Given a vector field F on R3 and non-intersecting curves s(t) and c(t) over t ∈ [0, 1] with s(0) = c(0) and s(1) = c(1), if we have ∮s F · d = ∮c F · d, then the value of F · dr where r is given by s(t) for t ∈ [0, 1] is the same as the value of F · dr where r is given by c(t) for t ∈ [0, 1].
The given equation, ∮s F · d = ∮c F · d, is a statement of the conservative nature of the vector field F. It means that the line integral of F around a closed curve is zero, which implies that F is a conservative vector field.
Since s(t) and c(t) are non-intersecting curves with the same endpoints, they form a closed loop. By the conservative property of F, the line integral of F along both curves will be equal.
The value of F · dr where r is given by s(t) for t ∈ [0, 1] is the line integral of F along the curve s(t). Since s(t) and c(t) form a closed loop, we can apply the equation ∮s F · d = ∮c F · d to conclude that the value of F · dr along s(t) is the same as the value of F · dr along c(t). Thus, we can evaluate F · dr by choosing either curve and applying the line integral formula.
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what is the probability that two people chosen at random were born during the same month of the year?
To calculate the probability that two people chosen at random were born during the same month of the year, we need to consider the total number of possible outcomes and the favorable outcomes.
There are 12 months in a year, so the total number of possible outcomes is 12 (one for each month).
Now, let's consider the favorable outcomes. To have two people born in the same month, we need to choose any one of the 12 months for the first person, and then the second person should also be born in the same month.
The probability that the second person is born in the same month as the first person is 1/12 since there is only one favorable outcome out of 12 possible outcomes.
Therefore, the probability that two people chosen at random were born during the same month of the year is 1/12 or approximately 0.0833 (rounded to four decimal places).
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Help how do I factor with the given zero!
y=x^4+2x^3-20x^2+64x-32
2+2i
The factored function is given as follows:
[tex]x^4 + 2x^3 - 20x^2 + 64x - 32 = (x^2 + 6x - 4)(x^2 - 4x + 8)[/tex]
How to factor the function?The function for this problem is defined as follows:
[tex]y = x^4 + 2x^3 - 20x^2 + 64x - 32[/tex]
The zeros are given as follows:
x = 2 + 2i.x = 2 - 2i. -> complex conjugate theorem, if a complex number is a zero, the conjugate also is:Hence the function is factored as follows:
[tex]x^4 + 2x^3 - 20x^2 + 64x - 32 = (ax^2 + bx + c)(x - 2 + 2i)(x - 2 - 2i)[/tex]
(we have the multiplication of two second degree polynomials resulting in a fourth degree polynomial, we must obtain the other second degree polynomial).
[tex]x^4 + 2x^3 - 20x^2 + 64x - 32 = (ax^2 + bx + c)(x^2 - 4x + 8)[/tex]
[tex]x^4 + 2x^3 - 20x^2 + 64x - 32 = ax^4 + (-4 + b)x^3 + \cdots + 8c[/tex]
(it is not necessary to make the calculations in the middle of the function as they are not needed to obtain the constants).
Hence the value of a is given as follows:
a = 1.
The value of b is given as follows:
-4 + b = 2
b = 6.
The value of c is given as follows:
8c = -32
c = -4.
Hence the factored expression is of:
[tex]x^4 + 2x^3 - 20x^2 + 64x - 32 = (x^2 + 6x - 4)(x^2 - 4x + 8)[/tex]
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 From the attachment, what is the measure of Arc CDE?
The value of the measure of Arc CDE is,
⇒ Arc CDE = 128 degree
Since, An angle is a combination of two rays (half-lines) with a common endpoint. The latter is known as the vertex of the angle and the rays as the sides, sometimes as the legs and sometimes the arms of the angle.
Here, A circle is shown in figure.
And, By circle we have;
The measure of Arc CDE is,
⇒ Arc CDE = 128 degree
Thus, The value of the measure of Arc CDE is,
⇒ Arc CDE = 128 degree
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The ratio of boys to girls in Mr. Johnson's class is 2 to 3. There are 15 girls in the class.
There are 10 boys in Mr. Johnson's class, and the total number of Students in the class is:10 + 15 = 25
If the ratio of boys to girls in Mr. Johnson's class is 2 to 3, this means that for every 2 boys, there are 3 girls. Let's represent the number of boys in the class as "b". Then we can set up the following proportion:2/3 = b/15
To solve for "b", we can cross-multiply:2 x 15 = 3 x b
30 = 3b
b = 10
Therefore, there are 10 boys in Mr. Johnson's class, and the total number of students in the class is:10 + 15 = 25
It's worth noting that we could have also found the number of girls in the class by using the ratio. Since the ratio of boys to girls is 2 to 3, this means that the total number of parts in the ratio is 2 + 3 = 5. To find the number of girls, we can divide the total number of students (25) by the total number of parts (5) and then multiply by the number of parts representing girls (3):(25/5) x 3 = 15
So we can see that there are indeed 15 girls in the class.
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can someone help real fast?
Answer:
51
Step-by-step explanation:
Sum of interior angles in a triangle is 180.
x + 90 + 39 = 180
x + 129 = 180
x = 180 - 129
x = 51
Answer:
51 degrees
Step-by-step explanation:
We are asked to find angle x.
To find angle x, we have to write an equation to find it.
We know that one angle is 39 degrees, and the other is 90 degrees because of the small box. We also know that all 3 angles in a triangle have to add up to 180.
Here's the equation for this:
180=39+90+x
simplify
180=129+x
subtract 129 from both sides
51=x
So, angle x is 51 degrees.
Hope this helps! :)
We are interested in whether the mean blood pressure for women is equal to the mean blood pressure for men or whether these two means are different. The null hypothesis is that these two means are equal. If they are different we have no prior view as to whether women or men have the higher mean blood pressure. We took a sample of the blood pressures of 16 women (group 1) and found an average blood pressure x 1 of 119.4. We also measured the blood pressures of their respective brothers (group 2) and found an average blood pressure x 2 of 121.2. In order to carry out the relevant t test we calculated s2d to be 25. We choose a Type I error value a = 0.05. Calculate the numerical value of the test statistic. [5] State the relevant critical point(s). [3] Carry out the test, indicating whether you accept or reject the null hypothesis.
There is insufficient evidence to conclude that the mean blood pressure for women is different from the mean blood pressure for men. To determine whether the mean blood pressure for women is equal to the mean blood pressure for men, a t-test is conducted using the sample data of 16 women (group 1) and their respective brothers' blood pressures (group 2).
The null hypothesis states that the means are equal, and the alternative hypothesis suggests they are different. The test statistic is calculated, critical points are identified, and the null hypothesis is either accepted or rejected based on the test results.
To carry out the t-test, we first calculate the test statistic. The formula for the test statistic (t) in this case is:
t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))
where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
Given:
x1 = 119.4 (average blood pressure for women)
x2 = 121.2 (average blood pressure for men)
s2d = 25 (pooled sample variance)
n1 = n2 = 16 (sample sizes)
α = 0.05 (Type I error value)
Now, we can calculate the test statistic:
t = (119.4 - 121.2) / sqrt((25/16) + (25/16))
= -1.8 / sqrt(3.125 + 3.125)
= -1.8 / sqrt(6.25)
= -1.8 / 2.5
= -0.72
Next, we determine the relevant critical point(s) for the t-test. Since the sample size is small (n1 = n2 = 16), we refer to the t-distribution with degrees of freedom equal to n1 + n2 - 2 = 30 - 2 = 28. Using a significance level (α) of 0.05, the critical value for a two-tailed test is approximately ±2.048.
Since the absolute value of the test statistic (0.72) is less than the critical value (2.048), we fail to reject the null hypothesis.
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which of the following bit arrays below is the correct 4-bit combination for the decimal number 9?
The correct 4-bit combination for the decimal number 9 is 1001. To explain it in a long answer, we need to understand binary representation. In binary, each digit can either be 0 or 1, and the value of the digit depends on its position.
The rightmost digit represents the value 2^0 (which is 1), the next digit to the left represents the value 2^1 (which is 2), the next represents 2^2 (which is 4), and so on. To convert decimal number 9 to binary, we can start by finding the highest power of 2 that is less than or equal to 9, which is 2^3 (which is 8). We can subtract 8 from 9, and the remainder is 1. This means the leftmost digit in the binary representation is 1.
We repeat the same process with the remainder, which is 1, and find the highest power of 2 that is less than or equal to 1, which is 2^0 (which is 1). We subtract 1 from 1, and the remainder is 0. This means the rightmost digit in the binary representation is 0. Thus, the binary representation of decimal number 9 is 1001, which is the correct 4-bit combination.
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Write a quadratic function f whose zeros 3 are and -8.
Answer: y=x²+5x−24
Step-by-step explanation:
Fill in this table as you work through the lesson. You may also use the glossary to help you.
absolute value
additive inverse
horizontal
integer
vertical
zero pair
the ____ a number is from 0 on a number line
The ____ of a number
_______ left and right
A ____ number or its opposite
Straight _____ and down
a set of two opposite numbers that equal ____ when combined
All the correct statements are,
⇒ the absolute value a number is from 0 on a number line
⇒ The additive inverse of a number
⇒ Horizontal left and right
⇒ An integer number or its opposite
⇒ Straight vertical and down
⇒ A set of two opposite numbers that equal zero when combined.
We have to given that;
To fill the blanks in all the statement.
Hence, We get;
We know that;
⇒ The absolute value a number is from 0 on a number line
⇒ The additive inverse of a number
⇒ Horizontal left and right
⇒ An integer number or its opposite
⇒ Straight vertical and down
⇒ A set of two opposite numbers that equal zero when combined.
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dylan says he has a polyhedron with 8 faces, 7 vertices and 10 edges. dylan has made a mistake, two of his values are correct, state the possible correct number of faces, vertices and edges.
Answer:
5 faces4 vertices13 edgesStep-by-step explanation:
Given two of three numbers correct, you want to find the correct value for the third number of 8 faces, 7 vertices, and 10 edges.
Euler's formulaThe relation between faces, vertices, and edges is ...
F + V = E + 2
The given numbers are off by 3:
8 + 7 = 15 ≠ 12 = 10 + 2
ApplicationWe can decrease the numbers of Faces or Vertices by 3, or we can increase the number of Edges by 3.
The numbers will be correct if we change to ...
5 faces, or4 vertices, or13 edges#95141404393
suppose it has been determined that the probability is 0.8 that a rat injected with cancerous cells will live. if 45 rats are injected, how many would be expected to die?
We can expect approximately 9 rats to die after being injected with cancerous cells based on probability.
To answer your question, we will use the given probability and the number of rats injected to find the expected number of rats that would die.
1. The probability that a rat injected with cancerous cells will live is 0.8.
2. Therefore, the probability that a rat will die is 1 - 0.8 = 0.2 (since the sum of probabilities of all possible outcomes should be equal to 1).
3. We have 45 rats injected with cancerous cells.
4. To find the expected number of rats that would die, multiply the total number of rats by the probability of a rat dying: 45 rats * 0.2 = 9 rats.
So, we can expect approximately 9 rats to die after being injected with cancerous cells.
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(q77) For what constant k is f(x) = ke-^(3x+2) a probability density function on [0 1]?
The probability density function on [0 1] is 36, the correct option is C.
We are given that;
Function f(x) = ke-^(3x+2)
Now,
The function f(x) = ke^(-3x+2) is a probability density function on [0 1] if and only if the integral of f(x) from 0 to 1 is equal to 1.
∫(0 to 1) ke^(-3x+2) dx = -1/3ke^(-3x+2) from 0 to 1 = -1/3ke^-1 + 1/3ke^2
For f(x) to be a probability density function, the above expression must be equal to 1. Therefore,
-1/3ke^-1 + 1/3ke^2 = 1
Solving for k gives us k = 36
Therefore, by probability the answer will be 36.
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Unit 5 progress check: mcq part a ap calculus ab Let f be the function given by f(x)=5cos2(x2)+ln(x+1)−3. The derivative of f is given by f′(x)=−5cos(x2)sin(x2)+1x+1. What value of c satisfies the conclusion of the Mean Value Theorem applied to f on the interval [1,4] ?
By the Mean Value Theorem, there exists a value c in the interval [1,4] such that f'(c) is equal to the average rate of change of f on the interval [1,4], which is (f(4) - f(1))/(4-1).
We can start by computing f(4) and f(1):
f(4) = 5cos(2(4^2)) + ln(4+1) - 3 = -0.841 + 1.609 - 3 = -1.232
f(1) = 5cos(2(1^2)) + ln(1+1) - 3 = 2.531 - 0.693 - 3 = -1.162
Then, we can compute the average rate of change:
(f(4) - f(1))/(4-1) = (-1.232 - (-1.162))/3 = -0.023
To satisfy the conclusion of the Mean Value Theorem, we need to find a value c in the interval [1,4] such that f'(c) = -0.023. From the given expression for f'(x), we can see that there is no value of c that satisfies this equation, since f'(x) can never be negative. Therefore, there is no value of c that satisfies the conclusion of the Mean Value Theorem applied to f on the interval [1,4].
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Let R be the set of real numbers, C = (0, 10], D = (9, 15], E = {1, 2, 3} and F = (7, 10). Find: (i) (CUD-E) specified using set builder notation without any words. (ii) (CE) specified using interval notation and set operations concisely without any words. (iii) (CDF) specified using the most concise notation. (3 marks) (b) Use element argument method to prove that if A and B are sets such that P(A) ≤ P(B), then A ≤ B, where P(A) and P(B) are power sets of A and B respectively. You must state your reasons clearly for every statement in your proof.
(i) (CUD-E) specified using set builder notation:
(CUD-E) = {x ∈ R | (x > 0 ∧ x ≤ 10) ∨ (x > 9 ∧ x ≤ 15) ∧ x ∉ {1, 2, 3}}
(ii) (CE) specified using interval notation and set operations concisely:
(CE) = (0, 10] ∩ {1, 2, 3} = {1, 2, 3}
(iii) (CDF) specified using the most concise notation:
(CDF) = (C ∩ D) ∩ F
(b) Proof using the element argument method:
Given: A and B are sets such that P(A) ≤ P(B).
To prove: A ≤ B.
Proof:
1. Let x be an arbitrary element in A.
2. Since x is in A, by definition, x is a subset of A. Hence, x ⊆ A.
3. Since x ⊆ A and A ≤ B, by the definition of ≤, x ⊆ B.
4. Therefore, x is a subset of B. Hence, x ∈ P(B), where P(B) is the power set of B.
5. Since x ∈ P(B), by definition, x is a subset of B. Hence, x ⊆ B.
6. Since x is an arbitrary element in A and x ⊆ B, by definition, A ≤ B.
7. Therefore, if P(A) ≤ P(B), then A ≤ B.
In this proof, we used the fact that if x is an element of A, then x is a subset of A. Also, if x is a subset of A and A ≤ B, then x is a subset of B. These properties are based on the definitions of subsets and the order relation between sets.
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The volume of a sphere with a diameter of 6cm, rounded to the nearest tenth
Answer:
113.1 cm³
Step-by-step explanation:
diameter = 2 X radius
Volume of sphere = (4/3) X π X r ³
= (4/3) π (3)³
= 36π
= 113.1 cm³ to nearest tenth
Find the exact length of the curve.x = 5 + 12t2, y = 1 + 8t3, 0 ≤ t ≤ 2
The exact length of the curve is 8/3 (5sqrt(26) - 1).
What is the exact length of the curve x = 5 + 12t2, y = 1 + 8t3, 0 ≤ t ≤ 2?To find the length of the curve, we can use the arc length formula:
L = ∫[tex][a,b]sqrt(dx/dt)^2 + (dy/dt)^2 dt[/tex]
where a and b are the starting and ending values of the parameter t, and dx/dt and dy/dt are the derivatives of x and y with respect to t, respectively.
Plugging in the given equations, we get:
[tex]dx/dt = 24t[/tex]
[tex]dy/dt = 24t^2[/tex]
Therefore,
[tex](sqrt(dx/dt)^2 + (dy/dt)^2) = sqrt((24t)^2 + (24t^2)^2) = sqrt(576t^2 + 576t^4)[/tex]
Substituting these expressions into the arc length formula, we get:
L = ∫[tex][0,2]sqrt(576t^2 + 576t^4) dt[/tex]
We can factor out 576t^2 from the square root:
L = ∫[tex][0,2]sqrt(576t^2(1 + t^2)) dt[/tex]
And then simplify the expression inside the square root:
L = ∫[tex][0,2]24t sqrt(1 + t^2) dt[/tex]
This integral can be evaluated using the substitution[tex]u = 1 + t^2, du/dt = 2t, dt = du/2t:[/tex]
L = ∫[tex][1,5]12 sqrt(u) du[/tex]
Now we can use the power rule of integration to evaluate this integral:
[tex]L = [8/3 u^(3/2)]_1^5 = 8/3 (5sqrt(26) - 1)[/tex]
Therefore, the exact length of the curve is 8/3 (5sqrt(26) - 1).
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The probabilities of hitting the next bat are Mitchell = 5/12 and Travis= 9/20
How to determine the probabilities of hitting the next batFrom the question, we have the following parameters that can be used in our computation:
Mitchell hits 5 out of 12 times
Travis hits 9 out of 20 times
The probabilities of hitting the next bat is calculated as
P(Hit) = Hit/Total number of times
using the above as a guide, we have the following:
P(Mitchell Hit) = 5/12
P(Travis Hit) = 9/20
Hence, the probabilities of hitting the next bat are 5/12 and 9/20
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DS
Plans for a new storage unit are
shown. If 0.125 in = 1 ft, what is the area
of the new unit?
E
Answer:
.125 = 3.25/l
.125l = 3.25, so l = 26 feet
.125 = 2.5/w
.125w = 2.5, so w = 20 feet
A = lw = (26 feet)(20 feet)
= 520 square feet
The correct answer is A.
Pets Survey
Pets No Pets Total
6th grade
28 23
7th grade 20 29
8th grade 12
Total 60
How many 7th graders were surveyed?
22
74
51
49
34
134
aider moi svp, merci.
Answer:
Step-by-step explanation:
a= 3(x+11)
B=9(x+8)
C=5(x+5)
D=3(3x+2)
write each of the following polynomials as a product of irreducible polynomials over the given field. (a) 2x3 x2 2 over f3 (d) x4 x3 2x2 x 2 over f3
A quartic polynomials is more complicated than quadratic or cubic polynomials. There are no obvious linear or quadratic factors. It is possible that the polynomial x^4 + x^3 + 2x^2 + x + 2 is irreducible over F3.
The polynomial 2x³ + 2x² + 2 over the field F₃, we first notice that we can factor out a 2 from all three terms to obtain:
2(x³ + x² + 1)
Now we need to factor the polynomial x³ + x² + 1 over F₃. One way to do this is to simply plug in all possible values for x (which are 0, 1, and 2 in F₃) and see if any of them result in a zero polynomial. We find that none of them do, so we know that x³ + x² + 1 is irreducible over F₃.
Therefore, our final factorization of 2x³ + 2x² + 2 over F₃ is:
2(x³ + x² + 1)
(b) To factor the polynomial x⁴ + x³ + 2x² + x + 2 over F₃, we can start by plugging in all possible values for x and checking if any of them result in a zero polynomial. Doing so, we find that x = 1 is a root of the polynomial, which means that x - 1 is a factor. Using polynomial long division or synthetic division, we can divide x⁴ + x³ + 2x² + x + 2 by x - 1 to obtain:
x⁴ + x³ + 2x² + x + 2 = (x - 1)(x³ + 2x² + 4x + 2)
Now we need to factor the cubic polynomial x³ + 2x² + 4x + 2 over F₃. Again, we can plug in all possible values for x and check for roots, but we won't find any in this case. However, we can use the fact that the sum of the coefficients of the polynomial is zero (1 + 2 + 4 + 2 = 0 in F₃) to infer that x = 1 is a root mod 3, and therefore x - 1 is a factor. Dividing x³ + 2x² + 4x + 2 by x - 1 using polynomial long division or synthetic division, we obtain:
x³ + 2x² + 4x + 2 = (x - 1)(x² + 3x + 2)
Now we need to factor the quadratic polynomial x² + 3x + 2 over F₃. We can do this by factoring it as (x + 1)(x + 2), since (x + 1)(x + 2) = x² + 3x + 2 mod 3.
Therefore, our final factorization of x⁴ + x³ + 2x² + x + 2 over F₃ is:
(x - 1)(x + 1)(x + 2)²
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in a hypothesis testing context, before examining the data, one should a. compute the p-value for the test. b. decide whether or not to reject the null hypothesis. c. decided whether the alternative hypothesis is one-sided or two-sided. d. all of the above.
In a hypothesis testing context, before examining the data, one should typically decide whether the alternative hypothesis is one-sided or two-sided. This decision is based on the specific research question and the expected direction of the effect being tested.
It helps determine the appropriate statistical test and the formulation of the null and alternative hypotheses.
The computation of the p-value and the decision of whether or not to reject the null hypothesis are made after examining the data and conducting the statistical analysis. The p-value is a measure of the strength of the evidence against the null hypothesis, and it is compared to a predetermined significance level to make a decision. If the p-value is below the significance level, the null hypothesis is typically rejected in favor of the alternative hypothesis.
Therefore, the correct answer is (c) decided whether the alternative hypothesis is one-sided or two-sided.
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this is due today im stuck on the last question
The association in this graph can best be described as C. Negative linear.
What is a negative linear association?A negative linear association is one that moves from the left to the right. In this kind of association, the predictor increases while the response decreases. The linear nature of this association is seen in the straight line formed from the plot.
A positive linear association would fall from the right towards the left side and a non-linear association will form a curve. So, the association in the table is negative linear.
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Given that
x
= 7.7 m and
θ
= 36°, work out BC rounded to 3 SF.
Answer:
Avg BC= 10.873 m
Step-by-step explanation:
See a picture
EXPLAIN PLEASEEEE, I need help
The value of angle x in the kite is 243 degrees.
How to find angle in a kite?The diagram above is a kite. The sum of angle in a kite is 360 degrees. Therefore, a kite kites have two sets of equivalent adjacent sides and one set of congruent opposite angles.
Therefore,
360 - 318 + 84 + 2y = 360
where
y are the inner congruent angles
Therefore,
42 + 84 + 2y = 360
2y = 360 - 126
2y = 234
divide both sides of the equation by 2
y = 234 /2
y = 117 degrees
Therefore,
x = 360 - 117
x = 243 degrees
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find a function r(t) for the line passing through the points p(8,3,3) and q(6,8,7)
The function r(t) for the line passing through the points p(8,3,3) and q(6,8,7) is r(t) = <8-2t, 3+5t, 3+4t>
We can use the vector form of the equation of a line to find a function r(t) for the line passing through the points p(8,3,3) and q(6,8,7).
Let's first find the direction vector of the line by subtracting the coordinates of the two points:
q - p = <6-8, 8-3, 7-3> = <-2, 5, 4>
Now, we can write the vector equation of the line in terms of a parameter t as:
r(t) = p + t(q - p)
Substituting the values of p and q, we get:
r(t) = <8, 3, 3> + t<-2, 5, 4>
Expanding, we get:
r(t) = <8-2t, 3+5t, 3+4t>
Therefore, the function r(t) for the line passing through the points p(8,3,3) and q(6,8,7) is:
r(t) = <8-2t, 3+5t, 3+4t>
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Los costos de fabricación de maquetas se modelan a la siguiente función. C(x) = 10 + 2x. El fabricante estima que el precio de venta en soles de cada maqueta viene dado por: P(x) = 20 6x2 800 ¿Qué cantidad de maquetas debe producir?
Models should be produced of the function C(x) = 10 + 2 x is 329.4 .
Cost of manufacturing is
C(x) = 10 + 2 x
Sale price in soles of each model is
P(x) = 20 - [tex]\frac{6x^{2} }{800}[/tex]
U(x) is the utility function
U(x) = x P(x) - C(x)
U(x) = x (20 - [tex]\frac{6x^{2} }{800}[/tex] ) - (10 +2x)
U(x) = 20x - [tex]\frac{6x^{3} }{800}[/tex] - 10 - 2x
U(x) = 18x - [tex]\frac{6x^{3} }{800}[/tex] - 10
U'(x) = 18 - 18x²/800
For maximum model U'(x) = 0
18 - 18x²/800 = 0
18x²/800 = 18
x² = 800
x = √800
x = 20√2
U(x) = 18(20√2 ) - [tex]\frac{6(20\sqrt{2} )^{2} }{800}[/tex] - 10
U(x) = 329.5
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The question is in Spanish question in English :
The manufacturing costs of models are modeled to the following function. C(x) = 10 + 2x. The manufacturer estimates that the sale price in soles of each model is given by: P(x) = 20- 6x2/800 How many models should be produced?
Pls help!!!!!!!! 50 POINTS !!!!! Divide
[tex]6\sqrt{5} \ cis(\frac{11\pi}{6}) \div 3\sqrt{6} \ cis (\frac{\pi}{2} )[/tex] can be expressed in polar form as[tex]6\sqrt{5} \ cis(\frac{11\pi}{6}) \div 3\sqrt{6} \ cis (\frac{\pi}{2} )=\underline{\frac{\sqrt{30} }{3} } \ cis\ (\underline{\frac{4\pi}{3}})[/tex] . Therefore the values to be dragged in the box are [tex]\frac{\sqrt{30} }{3}[/tex] and [tex]\frac{4\pi}{3}[/tex].
We have to express in polar form, polar form of complex number:
[tex]r(cos\theta+isin\theta) \rightarrow rcis\theta[/tex]
where, r = modulus of complex number
[tex]\theta[/tex] = argument of complex number
The division of two complex number, [tex]z=[/tex] [tex]r_{1} cis \theta_{1}[/tex] and [tex]x=[/tex] [tex]r_{2} cis \theta_{2}[/tex]
[tex]\frac{z}{x} = \frac{r_{1} }{r_{2} }\ cis(\theta_{1}- \theta_{2})[/tex]
Similarly, let a = [tex]6\sqrt{5} \ cis(\frac{11\pi}{6})[/tex]
b = [tex]3\sqrt{6} \ cis(\frac{pi}{2})[/tex]
[tex]\frac{a}{b}= \frac{6\sqrt{5} }{3\sqrt{6} } \ cis (\frac{11\pi }{6}- \frac{\pi}{2} )[/tex]
[tex]= \frac{{\sqrt{2}}\times\sqrt{2}\times\sqrt{5} }{\sqrt{2} \times\sqrt{3} } \ cis(\frac{11\pi-3\pi}{6} )[/tex]
[tex]=\sqrt{\frac{10}{3} }\ cis\ \frac{8\pi}{6}[/tex]
[tex]\frac{a}{b}= \sqrt{\frac{10}{3} } \ cis\ \frac{4\pi}{3}[/tex]
It can also be written as, [tex]\frac{a}{b}= \frac{\sqrt{30} }{3} \ cis\ \frac{4\pi}{3}[/tex]
⇒ [tex]6\sqrt{5} \ cis(\frac{11\pi}{6}) \div 3\sqrt{6} \ cis (\frac{\pi}{2} )= \frac{\sqrt{30} }{3} \ cis\ \frac{4\pi}{3}[/tex]
Comparing it with the question we get:
[tex]6\sqrt{5} \ cis(\frac{11\pi}{6}) \div 3\sqrt{6} \ cis (\frac{\pi}{2} )=\underline{\frac{\sqrt{30} }{3} } \ cis\ (\underline{\frac{4\pi}{3}})[/tex]
Therefore, the first blank is [tex]\frac{\sqrt{30} }{3}[/tex] and the second blank is [tex]\frac{4\pi}{3}[/tex].
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standard pair of six-sided dice is rolled. what is the probability of rolling a sum greater than 5 ?
The probability of rolling a sum greater than 5 when rolling a pair of standard six-sided dice is 5/6 or approximately 0.833.
To determine the probability of rolling a sum greater than 5, we can first find the total number of possible outcomes when rolling two dice, which is 36 (6 possible outcomes for each of the 6 sides on the first die). We can then count the number of outcomes where the sum of the two dice is greater than 5, which includes the outcomes (2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6), (4,2), (4,3), (4,4), (4,5), (4,6), (5,2), (5,3), (5,4), (5,5), (5,6), (6,2), (6,3), (6,4), (6,5), and (6,6). There are 21 such outcomes, so the probability of rolling a sum greater than 5 is 21/36, which simplifies to 5/6 or approximately 0.833.
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kyle is tossing bean bags at a target. so far, he has had 22 hits and 14 misses. what is the experimental probability that kyle's next toss will be a hit?
The experimental probability that Sue will hit the bullseye on her next toss is 2/7.
We have,
The proportion of outcomes where a specific event occurs in all trials, not in a hypothetical sample space but in a real experiment, is known as the empirical probability, relative frequency, or experimental probability of an event.
Here, we have
Given: Sue is playing darts. So far, she has hit the bullseye 4 times and missed the bullseye 10 times.
We have to find the experimental probability that Sue will hit the bullseye on her next toss.
experimental probability = 4/(4+10) = 4/14 = 2/7
The next toss = P = 2/7
Hence, the experimental probability that Sue will hit the bullseye on her next toss is 2/7.
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complete question:
Sue is playing darts. So far, she has hit the bullseye 4 times and missed the bullseye 10 times. What is the experimental probability that Sue will hit the bullseye on her next toss?