There amount are 504 different groups of 3 side dishes that a customer can choose from.
There are 9 possible side dishes, and the customer can choose any 3 of them. The number of different groups of 3 side dishes can be determined using the formula:
nCr = n!/(r!(n-r)!).
In this case,
n = 9 and r = 3,
so the number of possible groups is
9!/(3!(9-3)!) = 504.
When a customer buys a family-sized meal at a certain restaurant, they get to choose 3 side dishes from 9 options. The number of different groups of 3 side dishes possible for the customer to choose from can be determined using the formula for combination: nCr = n!/(r!(n-r)!). In this case, n = 9 (the number of possible side dishes) and r = 3 (the number of side dishes chosen). Applying the formula, the number of possible groups of 3 side dishes is 9!/(3!(9-3)!) = 504. This means that there are 504 different groups of 3 side dishes that a customer can choose from.
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A trapezium ABCD is inscribed into a semi-circle of radius l so that the base AD of the trapezium is diameter and the vertices B and C lie on the circumference. Then the value of base angle θ (in degree) of the trapezium ABCD which has the greatest perimeter, is
The value of base angle θ (in degree) of the trapezium ABCD which has the greatest perimeter, is 60°
Given, AD = 2I
From ΔABD,
cos∅=[tex]\frac{AB}{AD}[/tex]
⇒ AB = AD cos∅
⇒ AB = 2I cos∅
and x = ABcos∅ = 2I [tex]cos^{2}[/tex]∅
Now perimeter,
P = 2I+ 2I - 2x + 2AB
P = 4I - 4I [tex]cos^{2}[/tex]∅ + 4Icos∅
P = 4I (1 - cos²∅ + cos∅)
Differentiating with respect to ∅, we get
dP/d∅= 4I(2 cos∅ sin∅ - sin∅)
For max/min,
dP/d∅ = O
2 cos∅ sin∅ - sin∅ = 0
⇒ cos∅ = [tex]\frac{1}{2}[/tex] (∅ [tex]\neq[/tex] 0)
= 60°
[tex]d^{2}[/tex]P/d∅ = 4i(2 cos2∅ - cos∅)
[tex]d^{2}[/tex]P/d∅ < 0
∅ = 60°
Therefore, Perimeter is maximum when ∅ = 60°
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Find a vector (u ) with magnitude 4 in the opposite direction as v =⟨−2,3⟩ Give EXACT answer. You do NOT have to simplify your radicals!
u =
A vector (u) with magnitude 4 in the opposite direction as v =⟨−2,3⟩ is u = ⟨-2/√13, 3/√13⟩.
A vector in the opposite direction of a given vector can be found by multiplying that vector by -1. The magnitude of a vector is the length of the vector and it can be found by using the Pythagorean theorem on the coordinates of the vector.
The magnitude of the vector v = ⟨-2,3⟩ is:
|v| = √(-2)^2 + 3^2 = √4 + 9 = √13
To find a vector with magnitude 4 in the opposite direction as v, we can first normalize the vector v by dividing it by its magnitude,
then multiply it by 4:
u = (-1) * (1/|v|) * v
= (-1) * (1/√13) * ⟨-2,3⟩
= ⟨2/√13, -3/√13⟩
So, a vector (u) with magnitude 4 in the opposite direction as v =⟨−2,3⟩ is u = ⟨2/√13, -3/√13⟩
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What is the algebraic expression that represents the perimeter of the figure below?
A. 3x+31
b. 3x+43
c. 4x+31
d.4x+43
Find the 10th term of the sequence below
Answer:
10th term = 94
Step-by-step explanation:
Given equation,
→ Tn = n² - n + 4
Now the 10th term will be,
→ Tn = n² - n + 4
→ T10 = (10)² - 10 + 4
→ T10 = 100 - 6
→ [ T10 = 94 ]
Hence, the answer is 94.
given vector u equals open angled bracket negative 10 comma negative 3 close angled bracket and vector v equals open angled bracket 4 comma 8 close angled bracket comma what is projvu? open angled bracket negative 640 over 109 comma negative 192 over 169 close angled bracket open angled bracket negative 256 over 109 comma negative 512 over 109 close angled bracket open angled bracket negative 8 comma negative 3 close angled bracket open angled bracket negative 16 over 5 comma negative 32 over 5 close angled bracket
The correct answer is open angled bracket negative 16 over 5 comma negative 32 over 5 close angled bracket, as it is the vector u.
The projection of vector u onto vector v is given by the formula:
projv(u) = (u . v/||v||^2) * v
where u . v is the dot product of vectors u and v,and ||v|| is the magnitude of vector v
Given that u = <-10, -3> and v = <4, 8>, we can substitute these values into the formula and find projv(u)
projv(u) = ( (-10)(4) + (-3)(8) )/ (4^2 + 8^2) * <4,8>
projv(u) = ((-40 - 24)/ (16 + 64))* <4,8>
projv(u) = (-64/80) * <4,8>
projv(u) = (-4/5) * <4,8>
projv(u) = <-16/5,-32/5>
The vector u is already parallel to the vector v, that's why the projection of vector u on vector v is equal to vector u itself.
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Caitlyn needs to mail a USB drive to a friend. She uses 42 -cent stamps and 5 -cent stamps to pay $1.71 in postage. How many of each stamp did Caitlyn use?
Solving an equation we will see that Caitlyn used 3 of the 42-cent stamps and 9 of the 5-cent ones.
How many of each stamp does she use?Let's define the variables that we will be using:
x = number of 42-cent stapms.y = number of 5-cent stamps.Then the total value of the stamps, in dollars, is:
x*0.42 + y*0.5
And we know that the value must be $1.71, then we can write the equation:
x*0.42 + y*0.05 = 1.71
y = (1.71 - x*0.42)/0.05
To find the values of x and y, we can just evaluate in different whole values of x, until we get a whole value of y.
using x = 1 we get:
y = (1.71 - 0.42)/0.05 = 2.58
This is not a solution.
if x = 3
y = (1.71 - 0.42*3)/0.05 = 9
This is the solution, she used 3 of the 42-cent and 9 of the 5-cent.
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Solve the differential equation dy/dx=x/(25*y)1. Find an implicit solution and put your answer in the following form: = constant.2. Find the equation of the solution through the point (x,y)=(-5,1).3. Find the equation of the solution through the point (x,y)=(0,-6). Your answer should be of the form x=f(y) or y=f(x), whichever is appropriate.4. Find the equation of the solution through the point (x,y)=(6,0). Your answer should be of the form x=f(y) or y=f(x), whichever is appropriate.
The implicit solution of the differential equation dy/dx=x/(25*y) is y = (3(25x^2/2 + c))^(1/3) = constant. The equation of the solution through the points (x,y)=(-5,1), (x,y)=(0,-6), and (x,y)=(6,0) are y = (3(25x^2/2 - 313.75))^(1/3), y = (3(25x^2/2 - 1875))^(1/3), and y = (3(25x^2/2 - 2025))^(1/3) respectively.
1. 25y^2dx = xdy → y^2 dy = 25x dx → ∫y^2 dy = ∫25x dx → y^3/3 = 25x^2/2 + c → y = (3(25x^2/2 + c))^(1/3) = constant
2. Substituting x=-5 and y=1 in the implicit solution, we get (3(25(-5)^2/2 + c))^(1/3) = 1 → c = -313.75 → y = (3(25x^2/2 - 313.75))^(1/3)
3. Substituting x=0 and y=-6 in the implicit solution, we get (3(25(0)^2/2 - 313.75))^(1/3) = -6 → c = -1875 → y = (3(25x^2/2 - 1875))^(1/3)
4. Substituting x=6 and y=0 in the implicit solution, we get (3(25(6)^2/2 - 1875))^(1/3) = 0 → c = -2025 → y = (3(25x^2/2 - 2025))^(1/3)
The implicit solution of the differential equation dy/dx=x/(25*y) is y = (3(25x^2/2 + c))^(1/3) = constant. The equation of the solution through the points (x,y)=(-5,1), (x,y)=(0,-6), and (x,y)=(6,0) are y = (3(25x^2/2 - 313.75))^(1/3), y = (3(25x^2/2 - 1875))^(1/3), and y = (3(25x^2/2 - 2025))^(1/3) respectively.
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The pre-image, quadrilateral ABCD was dilated to produce the image, quadrilateral A'B'C'D'. The length of the side between the vertices A and B is 5 inches. What is the length of the side between the vertices A' and B', when vertices B'C' is 9 inches and vertices BC is 12 inches?
The length of the side between the vertices A' and B' is 3.75 inches
How to find the length of the side between the vertices A' and B'?In order to find the length of the side between the vertices A' and B', we need to use the fact that the ratio of corresponding side lengths between the pre-image and the image is the same as the ratio of the scale factor used in the dilation.
The ratio of the side length between the vertices B'C' and BC is 9/12. If we call this ratio k, then the ratio of the side length between the vertices A'B' and AB is also k.
So we can use the side length between AB = 5 inches and k = 9/12 to find the side length between A'B'
A'B' /AB = k
A'B' = k × AB
A'B' = (9/12) × 5
A'B' = 3.75 inches
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The acceleration function (in m/s2) and the initial velocity are given for a particle moving along a line.
a(t) = t + 4, v(0) = 3, 0 ≤ t ≤ 11(a) Find the velocity at time t. 3 m/s (b) Find the distance traveled during the given time interval.
The distance traveled during the given time interval be 2024 m.
What is meant by acceleration function?An object's rate of change in velocity is referred to as its acceleration. The rate of change in velocity is known as acceleration. Acceleration typically signals a change in speed, though this is not always the case. Because of the shifting direction of its velocity, an item moving on a circular path at a constant speed is still accelerating.
By integrating the velocity v in terms of time t, v = gt, we can determine acceleration, which is the rate of change of velocity with respect to time.
Let the function be a(t) = t + 4
time = 11
substitute the value of t in the above equation, we get
a(11) = 11 + 6 = 17 m/s²
a = v/t, so v = at
finally v = (17 m/s²) (11s)
v = 187 m/s
b) Find the distance traveled during the given time interval
[tex]$$V_{total} = V_{final} - V_{ initial[/tex]
V final = 187 m/s, Vinitial = 3 m/s
V total = 187 - 3 = 184 m/s
Let the equation of velocity be
Velocity = distance/time
distance = velocity × time
distance = 184 m/s × 11 s
distance = 2024 m
Therefore, the distance traveled during the given time interval be 2024 m.
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In an interval whose length is z seconds, a body moves (32z + 2z^ 2 ) ft. What is the average speed v of the body in this interval?
The average speed of the body in the given interval is 32 + 2z ft/s.
The average speed v of a body in an interval of z seconds can be calculated using the formula v = d/t, where d is the distance traveled and t is the time taken.
In this case, the distance d is (32z + 2z^ 2 ) ft and the time taken is z seconds.
Therefore, the average speed v of the body in this interval is:
v = (32z + 2z^ 2 ) ft / zs
v = 32 + 2z ft/s
Therefore, the average speed of the body in the given interval is 32 + 2z ft/s.
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solve the following equation for g:
m+n^2 = g5M
The solution to the equation m+[tex]n^{2}[/tex] = g5M is g = m[tex]n^{2}[/tex]/5M, which can be solved by dividing both sides of the equation by 5M and rearranging the equation to isolate the g on the right side.
What is an equation?A mathematical declaration that two expressions are equal is known as an equation. The equal symbol (=) is used to denote the separation of two phrases.
In the given question, we must isolate the g on one side of the equation in order to solve this equation for g. We may start by multiplying both sides of the equation by 5M to accomplish this. This will result in the left side of the equation having m/5M + ([tex]n^{2}[/tex]/5M). It will only provide us with g on the right side.
The terms on the left side of the equation can then be combined by multiplying them together. This will result in the equation shown below: (m/5M)([tex]n^{2}[/tex]/5M) = g.
The fractions can then be eliminated and the denominators removed by multiplying both sides of the equation by 5M. This will result in the formation of 5Mg on the right side and mn2/5M on the left.
We can then rearrange the equation to isolate the g on the right side, giving us: m[tex]n^{2}[/tex] = 5Mg. Finally, we can divide both sides of the equation by 5M to get g by itself on the right side, giving us the final equation: g = m[tex]n^{2}[/tex]/5M.
Therefore, the solution to the equation m+[tex]n^{2}[/tex] = g5M is g = m[tex]n^{2}[/tex]/5M.
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Interpreting Polynomials
5x+9
A. 5
B. 9
C.
O
D. 1
What is the degree of the
polynomial?
Choice A
O Choice B
Choice C
O Choice D
Answer:
Give the degree of the polynomial.
6x5+5x4−3x2+x7
Answers:
18
5
6
9
7
Hope it helps :)
Based on the information provided, what are the constraints on the mining problem? Let x represent the bins of Metalite and y represent the bins of Helotite.
The constraints on the mining problem are 8.3x + 7.2y = 180
and 4x + 3y = 24
How to determine the constraints on the mining problem?From the question, we have the following parameters that can be used in our computation:
x represent the bins of Metalitey represent the bins of Helotite.This means that
Weight: 8.3x + 7.2y
Number of bins: 4x + 3y
From the question, we have
Capacity = 180 tons ot 24 bins
So, the constraints are
8.3x + 7.2y = 180
4x + 3y = 24
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Divide the following and then check by multiplying.
7)84
Select the correct choice below and, if necessary, fill in the answer box to complete your choice
OA. The quotient does not have a remainder. The quotient is
B. The quotient has a remainder not equal to 0. The quotient is
OC. The quotient is undefined.
When dividing 84 by 7 then the quotient does not have a reminder and the quotient will be 12.
What is division?
A division in mathematics is the process of dividing a specific amount into equal parts. The division is the inverse of multiplication.
We can find a division fact if we know a multiplication fact: For instance, 3 5 = 15, so 15 / 5 = 3.
Also 15 / 3 = 5.
Here given that,
We have to divide 87/7,
By dividing,
we know that 8/7 gives reminder 1, and a quotient of 1.
Then divide 14 by 7,
which gives reminder 0 and quotient 2.
So by dividing 84 / 7 we get reminder 0 with a quotient of 12.
Therefore the correct option is option A. With a quotient of 12.
Hence, When dividing 84 by 7 then the quotient does not have a reminder and the quotient will be 12.
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A school is arranging a field trip to the zoo. The school spends 885.69 dollars on passes for 37 students and 2 teachers. The school also spends 325.23 dollars on lunch for just the students. How much money was spent on a pass and lunch for each student?
The $31.309 money was spent on a pass and lunch for each student
What is Equation?Two or more expressions with an Equal sign is called as Equation.
School spends on a pass for each student,885.69/39
$22.97
School spends on a lunch for each student,
325.23/39
$8.339
Therefore, total money spent on a pass and lunch for each student
= 22.97 + 8.339
= $31.309
Hence, the $31.309 money was spent on a pass and lunch for each student
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Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle.
The area of the region enclosed by the given curves is 32 units2.
The region enclosed by the given curves is bounded by the equation y = x2 and y = 4. The region is a parabolic region and needs to be integrated with respect to y. A typical approximating rectangle for this region is shown in the diagram below.
The area of this region can be expressed as the integral:
A = ∫y=x2→4 y dx
This integral can be evaluated by splitting the integral into two parts: A = ∫y=x2→4 x2 dx + ∫y=x2→4 4dx
After evaluating the integrals, the area of the region is:
A = (1/3)x3 + 4x |y=x2→4
A = (1/3)(42) + 4(4 - 22)
A = 32
Therefore, the area of the region enclosed by the given curves is 32 units2.
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2/3 < 5/12
true or false
Answer:
False
Step-by-step explanation:
Converting
[tex] converting \: \frac{2}{3} \: to \: decimal \\ \frac{2}{3 } = 0.6666666 .....[/tex]
[tex]converting \: \frac{5}{12} \: to \: decimal \\ \frac{5}{12} = 0.4166666....[/tex]
[tex]therefore \: \frac{2}{3} > \: \frac{5}{12} \\ because \: 0.66666... > 041666...[/tex]
find the indicated probability round to three decimal places
The probability of randomly selecting a blue ball is 3/14, or 0.214 rounded to three decimal places.
Formula: P(blue ball) = n(blue ball)/n(total balls)
P(blue ball) = 3/14 = 0.214
The probability of randomly selecting a blue ball is 0.214. To calculate this probability, the formula P(blue ball) = n(blue ball)/n(total balls) was used. This formula is used to find the probability of an event occurring, in this case randomly selecting a blue ball. The numerator of the equation is the number of blue balls in the bag, which is 3, while the denominator is the total number of balls in the bag, which is 14. Therefore, the probability of randomly selecting a blue ball is 3/14, or 0.214 rounded to three decimal places.
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Complete question:A bag contains 6 yellow, 3 blue, and 5 red balls. Find the probability of randomly selecting a blue ball.
Three softball players discussed their batting averages after a game.
Probability
Player 1 six tenths
Player 2 five ninths
Player 3 four sevenths
Compare the probabilities and interpret the likelihood. Which statement is true?
.
.
A. Player 1 is more likely to hit the ball than Player 2 because P(Player 1) > P(Player 2)
B. Player 2 is more likely to hit the ball than Player 1 because P(Player 2) > P(Player 1)
C. Player 3 is more likely to hit the ball than Player 1 because P(Player 3) > P(Player 1)
D. Player 2 is more likely to hit the ball than Player 3 because P(Player 2) > P(Player 3)
Comparibg the probabilities and interpreting the likelihood, the true statement is that:
A. Player 1 is more likely to hit the ball than Player 2 because P(Player 1) > P(Player 2)
How to calculate the probability?Probability simply means the chance that a particular thing or event will happen. It is the occurence of likely events. It is simply the area of mathematics that deals with the numerical estimates of the chance that an event will occur.
Player 1 = six tenths = 0.6
Player 2 = five ninths = 0.56
Player 3 four sevenths = 0.57
This shows that player 2 has the highest probability. The correct option is A.
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How do you find a arc length without a radius? Need a answer asap.
the length of the arc will be C/2π*Ф.
What is circle?
The measurement of the circle's boundaries is called as the circumference or perimeter of the circle. whereas the circumference of a circle determines the space it occupies. The circumference of a circle is its length when it is opened up and drawn as a straight line. Units like cm or unit m are typically used to measure it. The circle's radius is considered while calculating the circumference of the circle using the formula. As a result, in order to calculate the circle's perimeter, we must know the radius or diameter value.
The radius of a circle and its centre angle both affect how long an arc is. We are aware that the arc length and circumference are equivalent at an angle of 360 degrees (2). Due to the continuous relationship between angle and arc length, we can therefore state that:
L/Ф = C/2π
L = C/2π*Ф
Hence the length of the arc will be C/2π*Ф.
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Evaluate the integral below by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using geometry.-5 ∫5 sqrt(5^2−x^2) dx my lower limit is -5 and my upper is 5. How do I solve this problem? do i graph this equation and use left or right approximation?
Answer:
[tex]\frac{25\pi}{2}\approx39.27\text{ units}^2[/tex]
Step-by-step explanation:
I assume your problem is [tex]\displaystyle \int\limits^{5}_{-5} {\sqrt{5^2-x^2}} \, dx[/tex]. Recognize that [tex]\sqrt{5^2-x^2}=\sqrt{25-x^2}[/tex] is a semi-circle directly above the x-axis with a radius of 5. The bounds cover the whole area of the semicircle, so its area is just half the area of a circle with a radius of 5. You probably know that the area of a circle is [tex]A=\pi r^2[/tex], so the area of a semicircle would be [tex]A=\frac{\pi r^2}{2}[/tex]. Thus, the area using geometry is [tex]A=\frac{\pi (5)^2}{2}=\frac{25\pi}{2}\approx39.27\text{ units}^2[/tex].
Consider an election with 521 votes
If there are 8 candidates, what is the smallest number of first-place votes a candidate could win with under the Plurality method?
4x-3y<9 x-3y>6 .... how does it look on a graph
A graph of the solution to this system of inequalities on a coordinate plane is shown in the image attached below.
How to graph the solution to this system of inequalities?In order to to graph the solution to the given system of inequalities on a coordinate plane, we would use an online graphing calculator to plot the given system of inequalities and then take note of the point of intersection;
4x - 3 < 9
x - 3y > 6
Based on the graph (see attachment), we can logically deduce that the solution to the given system of inequalities is the shaded region and the point of intersection of the lines on the graph representing each, which is given by the ordered pair (3, -1).
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PLEASE HELPPPP
An equilateral triangle has sides of length (3x+7). Find the perimeter of triangle.
Answer:
9x + 21
Step-by-step explanation:
An equilateral triangle is a triangle with three equal sides. If we know the length of one side, we can find the perimeter by multiplying by three.
3(3x + 7) = 9x + 21
What number should be added to −96 to get a sum of 9 ?
The number 105 should be added to - 96 to get a sum of 9.
What is Addition?The process of combining two or more numbers is called the Addition. The 4 main properties of addition are commutative, associative, distributive, and additive identity.
Now,
Let the value of number = x
So, We can formulate;
⇒ x + (-96) = 9
Solve for x;
⇒ x - 96 = 9
Add 96 both side,
⇒ x - 96 + 96 = 9 + 96
⇒ x = 105
Thus, The number 105 should be added to - 96 to get a sum of 9.
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Copy and fill in the chart.
Answer:
Step-by-step explanation:
Find the length and width of the actual room, shown in the scale drawing. Then find the area of the actual room. Round the final answer, of necessary, to the nearest tenth.
4.5 in
⬛️ 3 in
5 in:8 ft
The length is about _____ feet, the width is about _____ feet, and the area is about ______ square feet.
The length is about 7.2 feet, the width is about 4.8 feet, and the area is about 34.56 square feet.
What is Area?The area is the entire amount of space occupied by a flat (2-D) surface or an object's form. On a sheet of paper, draw a square using a pencil. It has two dimensions. The area of a form on paper is the area that it occupies.
Given that, The length and width of the model of the room are 4.5 in and 3 inches respectively.
Since Scale is 5 inches = 8ft
Thus the actual length of the room = 4.5*8/5 = 7.2 feet
the actual width of the room = 3*8 /5 = 4.8 feet
Since,
Area = length * width
Area of the room = 7.2 * 4.8
Area of the room = 34.56 square feet
Therefore, The dimensions are roughly 7.2 feet long, 4.8 feet wide, and 34.56 square feet in area.
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Identify a pattern in this list of numbers. Then use this pattern to find the next number. (More than one pattern might exist, so it is possible that there is more than one correct answer. (-8,-13),(4 1/2, -1/2), (8,3), (1,-4) , (0, ) Find the missing number.
The pattern in this sequence is given as follows:
When x increases by one, y increases by one.
Hence the missing number y, for the point (0,y), is given as follows:
y = -5.
How to define a linear function?The slope-intercept definition of a linear function is given as follows:
y = mx + b.
For which the parameters are given as follows:
m is the slope, representing the rate of change.b is the intercept, representing the value of y when x = 0.The pattern of the sequence is that when x increases by one, y also increases by one, hence the slope m is given as follows:
m = 1/1 = 1.
Hence:
y = x + b.
When x = -8, y = -13, hence the intercept b, which is also the missing value, is given as follows:
-13 = -8 + b
b = -5.
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find the equations of the lines that pass through the point (5,6) and are parallel to and perpendicular to the line with equation y =4x
The equation of this parallel line is y = 4x - 22. Railroad tracks, sidewalk edges, ladder rails, endless rail tracks, opposing sides of a ruler, opposite edges of a pen, an eraser, and other real-world objects are instances of parallel lines.
What is meant by parallel line?In a plane, lines that are consistently spaced apart from one another are referred to as parallel. Parallel lines don't ever intersect. Perpendicular lines are those that intersect at a 90-degree angle.Parallel lines are two lines that are never crossed and are equally spaced apart from one another.They could be horizontal or vertical. Parallel lines can be found in our daily life in the form of nearby train tracks, rows of notebooks, and zebra crossings.Since there is no "b" in the conventional equation y = mx + b, the line y = 4x has a slope of 4 and an intercept of 0. Any parallel line will have the same slope as the slope (m = 4) in this case. The slope of all parallel lines will be the same.
So let's return to the conventional format and replace the m with a "4".
y = 4x + b
Now that the values have been entered into the equation, let's solve for "b" using the point that sits on the line.
-6 = 4(4) + b
-6 - 16 = b
b = -22
The equation of this parallel line is y = 4x - 22
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A population proportion is 0.30. A random sample of size 150 will be taken and the sample proportion p will be used to estimate
the population proportion. Use the z-table.
Round your answers to four decimal places.
a. What is the probability that the sample proportion will be within ±0.03 of the population proportion?
b. What is the probability that the sample proportion will be within ±0.08 of the population proportion?
In both cases, the sample proportion is very likely to be within a certain range of the population proportion, with a probability of 1.0000 or 100%.
What is probability?
Probability is a measure of the likelihood that an event will occur, it is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
In this case, we are interested in the probability that the sample proportion (p) will be within a certain range of the population proportion (0.30).
a. To find the probability that the sample proportion will be within ±0.03 of the population proportion, we can use the standard normal distribution (z-table). The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
The formula for the standard normal distribution is:
z = (p - 0.30) / (standard deviation of p)
The standard deviation of p is given by the formula:
(population proportion * (1 - population proportion)) / sample size
In this case, we have:
(0.30 * (1 - 0.30)) / 150 = 0.0006
So, the standard deviation of p is 0.0006
The probability that the sample proportion will be within ±0.03 of the population proportion is the same as the probability that the sample proportion will be between 0.27 and 0.33.
Therefore, we can calculate the z-score for the lower and upper bounds of the range:
z1 = (0.27 - 0.30) / 0.0006 = -5
z2 = (0.33 - 0.30) / 0.0006 = 5
Using the z-table, we can find the probability that a z-score falls between -5 and 5.
The probability that the sample proportion will be within ±0.03 of the population proportion is:
P(z1 <= z <= z2) = P(-5 <= z <= 5) = 1 - 0.0000 = 1.0000
b. To find the probability that the sample proportion will be within ±0.08 of the population proportion, we can use the same formula as before. The probability that the sample proportion will be within ±0.08 of the population proportion is the same as the probability that the sample proportion will be between 0.22 and 0.38.
Therefore, we can calculate the z-score for the lower and upper bounds of the range:
z1 = (0.22 - 0.30) / 0.0006 = -10
z2 = (0.38 - 0.30) / 0.0006 = 10
Using the z-table, we can find the probability that a z-score falls between -10 and 10.
The probability that the sample proportion will be within ±0.08 of the population proportion is:
P(z1 <= z <= z2) = P(-10 <= z <= 10) = 1 - 0.0000 = 1.0000
Hence, In both cases, the sample proportion is very likely to be within a certain range of the population proportion, with a probability of 1.0000 or 100%.
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