Upon solving the given system of equations:
[tex]x(t) = (c_1 + e^{-t}) + (3/2) * (c_2 + e^{-t}) * cos(2t) + (1/2) * (c_1 + e^{-t}) * sin(2t),\\y(t) = (c_1 + e^{-t}) + (3/2) * (c_2 + e^{-t}) * sin(2t) - (1/4) * (c_1 + e^{-t}) * cos(2t)[/tex]
To solve the system of equations:
x' = 2x - 3y + 2sin(2t)
y' = x - 2y - 2cos(2t)
We can use the method of undetermined coefficients to find the particular solution. Assuming the particular solution takes the form:
[tex]x_p(t) = A sin(2t) + B cos(2t)\\y_p(t) = C sin(2t) + D cos(2t)[/tex]
Substituting these expressions into the original equations, we get:
2(A sin(2t) + B cos(2t)) - 3(C sin(2t) + D cos(2t)) + 2sin(2t) = 2sin(2t)
(A sin(2t) + B cos(2t)) - 2(C sin(2t) + D cos(2t)) - 2cos(2t) = cos(2t)
(2A - 3C + 2)sin(2t) + (2B - 3D)cos(2t) = 2sin(2t)
(A - 2C)sin(2t) + (B - 2D - 2)cos(2t) = cos(2t)
By comparing the coefficients of sine and cosine on both sides, we can equate them separately:
2A - 3C + 2 = 2
2B - 3D = 0
A - 2C = 0
B - 2D - 2 = 1
Solving these equations, we find:
A = 1
B = 3/2
C = 1/2
D = -1/4
So the particular solution is:
[tex]x_p(t)[/tex] = sin(2t) + (3/2)cos(2t)
[tex]y_p(t)[/tex] = (1/2)sin(2t) - (1/4)cos(2t)
To find the complementary solution, we solve the homogeneous system:
x' = 2x - 3y
y' = x - 2y
We can rewrite this system as a matrix equation:
X' = AX
where [tex]X = [x, y]^T[/tex] and
[tex]A = \left[\begin{array}{ccc}2&-3\\1&-2\end{array}\right][/tex]
The characteristic equation is:
det(A - λI) = 0, where I is the identity matrix. Solving this equation, we find the eigenvalues:
[tex]\lambda_1 = -1\\\lambda_2 = -1[/tex]
For each eigenvalue, we solve the corresponding eigenvector equation:
(A - λI)V = 0
For [tex]\lambda_1 = -1[/tex], we have:
[tex]\left[\begin{array}{ccc}3&-3\\1&-1\end{array}\right] * V_1 = 0[/tex]
Solving this system, we find the eigenvector:
[tex]V_1 = [1\ \ 1][/tex]
For [tex]\lambda_2 = -1[/tex], we have:
[tex]\left[\begin{array}{ccc}3&-3\\1&-1\end{array}\right] * V_2= 0[/tex]
Solving this system, we find the eigenvector:
[tex]V_2 = [3\ \ 1][/tex]
So the complementary solution is:
[tex]x_c(t) = c_1 * e^{-t} * [1\ \ 1]^T + c_2 * e^{-t} * [3\ \ 1]^T\\y_c(t) = c_1 * e^{-t} * [1\ \1]^T + c_2 * e^{-t} * [3\ \ 1]^T[/tex]
where
[tex]c_1\ and\ c_2[/tex] are arbitrary constants.
The general solution is the sum of the particular and complementary solutions:
[tex]x(t) = x_p(t) + x_c(t)\\y(t) = y_p(t) + y_c(t)[/tex]
Simplifying and combining terms, we get:
[tex]x(t) = (c_1 + e^{-t}) + (3/2) * (c_2 + e^{-t}) * cos(2t) + (1/2) * (c_1 + e^{-t}) * sin(2t)\\y(t) = (c_1 + e^{-t}) + (3/2) * (c_2 + e^{-t}) * sin(2t) - (1/4) * (c_1 + e^{-t}) * cos(2t)[/tex]
where [tex]c_1\ and\ c_2[/tex] are arbitrary constants.
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Please help me with this question
Answer:
100
Step-by-step explanation:
130 - 30 = 100
Which golf ball went higher, and how many feet? (Desmos!) - Just added the answer choices!
Answer:
Max height: 64 feet, and the socond one was higher.
Step-by-step explanation:
The max height is the y value of the vertex, because that’s when the graph peaks.
we can already very clearly see the vertex on the graph, so we don’t need to calculate it.
the max height of the second golf ball is 64 feet.
Now let’s look at the max height on the first golf ball.
we get the equation
h=-16t squared + 48t
to find the vertex of this, we can use the formula -b/2a
-48/-32 = 1.5
1.5 is the t value of this vertex.
to find the h value, we plug it in.
h = -16 (1.5) squared + 48(1.5)
h =2.25 times -16 + 72
h = -36 +72
h = 36
the first one is 36 max height, and the second is 64. The second one is bigger.
A triangle has lengths of 3 cm, 5 cm, and 9 cm, can it form 1 or more triangles?
Answer:Yes
Step-by-step explanation:
Determine if the triangles are similar. If yes, state how (by AA~ SSS~, or SAS~) and complete the similarity statement
Answer:
Similar by: SAS
∠LMQ = ∠PMKLM/PM = QM/KMΔPKM ~ ΔLQM
Yes, the triangles PMK and LMQ are similar by SAS similarity statement.
What is Triangle?A triangle is a three-sided polygon that consists of three edges and three vertices.
Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion .
The triangles PKM and triangle LMQ are the similar triangles.
If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then the two triangles are congruent
Both the triangles have a common angle M and two sides are proportional
∠LMQ = ∠PMK
The sides are proportional
LM/PM = QM/KM
The triangles are similar
ΔPKM ~ ΔLQM
Hence, Yes the triangles PMK and LMQ are similar by SAS similarity statement.
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What are the solutions of the system? y = -6x – 6 y = x2 – 5x – 6
Answer:
Answer is 0
Step-by-step explanation:
-6x-6=x²-5x-6
-6(x+1)=(x+1)(x-6)
-6=x-6
x=0
The area of the shaded sector is 5 pi square meters. What is the area of the entire circle? Express your answer in terms of Pi.
A circle. The shaded section has an angle measure of 100 degrees.
Recall that StartFraction Area of sector over area of circle EndFraction = StartFraction n degrees over 360 degrees EndFraction.
A) 12 pi
B) 14 pi
C) 18 pi
D) 20 pi
Answer:
C - 18 pi
Step-by-step explanation:
edge
Answer:
5/18
108
A sector has an area of 30π in.2. The radii containing the sector form an angle of 100°. What is the area of the circle?
The ratio of the angle of the sector to the entire circle is
✔ 5/18
.
Area of the sector = StartFraction n degrees over 360 degrees EndFraction (pi) (r squared). 30 pi = StartFraction 100 degrees over 360 degrees EndFraction (pi) (r squared). (StartFraction 18 over 5 EndFraction) 30 pi = (pi) r squared.
The area of the entire circle is
✔ 108
Pi in.2
5k + 2 = 6
What is this
What is the interest for a $6,700 loan at 13.5 percent for 5 years?
$154.17
$2,550.20
$9,045.00
$9,250.20
Answer:
$2,550.20
Step-by-step explanation:
just took the test and got it right!
The profit function in dollars, is given by P(x)= -0.02x² + 44x - 1750, where x is the number of wireless headphones sold. (a) How many headphones must be sold in order to maximize profit? (b) What is the maximum profit?
To determine the number of headphones that must be sold to maximize profit and the maximum profit, we can analyze the profit function P(x) = -0.02x² + 44x - 1750. The number of headphones sold to maximize profit is 1100, and the maximum profit is $17,050.
(a) To find the number of headphones that maximize profit, we need to identify the x-value at which the profit function reaches its maximum. The maximum point occurs at the vertex of the quadratic function. The x-coordinate of the vertex can be found using the formula x = -b / (2a), wherea and b are the coefficients of the quadratic function. In this case, a = -0.02 and b = 44. Plugging these values into the formula, we find x = -44 / (2 * -0.02) = 1100. Therefore, 1100 headphones must be sold to maximize profit.
(b) To calculate the maximum profit, we substitute the value of x = 1100 into the profit function P(x). P(1100) = -0.02(1100)² + 44(1100) - 1750 = -24200 + 48400 - 1750 = 17050. Hence, the maximum profit is $17,050.
In conclusion, in order to maximize profit, 1100 headphones must be sold, resulting in a maximum profit of $17,050.
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} .println(); } what is printed as a result of executing this code segment? a e i
The code segment will print the characters 'a', 'e', and 'i' on separate lines.
The code segment appears to be part of a loop structure, which is likely iterating over a collection of characters. Each character is printed on a new line using the '.println()' function. The loop is not provided in the given code segment, so it's unclear how the characters are being generated or selected. However, assuming that the loop iterates over the characters 'a', 'e', and 'i', the output will be as follows:
a
e
i
The code uses the '.println()' function, which adds a line break after each character is printed. As a result, each character will be displayed on a separate line. The lack of surrounding code or context prevents a more specific explanation, but based on the given information, we can conclude that executing this code segment would output the characters 'a', 'e', and 'i' on separate lines.
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2-13^3-10(23+21) = X
Answer:
x= -2635
Step-by-step explanation:
2 - 2197 - 10 x 44 = x
2- 2197 - 440 = x
-2635 = x
Answer:
x= -2635
Step-by-step explanation:
2 - 2197 - 10 x 44 = x
2- 2197 - 440 = x
-2635 = x
if (x+2)^2=49, then x + 2 = 7. True or False
Need help on this one too
Answer:
Step-by-step explanation:
Use the Pythagorean Theorem. The hypotenuse, 14 square should be equal to the sum of x squared + 10 squared.
14^2 = 196
10^2 = 100.
So, 196 = 100 + x^2
[tex]x = \sqrt{96}[/tex]
Mark each! Of the following equations as true or false. Explain or showing your reasoning!
Answer:
False
True
True
Step-by-step explanation:
Multiplying add exponents
Divide subtract exponents
Exponent multiply exponents
The statement and reason for each equation is written below:
False (Multiplication law and exponential law)True (Multiplication law and exponential law)True (division law and exponential law)Meaning of Multiplication and division lawMultiplication law is a law of indices that governs the multiplication of variable. if they are of the same base they add up their powers.
Division law is also a law of indices that governs division of variables, and it states that for every division the bases are equal we subtract their powers
In conclusion, The statement and reason for each equation is written above.
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Given are five observations collected in a regression study on two variables:
xi 2 6 9 13 20
yi 7 18 9 26 23
a) Develop a scatter diagram for these data.
b) Develop the estimated regression equation for these data
c) Use the estimated regression equation to predict the value of y when x = 6.
d) What percentage of the total sum of squares can be accounted for by the estimated regression equation?
e) What is the sample correlation coefficient?
f) What is the value of the standard error of the estimate?
g) Test for a significant relationship by using the t test. UseImage for Given are five observations collected in a regression study on two variables: a) Develop a scatter diagram f?=.05.
a) The scatter diagram is shown below.
b) The estimated regression equation for the given data is given as [tex]y = 5.47 + 1.18x[/tex].
c) When x = 6, the estimated value of y is approximately 12.65.
d) T
e) The sample correlation coefficient is approximately 0.8107.
f) In this case, the standard error of the estimate is approximately 5.10.
g) Without these details, it is not possible to conduct the t-test.
What is a scatter diagram?Scatter diagram is a visual representation of data points plotted on a Cartesian coordinate system. It is used to display the relationship between two variables.
a) To develop a scatter diagram, plot the given data points (xi, yi) on a graph, where xi represents the values of the independent variable and yi represents the values of the dependent variable. The scatter diagram is shown below.
b) To develop the estimated regression equation, we need to find the equation of the line that best fits the data points. This line represents the relationship between the independent variable (x) and the dependent variable (y). The estimated regression equation is given by:
y = a + bx
where "a" represents the y-intercept and "b" represents the slope of the line.
Using statistical methods such as least squares regression, the estimated regression equation can be calculated. In this case, the estimated regression equation is:
y = 5.47 + 1.18x
c) To predict the value of y when x = 6, substitute x = 6 into the estimated regression equation:
y = 5.47 + 1.18(6)
y = 12.65
Therefore, when x = 6, the estimated value of y is approximately 12.65.
d) The percentage of the total sum of squares accounted for by the estimated regression equation is given by the coefficient of determination (R-squared). It represents the proportion of the total variation in the dependent variable (y) that can be explained by the independent variable (x) through the estimated regression equation.
In this case, the coefficient of determination (R-squared) is 0.6563, or approximately 65.63%. This means that the estimated regression equation can account for about 65.63% of the total variation in the dependent variable.
e) The sample correlation coefficient (r) measures the strength and direction of the linear relationship between the two variables. In this case, the sample correlation coefficient is approximately 0.8107, indicating a strong positive linear relationship between the variables.
f) The standard error of the estimate measures the average distance between the observed data points and the predicted values from the regression equation. In this case, the standard error of the estimate is approximately 5.10.
g) To test for a significant relationship using the t-test, you would need additional information such as the sample size and the significance level (α) specified in the question. Without these details, it is not possible to conduct the t-test.
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Two sides of a right triangle have lengths of 46 centimeters and 23 centimeters. The third side is not the hypotenuse. How long is the third side? Round your answer to the nearest centimeter.
Answer:
about 40 cm.
Step-by-step explanation:
I know the length of the third side is 40cm because I used the Pythagorean theorem.
a^2+b^2=c^2 The "a" and "b" values are the lengths of the legs of the triangle, while "c" is the length of the hypotenuse. We know the third side of this triangle is not the hypotenuse.
* The longest side of a right triangle is the hypotenuse, so we know the length of the hypotenuse is 46cm.
Therefore, we plug our values into the Pythagorean theorem.
23^2+b^2=46^2
529+b^2=2116
Next, we subtract 529 on both sides.
b^2=1587
Next, find the square root of 1587, so we can find the true value of b.
b=39.8371685741
Rounded to the nearest centimeter is 40.
In conclusion, the length of the third side is 40cm.
Solve the following system of linear equations using Gaussian Elimination Method with Partial Pivoting. Show all steps of your calculations. 0.5x - 0.5y + z = 1 -0.5x + y - 0.5z = 4 X - 0.5 + 0.5z = 8
the solution of system of linear equations is x = 27/2, y = 21/2, z = -1/2
To solve the system of linear equations using the Gaussian Elimination Method with Partial Pivoting, we'll perform the following steps:
Step 1: Set up the augmented matrix for the system of equations.
Step 2: Perform row operations to eliminate variables below the main diagonal.
Step 3: Back-substitute to find the values of the variables.
Let's proceed with the calculations:
Step 1: Augmented matrix setup
The augmented matrix for the system of equations is:
[ 0.5 -0.5 1 | 1 ]
[-0.5 1 -0.5 | 4 ]
[ 1 -0.5 0.5 | 8 ]
Step 2: Row operations
[ 0.5 -0.5 1 | 1 ]
[-0.5 1 -0.5 | 4 ]
[ 1 -0.5 0.5 | 8 ]
R₂ -> R₂ + R₁
[ 0.5 -0.5 1 | 1 ]
[0 0.5 0.5 | 5 ]
[ 1 -0.5 0.5 | 8 ]
R₃ -> R₃ - 2R₁
[ 0.5 -0.5 1 | 1 ]
[0 0.5 0.5 | 5 ]
[ 0 0.5 -1.5 | 6 ]
R₃ -> R₃ - R₂
[ 0.5 -0.5 1 | 1 ]
[0 0.5 0.5 | 5 ]
[ 0 0 -2 | 1 ]
The new augmented matrix after the row operations is:
[ 0.5 -0.5 1 | 1 ]
[0 0.5 0.5 | 5 ]
[ 0 0 -2 | 1 ]
Step 3: Back-substitution
Now, we'll back-substitute to find the values of the variables. Starting from the last row, we can directly determine the value of z:
-2z = 1
z = - 1/2
Substituting z = - 1/2 into the second equation, we can find the value of y:
0.5y + 0.5z = 5
0.5y + 0.5(-1/2) = 5
y = 21/2
0.5x - 0.5y + z = 1
0.5x - 0.5(21/2) + (-1/2) = 1
x = 27/2
Therefore, the solution of system of linear equations is x = 27/2, y = 21/2, z = -1/2
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Can i get help please i will mark brainlest
Answer:
7. 78
8. 169
Step-by-step explanation:
Answer:
5
Step-by-step explanation:
the height y (in feet) of a ball thrown by a child is y = − 1/16 x^2 2 x + 3 where x is the horizontal distance in feet from the point at which the ball is thrown.
(a) How high is the ball when it leaves the child s hand?
(b) What is the maximum height of the ball?
(c) How far from the child does the ball strike the ground?
(a) The ball's height when it leaves the child's hand is 3 feet.
(b) The maximum height of the ball is 3.5625 feet.
(c) The ball strikes the ground approximately 32 feet away from the child.
The ball strikes the ground approximately -16 + 32√(22) feet away from the child in the forward direction.
(a) To find the height of the ball when it leaves the child's hand, we need to determine the value of y when x is zero. Plugging x = 0 into the equation y = -1/16x^2 + 2x + 3, we get:
y = -1/16(0)^2 + 2(0) + 3
y = 3
Therefore, the ball is 3 feet high when it leaves the child's hand.
(b) The maximum height of the ball can be found by finding the vertex of the quadratic function. The x-coordinate of the vertex can be found using the formula x = -b/2a, where a and b are the coefficients of the quadratic equation. In this case, the equation is y = -1/16x^2 + 2x + 3, so a = -1/16 and b = 2. Plugging these values into the formula, we get:
x = -(2)/(2(-1/16))
x = -16/32
x = -1/2
To find the corresponding y-coordinate, we substitute this value back into the equation:
y = -1/16(-1/2)^2 + 2(-1/2) + 3
y = -1/16(1/4) - 1 + 3
y = 1/64 - 64/64 + 192/64
y = 129/64
Therefore, the maximum height of the ball is 129/64 feet.
(c) The ball strikes the ground when its height is zero. To find the distance from the child where this occurs, we set y = 0 and solve for x:
0 = -1/16x^2 + 2x + 3
This equation can be solved using various methods such as factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula here:
x = (-2 ± √(2^2 - 4(-1/16)(3)))/(2(-1/16))
x = (-2 ± √(4 + 3/2))/(2(-1/16))
x = (-2 ± √(11/2))/(2(-1/16))
x = (-2 ± √(11/2))/(-1/8)
x = (-2 ± 4√(22))/(1/8)
Since we're interested in the positive value of x (the ball strikes the ground in the forward direction), we take the positive square root and simplify:
x = (-2 + 4√(22))/(1/8)
x = 8(-2 + 4√(22))
x = -16 + 32√(22)
Therefore, the ball strikes the ground approximately -16 + 32√(22) feet away from the child in the forward direction.
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Layne rode his bike from point a to b by using cherry st . How much further would his trip have been if he took orange drive and peach avenue instead ? Point a 85 point b 77
Answer:
cherry street is 85 yards, while orange dr. and peach ave. is 113 yards, meaning that cherry st. is 28 yards faster
Step-by-step explanation:
85 x 85) - (77 - 77) = 1296
square root of 1296 is 36
77 + 36 = 113
113 - 85 = 28
this means that cherry st. is 28 yds faster
using the Pythagorean Theorum
hope i helped
-lvr
What is 0.5 divided by 2.675
.186915888
that is the answer
if I meant 2.675 divided by .5 it is 5.35
Answer:
you can use a calculator but, its 0.1869158878504672897196261682243
Step-by-step explanation:
Pls help! question is on picture, will mark brainlyest if its right
Answer:
sine=opposite/hypotenuse
sine=6/10
sine=0.6
sine^-1 0.6
=36.87
Kenzie will save 15% if she opens a credit card. She wants to buy a jacket for $120. How much will she save if she opens a credit card?
Answer:
i think its 102 or 120-18
Step-by-step explanation:
i think cauee 10% of 120 is 12 and half of ten percent is 5 and 5% is 6
Answer:
she will save $18
Step-by-step explanation:
find 10% of the number. then find 5%. add the two answers together and you get $18.
10% of $120 = $12
5% of $120 = $6. (half 10%)
$12 + $6 = $ 18
pls answer this i have no clue
Answer:
90 degrees
Step-by-step explanation:
The three angles of a triangle always add up to 180 degrees, so knowing this, all three of the measures given should add up to 180.
First, you need to solve for x:
S(x+10) + R(2x+60) + T(50+x) = 180
Add up all of the like terms:
10 + 60 + 50 = 120
x + 2x + x = 4x
So now you are left with 4x + 120 = 180
And solve for x:
4x/4 + 120/4 = 180/4
x + 30 = 45
x + 30-30 = 45-30
x = 15
Now that we have the value of x, plug it into the measure given for angle R:
(2x+60)
= (2(15)+60)
= (30 + 60)
= 90
So the measure of angle R is 90 degrees.
I hope this helped! :)
Suppose that the individuals are divided into groups j = 1, ...,J each with nj observations respectively, and we only observe the reported group means y; and īj. The model becomes y; = Ba; +ūj, - with error terms ū; = ; Uij, where Ui; indicates error term ui of individual i belonging to group Show that the error terms ūj are heteroskedastic.
The error terms are heteroskedastic
Suppose that the individuals are divided into groups j = 1, ...,J each with nj observations respectively, and we only observe the reported group means y; and īj.
The model becomes y; = Ba; +ūj, with error terms ū; = ; Uij, where Ui; indicates error term ui of individual i belonging to group j.
Now we have to demonstrate that the error terms ūj are heteroskedastic.
The model becomes: y; = Ba; + ūj;
For each group j, the estimated variance of the error term is given by the sum of squared deviations divided by the sample size, and we can write it as follows:
S_j^2 = sum ( yij - īj )^2 / ( nj - 1 ) where yij denotes the observation for the ith individual in the jth group.
The variance of the error term is therefore different for each group j. In other words, the error terms ūj are heteroskedastic.
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Given m||n, find the value of x.
t
(8x-7)
(x+16)°
Answer:
82°
Step-by-step explanation:
(8x-7)=(x+16)
x+16+x=180
2x+16=180
2x=180-16
2x=164
2x/2=164/2
x=82°
What is this just just look at it HOW!
Answer:
WOAH!!!!
Step-by-step explanation:
Thx for sharing that!
Mark as brainlist plssss.
Answer:
Wow
Step-by-step explanation:
a winter coat was priced at 200. each month for three months, thr price was reduced by 15. how much was the coat reduced in price
Let T: R³ R³ be a linear transformation such that T(1, 0, 0) = (-1, 4, 2), 7(0, 1, 0) = (1, 3, -2), and 7(0, 0, 1) = (0, 2, -2). Find the indicated image. T(1, -3, 0). T(1, -3,0) =
The image of the vector (1, -3, 0) under the linear transformation T is (-4, -5, 8).
The linear transformation T: R³ → R³, defined by T(1, 0, 0) = (-1, 4, 2), T(0, 1, 0) = (1, 3, -2), and T(0, 0, 1) = (0, 2, -2), can be used to find the image of the vector (1, -3, 0) under T.
To find the image of the vector (1, -3, 0) under the linear transformation T, we can use the linearity property of the transformation. Since T is a linear transformation, we can express any vector v = (x, y, z) as a linear combination of the standard basis vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1).
The given information states that T(1, 0, 0) = (-1, 4, 2), T(0, 1, 0) = (1, 3, -2), and T(0, 0, 1) = (0, 2, -2). Using these values, we can express (1, -3, 0) as a linear combination:
T(1, -3, 0) = T(1, 0, 0) - 3T(0, 1, 0) + 0T(0, 0, 1)
= (-1, 4, 2) - 3(1, 3, -2) + 0(0, 2, -2)
= (-1, 4, 2) - (3, 9, -6) + (0, 0, 0)
= (-1 - 3 + 0, 4 - 9 + 0, 2 + 6 + 0)
= (-4, -5, 8)
Therefore, the image of the vector (1, -3, 0) under the linear transformation T is (-4, -5, 8).
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At a certain bus station, 47% of all arrivals are late. Suppose a random
sample of 12 bus arrivals is examined. Using the binomial function, give the
probability
Complete question :
At a certain bus station, 47% of all arrivals are late. Suppose a random
sample of 12 bus arrivals is examined. Using the binomial function, give the probability of ;
Atleast 8 late arrivals 2) At most 4 late arrivals
Answer:
P(x >= 8) = 0.1411 ;
P(x ≤ 4) = 0.2570
Step-by-step explanation:
Using the binomial probability distribution relation :
P(x =x) = nCx * p^x * (1 - p)^(n - x)
p = 47% = 0.47
1 - p = 0.53
n = 12
A.)
Atleast 8 late arrivals
P(x >= 8) :p(x = 8)+p(x =9)+p(x=10)+p(x=11)+p(x=12)
Using a binomial distribution calculator to save computation time :
P(x >= 8) = 0.141096
P(x >= 8) = 0.1411
B.)
P(x ≤ 4) = p(x=0)+p(x=1)+p(x=2)+p(x=3)+p(x=4)
Using a binomial probability calculator ;
P(x ≤ 4) = 0.25697
P(x ≤ 4) = 0.2570