The Cartesian equation ( y = f(x)) for the curve is y = -x.
To find the Cartesian equation of the curve, we need to eliminate the parameter θ from the given parametric equations. We can do this by rearranging the equations and solving for θ in terms of x or y, and then substituting that value into the other equation.
First, let's rearrange the first equation to solve for cotθ:
2x = cotθ - cscθ
2x + cscθ = cotθ
cscθ - cotθ = -2x
Next, let's rearrange the second equation to solve for cscθ:
4y = 2cscθ - 2cotθ
2cscθ = 4y + 2cotθ
cscθ = 2y + cotθ
Now, let's substitute the value of cscθ from the second equation into the first equation:
2y + cotθ - cotθ = -2x
2y = -2x
y = -x
Therefore, the Cartesian equation of the curve is y = -x.
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Question 16. Let A be an m×n matrix, X an n×r matrix, and B an m×r matrix. Show that AX =B if and only if Axj=bj, j=1,…,r
Axj = bj, j=1,…,r.
To show that AX = B if and only if Axj = bj, j=1,…,r, we can start by recognizing that AX is an m×r matrix and B is an m×r matrix. Since the dimensions of these two matrices are the same, they are equal if and only if each corresponding element is equal.
Therefore, we can conclude that AX = B if and only if each corresponding element of AX is equal to its corresponding element in B. Mathematically, this is expressed as Axj = bj, j=1,…,r.
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Abstract Algebra: Let ???? be the group of all real-valued functions with domain ℝ under addition. Let H be the subset of ???? consisting of all functions that are differentiable. Determine if H is a subgroup of ????.
Yes, H is a subgroup. We can prove this by using the subgroup criterion, which states that a subset H of a group G is a subgroup if and only if it satisfies the following three conditions:
1. The identity element of G is in H.
2. If h1 and h2 are in H, then h1*h2 is in H.
3. If h is in H, then h^(-1) is in H.
Let's check if these conditions are satisfied for H:
1. The identity element of ???? is the zero function, f(x) = 0, which is differentiable. Therefore, the identity element is in H.
2. If h1 and h2 are in H, then they are both differentiable functions. The sum of two differentiable functions is also differentiable, so h1 + h2 is in H.
3. If h is in H, then it is a differentiable function. The inverse of a differentiable function under addition is its negative, which is also differentiable. Therefore,[tex]h^{-1} = -h[/tex] is in H.Since all three conditions are satisfied, H is a subgroup.
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The volume of football is 4000pi/3 cm^3. Given that the volume of a sphere =4/3pi r^3 ,,find the radius r of the ball on cm
Answer:
r = 10
Step-by-step explanation:
just rearrange the equation to find r
Select the correct answer from each drop down menu.
Consider polygon JKLMNO on the coordinate grid.
The other vertices appears to be located in the second and fourth quadrants.
What is a polygon?
A polygon is a two-dimensional geometric shape that is made up of straight line segments connected end-to-end to form a closed shape.
Based on the image, it appears that we are dealing with a polygon JKLMNO on a coordinate grid.
First, we can see that the polygon is a hexagon (six-sided figure) with vertices at points J, K, L, M, N, and O. We can also see that the polygon is not a regular hexagon, since its sides are of different lengths and its angles are not all equal.
To determine the coordinates of the vertices of the polygon, we would need to know the scale and orientation of the coordinate grid. However, based on the image, we can make some general observations about the location of the vertices. For example, we can see that vertex J appears to be in the third quadrant (negative x and y values), while vertex N appears to be in the first quadrant (positive x and y values).
Therefore, The other vertices appear to be located in the second and fourth quadrants.
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Give an example of a compound interest problem. Then solve the
problem.
Write me a problem about compound interest. The simple interest on a sum of money for 3 years at 6²/₃ % per annum is $ 6750. What will be the compound interest on the same sum at the same rate for the same period, compounded annually?
An example of a compound interest problem and the solution problem is:
"Sarah deposits $10,000 into a savings account with a 5% annual interest rate compounded annually. How much will she have in the account after 3 years?"
Solution: Sarah will have $11,576.25 in the account after 3 years.
The compound interest on the same sum at the same rate for the same period, compounded annually, is:
$6,646.34
What is an example of compound interest?Sarah deposits $10,000 into a savings account with a 5% annual interest rate compounded annually. How much will she have in the account after 3 years?
To solve this problem, we can use the formula for compound interest: A = P(1 + r)^n, where A is the final amount, P is the principal amount, r is the annual interest rate, and n is the number of years.
Plug in the given values: A = 10,000(1 + 0.05)^3Simplify the equation: A = 10,000(1.05)^3Calculate the final amount: A = $11,576.25To find the compound interest on the same sum at the same rate for the same period, we can use the same formula for compound interest: A = P(1 + r)^n.
First, we need to find the principal amount. We can use the formula for simple interest: I = Prt, where I is the interest, P is the principal, r is the annual interest rate, and t is the time in years. Rearranging the formula to solve for P, we get P = I/rt.Plug in the given values: P = 6750/(0.0667)(3) = 33,750Now we can plug in the values into the compound interest formula: A = 33,750(1 + 0.0667)^3Simplify the equation: A = 33,750(1.0667)^3Calculate the final amount: A = 40,396.34To find the compound interest, subtract the principal amount from the final amount: 40,396.34 - 33,750 = $6,646.34More information about compound interest here: https://brainly.com/question/24924853
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REAL NUMBERS Combining like terms in a quadratic expression Simplify the following expression. 5x^(2)+2x-4-10x^(2)
To simplify the expression 5x2+2x-4-10x2, we must first combine like terms. Like terms are terms that have the same variable or variables with the same exponents.
In this expression, 5x2 and -10x2 are like terms. To combine them, we can add their coefficients (the numbers in front of the variable) together. So, 5x2 - 10x2 = (-5 + 5)x2 = 0x2.
Now, the expression has only two terms left. 2x and -4 are also like terms, since they both have the same variable. We can again add their coefficients together, which results in (2 + -4)x = -2x.
Therefore, the simplified expression is 0x2 - 2x. Note that there is no need to use the 0x2 term, so the expression can also be written as -2x.
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a) Consider the linear system :⎩⎨⎧x+2y−3z=43x−y+5z=24x+y+(a2−14)z=a+2, For which values ofadoes the linear system has (i) A unique solution. (ii) Infinitely many solutions. (iii) No solution.adgbehcfi=3, then find3fci+2c3ebh+2b3dag+2a
The linear system has a unique solution for all values of a except √(97/13), and infinitely many solutions or no solution for a = √(97/13).
To find the values of a for which the linear system has a unique solution, infinitely many solutions, or no solution, we can use the determinant of the coefficient matrix. The determinant of a 3x3 matrix is given by:
|A| = adg - beh + cfi - 3fci - 2c - 3ebh - 2b - 3dag - 2a
For a unique solution, the determinant of the coefficient matrix must be nonzero. For infinitely many solutions or no solution, the determinant must be zero.
For the given system, the coefficient matrix is:
⎡⎣⎢1 2 −32 −1 54 1 a2−14⎤⎦⎥
The determinant of this matrix is:
|A| = (1)(-1)(a2-14) - (2)(5)(4) - (-3)(-1)(1) - (3)(4)(a2-14) - (2)(1)(-3) - (3)(2)(4) - (2)(-1)(-3)
Simplifying this expression gives:
|A| = -a2 + 14 - 40 - 3 - 12a2 + 168 - 6 - 24 - 6
|A| = -13a2 + 97
For a unique solution, |A| ≠ 0:
-13a2 + 97 ≠ 0
13a2 ≠ 97
a2 ≠ 97/13
a ≠ √(97/13)
For infinitely many solutions or no solution, |A| = 0:
-13a2 + 97 = 0
13a2 = 97
a2 = 97/13
a = √(97/13)
Therefore, the linear system has a unique solution for all values of a except √(97/13), and infinitely many solutions or no solution for a = √(97/13).
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Amanda has $200 in her lunch account. She spends 25$ each week on lunch. The
equation y=200-25x represents the total amount in Amanda's school lunch accou
y for x weeks of purchasing lunches.
The x intercept and y intercept of the equation is 8 and 200 respectively.
What are x Intercepts and y Intercepts?x intercept is the point on the line where it touches the X axis.
y intercept is the point on the line where it touches the Y axis.
Given equation is,
y = 200 - 25x
where y is the total amount in Amanda's school lunch account after purchasing lunches for x weeks.
x intercept is the point on X axis. Any point on x axis has y coordinate 0.
So x intercept is the x coordinate when y coordinate is 0.
0 = 200 - 25x
25x = 200
x = 8
This means that the amount in the account will become 0, after purchasing lunch for 8 weeks.
y intercept is the y coordinate when x coordinate = 0.
y = 200 - (25 × 0)
y = 200
This indicates the amount in the account at the start before purchasing any lunch.
Hence the x intercept is 8 and y intercept is 200.
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Your question is incomplete. The complete question is as follows.
Amanda has $200 in her lunch account. She spends 25$ each week on lunch. The equation y=200-25x represents the total amount in Amanda's school lunch account, y for x weeks of purchasing lunches.
Find the x and y intercepts and interpret their meaning in the context of the situation.
Collins Middle School has 228 sixth grade students. If the sixth grade is 30% of the total school, how many students are in the middle school?
Answer:
760 students
Step-by-step explanation:
divide 228 ÷ 30%
30% = 0.30
228 ÷ 0.30 = 760
Which is the better buy?
Frozen Peas
Cost (dollars)
Weight (ounces)
O Brand A
A B
2
16
3
28
O Brand B
O The unit cost is the same.
The better buy is given by the following brand:
Brand A.
How to obtain the better buy?The better buy is obtained applying the proportions in the context of the problem.
A proportion is applied as the cost per ounce is given dividing the total cost by the number of ounces.
Then the better buy is given by the option with the lowest cost per ounce.
The cost per ounce for each brand is given as follows:
Brand A: 16/2 = $8 per ounce.Brand B: 28/3 = $9.3 per ounce.$8 per ounce is a lesser cost than $9.3 per ounce, hence the better buy is given by Brand A.
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Audrey’s university English teacher Mr. Dalton has told her she will have 28 assignments this semester. Audrey has discovered that she needs to spend 2.5 hours on each assignment.
How many hours will Audrey spend on her English assignments?
AND
How many days will it take Audrey to complete her assignments?
Answer:
70 and 2.91 days
hope this helped
Summary
Describe the relationship that exists between the independent and dependent variables for a relationship that
has a negative slope.
The two variables move in opposite directions.
What is a linear relationship?For a relationship that has a negative slope, there is an inverse relationship between the independent and dependent variables. This means that as the value of the independent variable increases, the value of the dependent variable decreases, and vice versa. In other words, the two variables move in opposite directions.
For example, suppose we have a dataset that shows the relationship between the number of hours of study and the score on a test.
If the slope of the line of best fit is negative, this means that as the number of hours of study increases, the score on the test decreases. This negative relationship indicates that the more time a student spends studying, the lower their score on the test is likely to be.
It is important to note that a negative slope does not necessarily imply a causal relationship between the variables. There may be other factors that affect the relationship between the variables, and further analysis is needed to determine the nature of the relationship.
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Can someone help me find angle E,angle D,angle A and angle F and explain how you got it
The measure of the angles e, d, a, and f will be 26°, 122°, 102°, and 58°, respectively.
What is an angle?The inclination is the separation seen between planes or vectors that meet. Degrees are another way to indicate the slope. For a full rotation, the angle is 360°.
By the supplementary property, the equation is given as,
f + 78° + 44° = 180°
f = 58°
By the complementary property, the equation is given as,
e + 64° = 90°
e = 26°
By the corresponding angles property, the equation is given as,
c = f
c = 58°
By the supplementary property, the equation is given as,
c + d = 180°
58° + d = 180°
d = 122°
By the corresponding angles property, the equation is given as,
b = 78°
By the supplementary property, the equation is given as,
a + b = 180°
a + 78° = 180°
a = 102°
The diagram is given below.
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select the 2 linear functions A) (y=6x+14), B) (y= x/4 + 1), C) (y=x^3), D) (y=3/x +2)
The two linear equations are y = 6x + 14 and y = x/4 + 1. Then the correct options are A and B.
What is a linear equation?A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.
The linear equation is given as,
y = mx + c
Let's check all the options, then we have
A) y = 6x + 14, the equation is a linear equation because the degree is one.
B) y = x/4 + 1, the equation is a linear equation because the degree is one.
C) y = x³, the equation is a cubic equation because the degree is three.
D) y = 3/x +2, the equation is a non-linear equation because the degree is negative one.
The two linear equations are y = 6x + 14 and y = x/4 + 1. Then the correct options are A and B.
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Find the values of x and y
The values of x and y in the right triangle are 4√2
How to determine the values of x and yThe attached right triangle represents the given parameter
The given triangle is a special triangle with an angle of 45 degrees
This means that the legs of the triangles are
legs = Hypotenuse/√2
From the question, the value of the hypotenuse to be 8 units
i.e. Hypotenuse = 8
So, the above equation becomes
x and y = 8/√2
Rationalize the equation
This gives
x and y = 8/√2 * √2/√2
So, we have the final values to be
x and y = 4√2
Hence, the values are 4√2
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Find each quotient. Write all final answers in simplest form. (28x^(5)-8x^(4)+40x^(2))/(4x^(3))
The final answer in simplest form is 7[tex]x^{2}[/tex] - 2x + 10/x.
To find the quotient of (28[tex]x^{5}[/tex]-8[tex]x^{4}[/tex]+40[tex]x^{(2)[/tex])/(4[tex]x^{3}[/tex]), we need to divide each term in the numerator by the denominator.
First, we divide 28[tex]x^{5}[/tex] by 4[tex]x^{3}[/tex]:
(28[tex]x^{5}[/tex])/(4[tex]x^{3}[/tex]) = (28/4)[tex]x^{(5-3)[/tex] = 7[tex]x^{2}[/tex]
Next, we divide -8[tex]x^{4}[/tex] by 4[tex]x^{3}[/tex]:
(-8[tex]x^{4}[/tex])/(4[tex]x^{3}[/tex]) = (-8/4)[tex]x^{(4-3)[/tex] = -2x
Finally, we divide 40[tex]x^{2}[/tex] by 4[tex]x^{3}[/tex]:
(40[tex]x^{2}[/tex])/(4[tex]x^{3}[/tex]) = (40/4)[tex]x^{(2-3)[/tex] = 10[tex]x^{(-1)[/tex] = 10/x
Putting it all together, we get:
(28[tex]x^{5}[/tex] - 8[tex]x^{4}[/tex] + 40[tex]x^{2}[/tex])/(4[tex]x^{3}[/tex]) = 7[tex]x^{2}[/tex] - 2x + 10/x
Therefore, the final answer in simplest form is 7[tex]x^{2}[/tex] - 2x + 10/x.
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20. lan used 9½ cups of flour to make 1 batches of oatmeal cookies. One batch makes 18
cookies. If he follows the same recipe, about how many cookies can he make with 15 cups of
flour?
Answer:
To solve this problem, we can set up a proportion:
9.5 cups of flour / 18 cookies = 15 cups of flour / x cookies
To solve for x, we can cross-multiply:
9.5 cups of flour * x cookies = 18 cookies * 15 cups of flour
Dividing both sides by 9.5 cups of flour gives us:
x cookies = 18 cookies * 15 cups of flour / 9.5 cups of flour
x cookies ≈ 28.42
Since we can't make a fraction of a cookie, we'll round down to the nearest whole number. Therefore, Lan can make about 28 cookies with 15 cups of flour using the same recipe.
Pls give simple working
Screen shot this and then mark when it should go tyy
Answer:
Step-by-step explanation:
i need help answer plssd
The number representing the 6th grader participated in the field event is 120
What is a sample space?A sample space is a collection or a set of possible outcomes of a random experiment.
The sample space is represented using the symbol, “S”. The subset of possible outcomes of an experiment is called events.
Given is a graph, showing the number of class students taking parts in different events,
We need to find the number of the 6th graders who participated in field event.
The sample size is 20 for who participated in field event, that means one unit is representing 20 participants
The sample number for 6th graders who participated in field event = 6
That means, the total number of the 6th graders who participated in field event = 20 x 6 = 120
Hence, the number representing the 6th grader participated in the field event is 120
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Logan invested $3,800 in an account paying an interest rate of 5% compounded
continuously. Qasim invested $3,800 in an account paying an interest rate of 6%
compounded annually. To the nearest hundredth of a year, how much longer would
it take for Logan's money to double than for Qasim's money to double?
Answer:
Step-by-step explanation:
The formula to calculate the doubling time for continuously compounded interest is:
t = ln(2) / (r * ln(1 + (r/n)))
where t is the time in years, r is the annual interest rate as a decimal, and n is the number of times the interest is compounded per year (which is infinity for continuous compounding).
For Logan's investment, we have:
r = 0.05
n = infinity
t1 = ln(2) / (0.05 * ln(1 + (0.05/infinity)))
t1 ≈ 13.86 years
For Qasim's investment, we have:
r = 0.06
n = 1
t2 = ln(2) / (0.06 * ln(1 + (0.06/1)))
t2 ≈ 11.55 years
To find the difference in the time it takes for their investments to double, we can subtract t2 from t1:
t1 - t2 ≈ 13.86 - 11.55 ≈ 2.31
So it would take Logan's money approximately 2.31 years longer to double than Qasim's money, to the nearest hundredth of a year.
find the value of x,
de=5
The solution is, the value of x is, x=10.
What is triangle?A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane.
here, we have,
from the given figure, we get,
the two triangles are similar
and, DE is the bisector of other two sides,
so, we know from the property of triangles, that,
DE = 1/2 * AB
we have,
DE = 5
and. AB =x
so, we get,
5 = x/2
or, x = 10
Hence, The solution is, the value of x is, x=10.
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Compute the inverseA−1of the following matrices (a)A=[111−4](b)A=211312−13−1Verify thatA−1A=IandAA−1=I
We have verified that A^-1A = I and AA^-1 = I for both matrices.
To compute the inverse of a matrix, we can use the formula: A^-1 = 1/det(A) * adj(A), where det(A) is the determinant of the matrix A and adj(A) is the adjoint of the matrix A.
(a) A = [1 1; 1 -4]
det(A) = (1*-4) - (1*1) = -4 - 1 = -5
adj(A) = [-4 -1; -1 1]
A^-1 = 1/-5 * [-4 -1; -1 1] = [4/5 1/5; 1/5 -1/5]
(b) A = [2 1 1; 3 1 2; -1 3 -1]
det(A) = (2*(1*-1) - 1*(2*3) + 1*(-1*3)) - (1*(1*-1) - 1*(2*-1) + 1*(3*3)) = -6 - 3 - 3 - 9 = -21
adj(A) = [-1 -2 7; 5 1 -8; 2 5 -7]
A^-1 = 1/-21 * [-1 -2 7; 5 1 -8; 2 5 -7] = [1/21 2/21 -7/21; -5/21 -1/21 8/21; -2/21 -5/21 7/21]
To verify that A^-1A = I and AA^-1 = I, we can simply multiply the matrices and check if the result is the identity matrix.
For (a):
A^-1A = [4/5 1/5; 1/5 -1/5] * [1 1; 1 -4] = [1 0; 0 1] = I
AA^-1 = [1 1; 1 -4] * [4/5 1/5; 1/5 -1/5] = [1 0; 0 1] = I
For (b):
A^-1A = [1/21 2/21 -7/21; -5/21 -1/21 8/21; -2/21 -5/21 7/21] * [2 1 1; 3 1 2; -1 3 -1] = [1 0 0; 0 1 0; 0 0 1] = I
AA^-1 = [2 1 1; 3 1 2; -1 3 -1] * [1/21 2/21 -7/21; -5/21 -1/21 8/21; -2/21 -5/21 7/21] = [1 0 0; 0 1 0; 0 0 1] = I
Therefore, we have verified that A^-1A = I and AA^-1 = I for both matrices.
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What is a possible measure of an angle that has a terminal side in Quadrant III and makes a 60º angle with the x-axis?
A. 120°
B. 210°
C. 150°
D. 240°
Answer: option D is correct
Step-by-step explanation
Angle makes Quadrant III is 60°
To find : terminal side angle .
solution : we know that sum of total angle
is 360° and half of it in 180°
here the possible angle of an
that has terminal side in quadrant III
and makes a 60 degree with the x axis,
=) 120° + 60° =240°
Hence the correct option is in D) 240°
10 points if someone gets right
Sam believes that 1/2 % is equivalent to 50%. Is he correct? Why or why not?
Step-by-step explanation:
just ignore the % sign and answer :
is 1/2 equivalent to 50 ? yes or no ?
no, of course.
1/2 = 0.5
50 = 50
clearly they are different.
Sam is confused, because 50% = 1/2 = 0.5.
but 50% is NOT 1/2%
1/2% is half of 1%, which by itself is 1/50 of 50%.
1/2% = 1/50/2 of 50% = 1/100 of 50% =
= 0.01 × 0.5 = 0.005
Mr. Cho reached his office at 9:10 am. He took 37 minutes to check his emails and 1 hour and 38 minutes to write a report. Then he attended a meeting until he left for lunch at 12:30pm. How long did the meeting last?
Answer: the meeting lasted for an hour and 5 min.
Step-by-step explanation:
9:10am + 37min = 9:47am9:47am + 1hour = 10:47am10:47am + 38min = 11:25am12:30pm - 11:25am = 1hour, 5minTherefore, the meeting lasted for an hour and five minutes.
The angles in a triangle are in a 1:2:3 ratio
show that the triangle is right angled
The triangle with measure of the angle is 30°, 60°, and 90° proves that it is right angled triangle.
Let us consider 'x' be the measure of the angles in a triangle.
Ratio of the angles in a triangle is 1 : 2 : 3
Measure of angle 1 = 1x
Measure of angle 2 = 2x
Measure of angle 3 = 3x
Sum of all the interior angles in a triangle = 180°
Substitute the value of the measure of the angles we get,
⇒ 1x + 2x + 3x = 180°
⇒ 6x = 180°
⇒ x = 180° / 6
⇒ x = 30°
Measure of angle 1 = 30°
Measure of angle 2 = 2× 30°
= 60°
Measure of angle 3 = 3× 30°
= 90°
Therefore, as measure of one of the interior angle is 90 degree this implies it is a right angled triangle.
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Does anyone know how to solve this
y=-2x^(2) + 12x - 15 ;x=3
Cause when I try it I get y=-27
but answer is supposed to be y=3 and x=3
Pls explain how
Answer: Order of operations
Step-by-step explanation:
Plug in 3 for x so...
y = -2(3)² + 12(3) - 15
Don't forget the order of operations, exponents go first
y = -2(9) + 12(3) - 15
Then multiply
y = -18 + 36 - 15
Simplify
y = 3
Suppose that 30% of the applicants for a certain industrial job possess advanced training in
computer programming. Applicants are interviewed sequentially and are selected at random from
the pool. A) Find the probability that the first applicant with advanced training in programming is
found on the fifth interview. B) What is the expected number of applicants who need to be interviewed in order to find
the first one with advanced training
After calculating the probability we get, there is a 7.20 percent chance that the first candidate will be identified during the fifth interview and it should take 3.33 interviews to discover the first candidate.
The quantity X of repeated trials needed to obtain r successes with p probability in a binomial experiment is known as the negative binomial distribution.
When x succeeds after n tries, the probability is given by:
P(X-x) = Cn-1, x-1*p(power x) * (1 - p)power (n-x)
The number of unique combinations of x items from a set of n elements, Cn,x, is determined by the formula below.
Cn,x = n! / x! ( n! - x! )
This problem involves that:
Let's say that 30 percent of those who apply for a certain industrial position have advanced expertise in computer programming. That follows that.
(a) We need to determine the probability that the first applicant with advanced programming training will be discovered during the fifth interview.
This is the likelihood that it will take 5 attempts to get 1 success.
So, n=5, x=1
P(X-x) = Cn-1, x-1*p(power x) * (1 - p)power (n-x)
P(X-5) = C 4,0* (0.30)(power 1) * (0.70)power (4)
= 0.0720.
During the sixth interview, there is a 7.20% chance that the first candidate with advanced programming training will be discovered.
(b) To find how many applicants should be expected to be interviewed in order to select the first candidate with advanced training, calculate
It is stated by how many trials are anticipated to result in r success:
E = r/p
Thus with the value r=1
E = r/p
= 1/0.3
= 3.33.
For the first candidate with advanced training, it will take an average of 3.33 interviews.
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A polynomial f(x) and one of its zeros are given. Find all the zeros. f(x)=x^(4)-8x^(3)+18x^(2)+16x-40;,4+2i is a zero
The zeros of the polynomial f(x) = x4 - 8x3 + 18x2 + 16x - 40 are 4 + 2i, 4 - 2i, -2, and -8.
Given that 4 + 2i is one of its zeros, we can use the fact that the product of the zeros of a polynomial is equal to the product of the coefficients of the polynomial.
We can use this fact to find all of the zeros of the polynomial:
1. We can calculate the product of the coefficients of the polynomial:
( -40 ) * ( 16 ) * ( 18 ) * ( -8 ) = -442368
2. We can calculate the product of the known zero and its conjugate:
( 4 + 2i ) * ( 4 - 2i ) = 16
3. We can divide the product of the coefficients by the product of the known zero and its conjugate:
-442368 / 16 = -27735
4. This is the product of the other zeros:
-27735 = x^(2) + 8x + 1135
5. We can use the quadratic formula to solve for the remaining zeros:
x = (-8 +/- sqrt(64 - 4*1*1135))/2
x1 = (-8 + sqrt(144 - 4640))/2
x2 = (-8 - sqrt(144 - 4640))/2
Therefore, the remaining zeros of the polynomial f(x) are:
x1 = -5 + i7
x2 = -5 - i7
To find all the zeros of the polynomial f(x) = x4 - 8x3 + 18x2 + 16x - 40, we can use the fact that 4 + 2i is a zero and apply the conjugate root theorem. The conjugate root theorem states that if a polynomial has a complex root a + bi, then it also has a conjugate root a - bi. Therefore, 4 - 2i is also a zero of the polynomial.
Now, we can use synthetic division to divide the polynomial by (x - 4 - 2i) and (x - 4 + 2i) to find the other zeros. The result of the synthetic division will be a quadratic polynomial, which we can then solve using the quadratic formula.
Synthetic division with (x - 4 - 2i):
4 + 2i | 1 -8 18 16 -40
| 0 4+2i -4+14i -44-8i 56+40i
----------------------------
1 -4+2i 14+14i -28-8i 16+40i
Synthetic division with (x - 4 + 2i):
4 - 2i | 1 -4+2i 14+14i -28-8i 16+40i
| 0 4-2i -4-14i 44+8i -56-40i
----------------------------
1 0 10 16 0
The result of the synthetic division is the quadratic polynomial x2 + 10x + 16. We can solve this using the quadratic formula:
x = (-10 ± √(102 - 4(1)(16)))/(2(1))
x = (-10 ± √(100 - 64))/2
x = (-10 ± √36)/2
x = (-10 ± 6)/2
The two solutions are x = -2 and x = -8.
Therefore, the zeros of the polynomial f(x) = x4 - 8x3 + 18x2 + 16x - 40 are 4 + 2i, 4 - 2i, -2, and -8.
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Riya is applying mulch to her garden. She applies it at a rate of
250
,
000
cm
3
250,000 cm
3
250, comma, 000, start text, space, c, m, end text, cubed of mulch for every
m
2
Riya applying mulch in m³/ m² at the rate of 0.25 m³
To calculate the rate of mulch application in m³/m², we need to convert the given rate of 250,000 cm³/m² into m³/m². We can do this by using the conversion factor: 1 m³ = 1,000,000 cm³.
So, 250,000 cm³/m² can be converted to m³/m² as follows:
250,000 cm³/m² × (1 m³/1,000,000 cm³) = 0.25 m³/m²
This means that for every square meter of garden space, Riya is applying 0.25 m³ of mulch.
It's important to note that rate refers to the amount of mulch applied per unit of garden space. In this case, the unit is m². By expressing the rate in m³/m², we're specifying the volume of mulch applied per unit of area.
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Complete Question:
Riya is applying mulch to her garden. She applies it at a rate of 250,000 cm³ of mulch for every m² of garden space. At what rate is Riya applying mulch in m³/ m²?
