).
The third digit is odd: 5 choices for the odd digit (1,3,5,7,9)
[tex]4*5*5*5=500\text{ choices}[/tex]
Scenario 2.2.3. The last digit is odd.
The first digit is even: 4 choices for the first digit (2,4,6,8)
The second and third digits are even: 5 choices for each even digit (2,4,6,8,0).
The last digit is odd: 5 choices for the odd digit (1,3,5,7,9)
[tex]4*5*5*5=500\text{ choices}[/tex]
Scenarios 2.2.1, 2.2.2, and 2.2.3 comprise all of Scenario 2.2, so [tex]500+500+500=1500\text{ choices}[/tex]
Part 4. ConclusionHow many ways can a 4 digit number be formed where at least 4 of the digits are even?
This is the sum of the choices from Scenario 1, Scenario 2.1, and Scenario 2.2, so [tex]500+625+1500=2625\text{ choices}[/tex].
Let [tex]sin\beta =\frac{2\sqrt[]{2} }{5} \\[/tex] and [tex]\frac{\pi }{2} \leq\beta \leq \pi \\[/tex].
Determine the exact value of [tex]sin(\frac{\beta }{2} )[/tex]
Since beta is in the first quadrant, the final answer will be positive.
To find cos(beta) so we can use the half angle identity, we can substitute into the Pythagorean identity. Doing so gives us that
[tex] \sin( \beta ) = \frac{ \sqrt{17} }{5} [/tex]
So, this means that
[tex] \sin( \frac{ \beta }{2} ) = \sqrt{ \frac{1 - \frac{ \sqrt{17} }{2} }{2} } = \sqrt{ \frac{2 - \sqrt{17} }{4} } = \frac{ \sqrt{2 - \sqrt{17} } }{2} [/tex]