Answer:
Step-by-step explanation:
The values that are not equivalent to 72% are:
C. 3 72 / 100 - 3
D. 36/50
F. 1 - 0.28
I need help with this as soon as possible and shown work as well
Answer: EF = 6.5 FG = 5.0
Step-by-step explanation:
Since this is not a right triangle, you must use Law of Sin or Law of Cos
They have given enough info for law of sin : [tex]\frac{a}{sin A} =\frac{b}{sinB}[/tex]
The side of the triangle is related to the angle across from it.
[tex]\frac{a}{sin A} =\frac{b}{sinB}[/tex] >formula
[tex]\frac{FG}{sin E} =\frac{EG}{sinF}[/tex] >equation, substitute
[tex]\frac{FG}{sin 39} =\frac{7.9}{sin86}[/tex] >multiply both sides by sin 39
[tex]FG =\frac{7.9}{sin86}sin39[/tex] >plug in calc
FG = 5.0
<G = 180 - 86 - 39 >triangle rule
<G = 55
[tex]\frac{a}{sin A} =\frac{b}{sinB}[/tex] >formula
[tex]\frac{EF}{sin G} =\frac{EG}{sinF}[/tex] >equation, substitute
[tex]\frac{EF}{sin 55} =\frac{7.9}{sin86}[/tex] >multiply both sides by sin 55
[tex]EF =\frac{7.9}{sin86}sin55[/tex] >plug in calc
EF = 6.5
Consider the Quadratic function f(x)=2x 2−13x−24. Its vertex is (______ , ______) its largest z-intercept is z= ____
its y-intercept is y= _____
For the given quadratic function f(x) = 2x² - 13x - 24 its Vertex = (13/4, -25/8), Largest z-intercept = -24, Y-intercept = -24.
The standard form of a quadratic function is:
f(x) = ax² + bx + c where a, b, and c are constants.
To calculate the vertex, we need to use the formula:
h = -b/2a where a = 2 and b = -13
therefore
h = -b/2a
= -(-13)/2(2)
= 13/4
To calculate the value of f(h), we need to substitute
h = 13/4 in f(x).f(x) = 2x² - 13x - 24
f(h) = 2(h)² - 13(h) - 24
= 2(13/4)² - 13(13/4) - 24
= -25/8
The vertex is at (h, k) = (13/4, -25/8).
To calculate the largest z-intercept, we need to set
x = 0 in f(x)
z = 2x² - 13x - 24z
= 2(0)² - 13(0) - 24z
= -24
The largest z-intercept is z = -24.
To calculate the y-intercept, we need to set
x = 0 in f(x).y = 2x² - 13x - 24y
= 2(0)² - 13(0) - 24y
= -24
The y-intercept is y = -24.
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Use either indirect proof or conditional proof to derive the conclusions of the following symbolized argument.
1. (x)Ax ≡ (∃x)(Bx • Cx)
2. (x)(Cx ⊃ Bx) / (x)Ax ≡ (∃x)Cx
Using either indirect proof or conditional proof, it is derived the conclusion is (x)Ax ≡ (∃x)Cx.
How to use indirect proof or conditional proof?To derive the conclusion of the given symbolized argument using either indirect proof or conditional proof, consider both approaches:
Indirect Proof:
Assume the negation of the desired conclusion: ¬((x)Ax ≡ (∃x)Cx)
Conditional Proof:
Assume the premise: (x)(Cx ⊃ Bx)
Now, proceed with the proof:
(x)Ax ≡ (∃x)(Bx • Cx) [Premise]
(x)(Cx ⊃ Bx) [Premise]
¬((x)Ax ≡ (∃x)Cx) [Assumption for Indirect Proof]
To derive a contradiction, assume the negation of (∃x)Cx, which is ∀x¬Cx:
∀x¬Cx [Assumption for Indirect Proof]
¬∃x Cx [Universal Instantiation from 4]
¬(Cx for some x) [Quantifier negation]
Cx ⊃ Bx [Universal Instantiation from 2]
¬Cx ∨ Bx [Material Implication from 7]
¬Cx [Disjunction Elimination from 8]
Now, derive a contradiction by combining the premises:
(x)Ax ≡ (∃x)(Bx • Cx) [Premise]
Ax ≡ (∃x)(Bx • Cx) [Universal Instantiation from 10]
Ax ⊃ (∃x)(Bx • Cx) [Material Equivalence from 11]
¬Ax ∨ (∃x)(Bx • Cx) [Material Implication from 12]
From premises 9 and 13, both ¬Cx and ¬Ax ∨ (∃x)(Bx • Cx). Applying disjunction introduction:
¬Ax ∨ ¬Cx [Disjunction Introduction from 9 and 13]
However, this contradicts the assumption ¬((x)Ax ≡ (∃x)Cx). Therefore, our initial assumption of ¬((x)Ax ≡ (∃x)Cx) must be false, and the conclusion holds:
(x)Ax ≡ (∃x)Cx
Therefore, using either indirect proof or conditional proof, we have derived the conclusion.
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The proof uses a conditional proof, which assumes the truth of (x)Ax and proves that (∃x)Cx is true, which means that (x)Ax ≡ (∃x)Cx is true.
Indirect proof is a proof technique that involves assuming the negation of the argument's conclusion and attempting to demonstrate that the negation is a contradiction.
Conditional proof, on the other hand, is a proof technique that involves establishing a conditional statement and then proving the antecedent or the consequent of the conditional.
We can use conditional proof to derive the conclusion of the argument.
The given premises are: 1. (x)Ax ≡ (∃x)(Bx • Cx)
2. (x)(Cx ⊃ Bx) / (x)Ax ≡ (∃x)Cx
We want to prove that (x)Ax ≡ (∃x)Cx. We can do so using a conditional proof by assuming (x)Ax and proving (∃x)Cx as follows:
3. Assume (x)Ax.
4. From (x)Ax ≡ (∃x)(Bx • Cx), we can infer (∃x)(Bx • Cx).
5. From (∃x)(Bx • Cx), we can infer (Ba • Ca) for some a.
6. From (x)(Cx ⊃ Bx), we can infer Ca ⊃ Ba.
7. From Ca ⊃ Ba and Ba • Ca, we can infer Ca.
8. From Ca, we can infer (∃x)Cx.
9. From (x)Ax, we can infer (x)Ax ≡ (∃x)Cx by conditional proof using steps 3-8.The conclusion is (x)Ax ≡ (∃x)Cx.
The proof uses a conditional proof, which assumes the truth of (x)Ax and proves that (∃x)Cx is true, which means that (x)Ax ≡ (∃x)Cx is true.
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2x + x+x+2yX3yXy pleas help me stuck on this question
The simplified expression is 4x + 6y^3.
To simplify the expression 2x + x + x + 2y × 3y × y, we can apply the order of operations, which is also known as the PEMDAS rule (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Let's break it down step by step:
1. Simplify the expression within the parentheses: 2y × 3y × y.
This can be rewritten as 2y * 3y * y = 2 * 3 * y * y * y = 6y^3.
2. Combine like terms by adding or subtracting coefficients of the same variable:
2x + x + x = 4x.
3. Now we can rewrite the simplified expression by substituting the values we found:
4x + 6y^3.
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The standard or typical average difference between the mean number of seats in the 559 full-service restaurants in delaware (µ = 99.2) and one randomly selected full-service restaurant in delaware is:
The standard deviation of the sampling distribution of the sample mean would be approximately 2.8284
To determine the standard deviation of the sampling distribution of the sample mean, we will use the formula;
σ_mean = σ / √n
where σ is the standard deviation of the population that is 20 and n is the sample size (n = 50).
So,
σ_mean = 20 / √50 = 20 / 7.07
σ_mean = 2.8284
The standard deviation of the sampling distribution of the sample mean is approximately 2.8284 it refers to that the sample mean would typically deviate from the population mean by about 2.8284, assuming that the sample is selected randomly from the population.
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The complete question is;
Another application of the sampling distribution of the sample mean Suppose that, out of a total of 559 full-service restaurants in Delaware, the number of seats per restaurant is normally distributed with mean mu = 99.2 and standard deviation sigma = 20. The Delaware tourism board selects a simple random sample of 50 full-service restaurants located within the state and determines the mean number of seats per restaurant for the sample. The standard deviation of the sampling distribution of the sample mean is Use the tool below to answer the question that follows. There is a.25 probability that the sample mean is less than
(a) Find the solutions of the recurrence relation an ·an-1-12an-2 = 0, n ≥ 2, satisfying the initial conditions ao = 1,a₁ = 1
(b) Find the solutions of the recurrence relation a_n = 10a_(n-1) - 25a_(n-2) + 32, n ≥ 2, satisfying the initial conditions ao = 3, a₁ = 7. (c) Find all solutions of the recurrence relation a_n + a_(n-1) - 12a_(n-2) = 2^(n) (d) Find all the solutions of the recurrence relation a_n = 4a_(n-1) - 4a_(n-2)
(e) Find all the solutions of the recurrence relation a_n = 2a_(n-1) - a_(n-2) + 2
(f) Find all the solutions of the recurrence relation a_n - 2a_(n-1) - 3a_(n-2) = 3^(n)
Solutions for the given recurrence relations:
(a) Solutions for an ·an-1-12an-2 = 0, n ≥ 2, with ao = 1 and a₁ = 1.
(b) Solutions for a_n = 10a_(n-1) - 25a_(n-2) + 32, n ≥ 2, with ao = 3 and a₁ = 7.
(c) Solutions for a_n + a_(n-1) - 12a_(n-2) = 2^(n).
(d) Solutions for a_n = 4a_(n-1) - 4a_(n-2).
(e) Solutions for a_n = 2a_(n-1) - a_(n-2) + 2.
(f) Solutions for a_n - 2a_(n-1) - 3a_(n-2) = 3^(n).
In (a), the recurrence relation is an ·an-1-12an-2 = 0, and the initial conditions are ao = 1 and a₁ = 1. Solving this relation involves identifying the values of an that make the equation true.
In (b), the recurrence relation is a_n = 10a_(n-1) - 25a_(n-2) + 32, and the initial conditions are ao = 3 and a₁ = 7. Similar to (a), finding solutions involves identifying the values of a_n that satisfy the given relation.
In (c), the recurrence relation is a_n + a_(n-1) - 12a_(n-2) = 2^(n). Here, the task is to find all solutions of a_n that satisfy the relation for each value of n.
In (d), the recurrence relation is a_n = 4a_(n-1) - 4a_(n-2). Solving this relation entails determining the values of a_n that make the equation true.
In (e), the recurrence relation is a_n = 2a_(n-1) - a_(n-2) + 2. The goal is to find all solutions of a_n that satisfy the relation for each value of n.
In (f), the recurrence relation is a_n - 2a_(n-1) - 3a_(n-2) = 3^(n). Solving this relation involves finding all values of a_n that satisfy the equation.
Solving recurrence relations is an essential task in understanding the behavior and patterns within a sequence of numbers. It requires analyzing the relationship between terms and finding a general expression or formula that describes the sequence. By utilizing the given initial conditions, the solutions to the recurrence relations can be determined, providing insights into the values of the sequence at different positions.
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Consider a finite field F with q elements. This means that F has q- 1 non-zero elements, and hence the F vector space Fn has (q-1)" non-zero vectors. How many unordered bases for Fn are there? (Consider different orderings of the same set of vectors to be different bases.)
Given, a finite field F with q elements. The number of non-zero elements is q - 1.Now, we have to find the number of unordered bases for Fn. Here, n is a natural number. The answer would be (q-1)^n.
To solve this question, we have to use the following formula for finding the number of bases of a vector space:
Let V be a vector space of dimension n. Then there are(q^n - 1)(q^(n-1) - 1)...(q - 1)unordered bases of V over F.
Using this formula, we can find the number of unordered bases of Fn over F.
So, applying the formula in this case, we get the following answer:
Number of unordered bases of Fn over F= (q^n - 1)(q^(n-1) - 1)...(q - 1)
Where n is the dimension of vector space, which is n = dim(Fn) = n elements of the basis for Fn.
Therefore, the number of unordered bases for Fn is(q^(n) - 1)(q^(n-1) - 1)...(q - 1) = (q^n - 1) (q^(n-1) - 1) ... (q^1 - 1)
Now, Fn has q non-zero elements, and hence (q-1) non-zero vectors, since there are n elements in a basis, there are (q-1) elements not in that basis.
Therefore, there are (q-1) choices for the first element, (q-1) choices for the second element, and so on. And the total number of bases for Fn is then given by:(q - 1)^(n) - 1
Hence, the number of unordered bases for Fn is given by(q^(n) - 1) (q^(n-1) - 1) ... (q^1 - 1)= (q-1)^n
Therefore, the answer is (q-1)^n.
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Use Simple Algorithm - Big M Method to solve the following questions.
(a)
Max Z =3x1 + 2x2 + x3
Subject to
2x1 + x2 + x3 = 12
3x1 + 4x2 = 11 and x1 is unrestricted
x2 ≥ 0, x3 ≥ 0
(b)
Min Z = 2x1 + 3x2
Subject to
x1 + x2 ≥ 5
x1 + 2x2 ≥ 6
and x1 ≥ 0, x2 ≥ 0
Application of Simple Algorithm - Big M Method to solve linear programming problems with given constraints and objective functions.
(a) Maximize Z = 3x1 + 2x2 + x3 subject to 2x1 + x2 + x3 = 12, 3x1 + 4x2 = 11, x1 unrestricted, x2 ≥ 0, and x3 ≥ 0.Minimize Z = 2x1 + 3x2 subject to x1 + x2 ≥ 5, x1 + 2x2 ≥ 6, x1 ≥ 0, and x2 ≥ 0.The Simple Algorithm - Big M Method is a technique used to solve linear programming problems with both equality and inequality constraints.
In problem (a), we have a maximization problem with three variables (x1, x2, x3) and two equality constraints and non-negativity constraints.
The algorithm involves introducing slack variables, converting the problem into standard form, and using a Big M parameter to handle unrestricted variables.
The objective function is maximized by iteratively improving the solution until an optimal solution is reached.
In problem (b), we have a minimization problem with two variables (x1, x2) and two inequality constraints.
The procedure is similar, where surplus variables are introduced to convert the problem into standard form, and the Big M method is used to handle non-negativity constraints.
The objective function is minimized by following the steps of the algorithm.
By applying the Simple Algorithm - Big M Method to these problems, we can find the optimal solutions that satisfy the given constraints and optimize the objective function.
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Does the Law of Cosines apply to a right triangle? That is, does c²=a²+b²-2 a b cos C remain true when ∠ C is a right angle? Justify your answer.
The Law of Cosines does not apply to a right triangle when ∠C is a right angle. In a right triangle, the Pythagorean theorem is used instead to find the relationship between the sides.
The Law of Cosines states that in a triangle with sides of lengths a, b, and c, and angle C opposite the side of length c, the following equation holds: c² = a² + b² - 2ab cos(C)
This formula is used to find the length of one side of a triangle when the lengths of the other two sides and the included angle are known.
However, in a right triangle, one of the angles is 90 degrees, making it a special case. In a right triangle, the side opposite the right angle (the hypotenuse) is always the longest side, and its length can be found using the Pythagorean theorem:
c² = a² + b²
Since the angle C in a right triangle is 90 degrees, the term -2ab cos(C) becomes 0 in the Law of Cosines formula. Therefore, there is no need to use the Law of Cosines in a right triangle because the Pythagorean theorem directly relates the lengths of the sides.
In summary, the Law of Cosines is not applicable to a right triangle when ∠C is a right angle. Instead, the Pythagorean theorem should be used to find the length of the hypotenuse in a right triangle.
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Find the measure of each angle
The value of indicated angle 1 is 70⁰.
The value of indicated angle 2 is 20⁰.
The value of indicated angle 3 is 50⁰.
The value of indicated angle 4 is 110⁰.
What is the value of the missing angles?The value of the missing angles is calculated by applying the principle sum of angles in a triangle.
The value of indicated angle 2 is calculated as follows;
angle 2 = 20⁰ (alternate angles are equal)
The value of indicated angle 1 is calculated as follows;
angle 1 = 90 - ( angle 2) (complementary angles )
angle 1 = 90 - 20⁰
angle 1 = 70⁰
The value of indicated angle 4 is calculated as follows;
angle 2 + angle 4 + 50 = 180 (sum of angles in a straight line )
angle 4 + 20 + 50 = 180
angle 4 = 180 - 70
angle 4 = 110⁰
The value of indicated angle 3 is calculated as follows;
angle 3 + 20 + angle 4 = 180 (sum of angles in a triangle )
angle 3 + 20 + 110 = 180
angle 3 = 180 - 130
angle 3 = 50⁰
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[3](6) Determine whether the following set of vectors is a basis. If it is not, explain why. a) S = {(6.-5). (6.4).(-5,4)} b) S = {(5.2,-3). (-10,-4, 6). (5,2,-3))
Set S is not a basis because it does not satisfy the requirements for linear independence and spanning the vector space.
For a set of vectors to be a basis, it must satisfy two conditions: linear independence and spanning the vector space.
a) Set S = {(6, -5), (6, 4), (-5, 4)}: To determine if this set is a basis, we need to check if the vectors are linearly independent and if they span the vector space. We can do this by forming a matrix with the vectors as columns and performing row reduction. If the row-reduced form has a pivot in each row, then the vectors are linearly independent.
Constructing the matrix [6 -5; 6 4; -5 4] and performing row reduction, we find that the row-reduced form has only two pivots, indicating that the vectors are linearly dependent. Therefore, set S is not a basis.
b) Set S = {(5, 2, -3), (-10, -4, 6), (5, 2, -3)}: Similar to the previous set, we need to check for linear independence and spanning the vector space. By forming the matrix [5 2 -3; -10 -4 6; 5 2 -3] and performing row reduction, we find that the row-reduced form has only two pivots, indicating linear dependence. Therefore, set S is not a basis.
In both cases, the sets of vectors fail to meet the criteria of linear independence. As a result, they cannot form a basis for the vector space.
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State the concept of closeness between the two curves u(t) and 2 same end points u(a) = 2(a) and (b) = 2(b)
The concept of closeness between the two curves u(t) and 2 is determined by the condition that they have the same end points u(a) = 2(a) and u(b) = 2(b).
When considering the concept of closeness between two curves, it is important to examine their behavior at the end points. In this case, we are comparing the curves u(t) and 2, and we have the condition that they share the same end points u(a) = 2(a) and u(b) = 2(b).
This condition implies that at the points a and b, the values of the curve u(t) are equal to the constant value 2 multiplied by the respective points a and b. Essentially, this means that the curve u(t) is directly proportional to the constant curve 2, with the proportionality factor being the respective points a and b.
In other words, the curve u(t) is a linear transformation of the curve 2, where the points a and b determine the scaling factor. This scaling factor determines how closely the curve u(t) follows the curve 2. If the scaling factor is close to 1, the two curves will closely align, indicating a high degree of closeness. Conversely, if the scaling factor deviates significantly from 1, the two curves will diverge, indicating a lower degree of closeness.
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Use a half-angle identity to find the exact value of each expression.
cos 90°
The exact value of cos(90°) using a half-angle identity, is 0.
The half-angle formula states that cos(θ/2) = ±√((1 + cosθ) / 2). By substituting θ = 180° into the half-angle formula, we can determine the exact value of cos(90°).
To find the exact value of cos(90°) using a half-angle identity, we can use the half-angle formula for cosine, which is cos(θ/2) = ±√((1 + cosθ) / 2).
Substituting θ = 180° into the half-angle formula, we have cos(90°) = cos(180°/2) = cos(90°) = ±√((1 + cos(180°)) / 2).
The value of cos(180°) is -1, so we can simplify the expression to cos(90°) = ±√((1 - 1) / 2) = ±√(0 / 2) = ±√0 = 0.
Therefore, the exact value of cos(90°) is 0.
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QUESTION 5 Which of the following statement is true in Z? x(x+y=0); xy(x+y=0); x(x+y=0); O None of these
None of these statements are true in Z (the set of integers). Let's analyze each statement:
1. x(x + y = 0): This equation is not well-formed; it appears to be missing an operator between x and (x + y). Assuming you meant x * (x + y) = 0, even so, this statement is not true in Z. For example, if x = 2 and y = -2, the equation becomes 2(2 - 2) = 0, which simplifies to 0 = 0, but this is not a true statement in Z.
2. xy(x + y = 0): Similarly, this equation is not well-formed. Assuming you meant x * y * (x + y) = 0, this statement is also not true in Z. For example, if x = 2 and y = -2, the equation becomes 2 * -2 * (2 - 2) = 0, which simplifies to 0 = 0, but again, this is not a true statement in Z.
3. x(x + y = 0): This equation is not well-formed either; it seems to be missing a closing parenthesis. Assuming you meant x * (x + y) = 0, this statement is not universally true in Z. It is true when x = 0, as any number multiplied by zero is zero. However, when x ≠ 0, the equation is not satisfied in Z. For example, if x = 2 and y = -2, the equation becomes 2 * (2 - 2) = 0, which simplifies to 0 = 0, but this is not true for all integers.
Therefore, none of the given statements are true in Z.
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Given a line x−2y+5=0, find its slope. A. −2 B. −1/2
C. 1/2 D. 2
Considering the definition of a line, the slope of the line x-2y+5=0 is 1/2.
Definition of linear equationA linear equation o line can be expressed in the form y = mx + b
where
x and y are coordinates of a point.m is the slope.b is the ordinate to the origin. The ordinate to the origin is the point where a line crosses the y-axis.Slope of the line x-2y+5=0In this case, the line is x-2y+5=0. Expressed in the form y = mx + b, you get:
x-2y=-5
-2y=-5-x
y= (-x-5)÷ (-2)
y= 1/2x +5/2
where:
the slope is 1/2.the ordinate to the origin is 5/2Finally, the slope of the line x-2y+5=0 is 1/2.
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Considering the definition of a line, the slope of the line x-2y+5=0 is 1/2.
A linear equation o line can be expressed in the form y = mx + b
where
x and y are coordinates of a point.
m is the slope.
b is the ordinate to the origin. The ordinate to the origin is the point where a line crosses the y-axis.
Slope of the line x-2y+5=0
In this case, the line is x-2y+5=0. Expressed in the form y = mx + b, you get:
x-2y=-5
-2y=-5-x
y= (-x-5)÷ (-2)
y= 1/2x +5/2
where:
the slope is 1/2.
the ordinate to the origin is 5/2
Finally, the slope of the line x-2y+5=0 is 1/2.
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Linear Algebra
Question about additive inverse of vector space
1. Determine whether the set R2 with the operations
(x1,y1) + (x2,y2) = (x1x2,y1y2)
and
c(x1,y1) = (cx1,cy1)
solution(1)
This set is not a vector space because Axiom 5(additive inverse) fails.
The additive identity is (1,1) and so (0,0) has no additive inverse.
Axioms 7 and 8 also fail.
- I understood about additive identity, but I couldn't understand why (0,0) has no additive inverse.
- is it possible to be additive inverse as (0,0)?
2. Let V be the set of all positive real numbers. Determine whether V is a vector space with the following operations.
x + y = xy
cx = xc
Solution(2)
It is vector space.
The zero vector is 1 and additive inverse of x is 1/x.
(additive inverse) x + 1/x = x(1/x) = 1
- I don't understand why additive inverse is 1/x.
please help me understanding this concept
If we choose 1/x as the additive inverse of x, their sum is:
x + 1/x = (x^2 + 1) / x = 1
which is the additive identity in this set.
The additive inverse of a vector (x, y) in this set is defined as another vector (a, b) such that their sum is the additive identity (1, 1):
(x, y) + (a, b) = (1, 1)
Substituting the definition of the addition operation, we get:
(xa, yb) = (1, 1)
This implies that xa = 1 and yb = 1. If x or y is zero, then there is no solution for a or b, respectively. So, the vector (0, 0) does not have an additive inverse in this set.
The additive inverse of a positive real number x is its reciprocal 1/x, because:
x + 1/x = (x * x + 1) / x = (x^2 + 1) / x
Since x is positive, x^2 is positive, and x^2 + 1 is greater than x, so (x^2 + 1) / x is greater than 1. Therefore, if we choose 1/x as the additive inverse of x, their sum is:
x + 1/x = (x^2 + 1) / x = 1
which is the additive identity in this set.
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Max Z = 5x1 + 6x2
Subject to: 17x1 + 8x2 ≤ 136
3x1 + 4x2 ≤ 36
x1 ≥ 0 and integer
x2 ≥ 0
A) x1 = 5, x2 = 4.63, Z = 52.78
B) x1 = 5, x2 = 5.25, Z = 56.5
C) x1 = 5, x2 = 5, Z = 55
D) x1 = 4, x2 = 6, Z = 56
The option B) yields the highest value for Z, which is 56.5. Therefore, the correct answer is B) x1 = 5, x2 = 5.25, Z = 56.5
To determine the correct answer, we can substitute each option into the objective function and check if the constraints are satisfied. Let's evaluate each option:
A) x1 = 5, x2 = 4.63, Z = 52.78
Checking the constraints:
17x1 + 8x2 = 17(5) + 8(4.63) = 85 + 37.04 = 122.04 ≤ 136 (constraint satisfied)
3x1 + 4x2 = 3(5) + 4(4.63) = 15 + 18.52 = 33.52 ≤ 36 (constraint satisfied)
B) x1 = 5, x2 = 5.25, Z = 56.5
Checking the constraints:
17x1 + 8x2 = 17(5) + 8(5.25) = 85 + 42 = 127 ≤ 136 (constraint satisfied)
3x1 + 4x2 = 3(5) + 4(5.25) = 15 + 21 = 36 ≤ 36 (constraint satisfied)
C) x1 = 5, x2 = 5, Z = 55
Checking the constraints:
17x1 + 8x2 = 17(5) + 8(5) = 85 + 40 = 125 ≤ 136 (constraint satisfied)
3x1 + 4x2 = 3(5) + 4(5) = 15 + 20 = 35 ≤ 36 (constraint satisfied)
D) x1 = 4, x2 = 6, Z = 56
Checking the constraints:
17x1 + 8x2 = 17(4) + 8(6) = 68 + 48 = 116 ≤ 136 (constraint satisfied)
3x1 + 4x2 = 3(4) + 4(6) = 12 + 24 = 36 ≤ 36 (constraint satisfied)
From the calculations above, we see that options B), C), and D) satisfy all the constraints. However, option B) yields the highest value for Z, which is 56.5. Therefore, the correct answer is: B) x1 = 5, x2 = 5.25, Z = 56.5.
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Which laws allow us to compute the value of lim x→2(x3− 2x2+x−7) ? Find the limit using these laws and the previous two exercises.
The limit of the function is given by:limx→2(x3−2x2+x−7)=0×5=0
To compute the value of limx→2(x3−2x2+x−7), we can use the following laws:
1. Direct substitution: This law states that we can substitute the value of the limit point directly into the function to evaluate the limit if the function is continuous at that point.2. Limit laws: There are several limit laws that we can use to evaluate limits. These include the limit laws for sums, products, quotients, powers, and composition.We will use these laws to evaluate the limit in the following way:
First, we can simplify the function as follows:x3−2x2+x−7=x2(x−2)+(x−2)=(x−2)(x2+1)
Using the limit laws for sums and products, we can rewrite the function as follows:
limx→2(x3−2x2+x−7)=limx→2(x−2)(x2+1)=limx→2(x−2)
limx→2(x2+1)
Using direct substitution, we can evaluate the limits of each factor as follows:
limx→2(x−2)=0limx→2(x2+1)=22+1=5
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the number √ 63 − 36 √ 3 can be expressed as x y √ 3 for some integers x and y. what is the value of xy ? a. −18 b. −6 c. 6 d. 18 e. 27
The value of xy is -54
To simplify the expression √63 − 36√3, we need to simplify each term separately and then subtract the results.
1. Simplify √63:
We can factorize 63 as 9 * 7. Taking the square root of each factor, we get √63 = √(9 * 7) = √9 * √7 = 3√7.
2. Simplify 36√3:
We can rewrite 36 as 6 * 6. Taking the square root of 6, we get √6. Therefore, 36√3 = 6√6 * √3 = 6√(6 * 3) = 6√18.
3. Subtract the simplified terms:
Now, we can substitute the simplified forms back into the original expression:
√63 − 36√3 = 3√7 − 6√18.
Since the terms involve different square roots (√7 and √18), we can't combine them directly. But we can simplify further by factoring the square root of 18.
4. Simplify √18:
We can factorize 18 as 9 * 2. Taking the square root of each factor, we get √18 = √(9 * 2) = √9 * √2 = 3√2.
Substituting this back into the expression, we have:
3√7 − 6√18 = 3√7 − 6 * 3√2 = 3√7 − 18√2.
5. Now, we can express the expression as x y√3:
Comparing the simplified expression with x y√3, we can see that x = 3, y = -18.
Therefore, the value of xy is 3 * -18 = -54.
So, the correct answer is not provided in the given options.
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In 6 521 253, the digit 6 has the value of 6 x . write your answer in numerals.
In 6 521 253, the digit 6 has the value of 6 x 1,000,000.
To determine the value of a digit in a number, we consider its position or place value. In the number 6 521 253, the digit 6 is located in the millions place. The value of a digit in the millions place is determined by multiplying the digit by the corresponding power of 10.
Since the millions place is the sixth place from the right, its corresponding power of 10 is 1,000,000 (10 to the power of 6). Therefore, to find the value of the digit 6, we multiply it by 1,000,000.
6 x 1,000,000 = 6,000,000
Hence, in the number 6 521 253, the digit 6 has a value of 6,000,000.
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12. Extend the meaning of a whole-number exponent. a n
= n factors a⋅a⋅a⋯a,
where a is any integer. Use this definition to find the following values. a. 2 4
b. (−3) 3
c. (−2) 4
d. (−5) 2
e. (−3) 5
f. (−2) 6
The result of the whole-number exponent expressions are
a. 16
b. -27
c. 16
d. 25
e. -243
f. 64
How to solve the expressionsUsing the definition of whole-number exponent, we can multiply the base integer by itself as many times as the exponent indicates.
For positive exponents, the result is a repeated multiplication of the base. For negative exponents, the result is the reciprocal of the repeated multiplication.
a. 2⁴ = 2 * 2 * 2 * 2 = 16
b. (-3)³ = (-3) * (-3) * (-3) = -27
c. (-2)⁴ = (-2) * (-2) * (-2) * (-2) = 16
d. (-5)² = (-5) * (-5) = 25
e. (-3)⁵ = (-3) * (-3) * (-3) * (-3) * (-3) = -243
f. (-2)⁶ = (-2) * (-2) * (-2) * (-2) * (-2) * (-2) = 64
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The values are 16, -27, 26, 25, -243, 64
Using the extended definition of a whole-number exponent, we can find the values as follows:
a. 2^4 = 2 × 2 × 2 × 2 = 16
b. (-3)^3 = (-3) × (-3) × (-3) = -27
c. (-2)^4 = (-2) × (-2) × (-2) × (-2) = 16
d. (-5)^2 = (-5) × (-5) = 25
e. (-3)^5 = (-3) × (-3) × (-3) × (-3) × (-3) = -243
f. (-2)^6 = (-2) × (-2) × (-2) × (-2) × (-2) × (-2) = 64
So the values are:
a. 2^4 = 16
b. (-3)^3 = -27
c. (-2)^4 = 16
d. (-5)^2 = 25
e. (-3)^5 = -243
f. (-2)^6 = 64
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What is the annual rate of interest if P400 is earned in three months on an investment of P20,000?
The annual rate of interest is 8%.
What is the annual rate?
Interest is the amount that is paid to an investor for the use of their funds. The interest that is paid is a function of amount invested, interest rate and the duration of the loan.
Interest = amount invested x interest rate x time
Annual rate = interest ÷ (amount invested x time)
= 400 ÷ (20,000 x 3/12) = 0.08 = 8%
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Use the second partial test to examine the relative extrema for function f(x,y)=x^2+3xy+y^3.
Using the Second Partial Test , the relative extrema for the function f(x, y) = x² + 3xy + y³ occur at the points (0, 0) and (9/4, -3/2).
How to Use the Second Partial Test?To examine the relative extrema for the function that is given as f(x, y) = x² + 3xy + y³, we would do the following explained below:
Compute the partial derivatives:
∂f/∂x = 2x + 3y
∂f/∂y = 3x + 3y²
Set the partial derivatives equal to zero and solve the system of equations accordingly:
2x + 3y = 0
3x + 3y² = 0
Simplifying the equations, we get:
x = -3y/2
x = -y²
Set the expressions for x equal to each other:
-y² = -3y/2
Solve the equation to get:
y = 0 or y = -3/2
Substituting x = -3y/2, we have:
For y = 0, x = 0
For y = -3/2, x = 9/4
Therefore, the relative extrema for the function f(x, y) = x² + 3xy + y³ occur at the points (0, 0) and (9/4, -3/2).
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<< <
1
WRITER
2
Use the inequality to answer Parts 1-3.
-3(x-2) ≤ =
Part 1: Solve the inequality. Leave answer in terms of a whole number or reduced improper fraction.
Part 2: Write a verbal statement describing the solution to the inequality.
Part 3: Verify your solution to the inequality using two elements of the solution set.
Use a word processing program or handwrite your responses to Parts 1-3. Turn in all three responses.
>
A
Part 1: The solution to the inequality -3(x - 2) ≤ 0 is x ≥ 2.
Part 2: The solution to the inequality is any value of x that is greater than or equal to 2.
Part 3: Verifying the solution, we substitute x = 2 and x = 3 into the original inequality and find that both values satisfy the inequality.
Part 1:
To solve the inequality -3(x - 2) ≤ 0, we need to isolate the variable x.
-3(x - 2) ≤ 0
Distribute the -3:
-3x + 6 ≤ 0
To isolate x, we'll subtract 6 from both sides:
-3x ≤ -6
Next, divide both sides by -3. Remember that when dividing or multiplying by a negative number, we flip the inequality sign:
x ≥ 2
Therefore, the solution to the inequality is x ≥ 2.
Part 2:
A verbal statement describing the solution to the inequality is: "The solution to the inequality is any value of x that is greater than or equal to 2."
Part 3:
To verify the solution, we can substitute two elements of the solution set into the original inequality and check if the inequality holds true.
Let's substitute x = 2 into the inequality:
-3(2 - 2) ≤ 0
-3(0) ≤ 0
0 ≤ 0
The inequality holds true.
Now, let's substitute x = 3 into the inequality:
-3(3 - 2) ≤ 0
-3(1) ≤ 0
-3 ≤ 0
Again, the inequality holds true.
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Fill in the blank by performing the indicated elementary row operation(s)
[2 0 -1|-7]
[1 -4 0| 3]
[-2 8 0|-0]
- 2R_{2} + R_{1}, R_{2} + R_{1}
?
The resulting matrix after performing the given elementary row operations is:
[2 0 -1|-7]
[0 4 -1|-1]
[0 8 -1|-0]
Performing the indicated elementary row operation(s), the given matrix can be transformed as follows:
[2 0 -1|-7]
[1 -4 0| 3]
[-2 8 0|-0]
2R₂ + R₁:
[2 0 -1|-7]
[0 4 -1|-1]
[-2 8 0|-0]
R₂ + R₁:
[2 0 -1|-7]
[0 4 -1|-1]
[0 8 -1|-0]
So, the resulting matrix after performing the given elementary row operations is:
[2 0 -1|-7]
[0 4 -1|-1]
[0 8 -1|-0]
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Kay buys 12$ pounds of apples.each cost 3$ if she gives the cashier two 20 $ bills how many change should she receive
Kay buys 12 pounds of apples, and each pound costs $3. Therefore, the total cost of the apples is 12 * $3 = $36 and thus she should receive $4 as change.
Kay buys 12 pounds of apples, and each pound costs $3. Therefore, the total cost of the apples is 12 * $3 = $36. If she gives the cashier two $20 bills, the total amount she has given is $40. To find the change she should receive, we subtract the total cost from the amount given: $40 - $36 = $4. Therefore, Kay should receive $4 in change.
- Kay buys 12 pounds of apples, and each pound costs $3. This means that the cost per pound is fixed at $3, and she buys a total of 12 pounds. Therefore, the total cost of the apples is 12 * $3 = $36.
- If Kay gives the cashier two $20 bills, the total amount she gives is $20 + $20 = $40. This is the total value of the bills she hands over to the cashier.
- To find the change she should receive, we need to subtract the total cost of the apples from the amount given. In this case, it is $40 - $36 = $4. This means that Kay should receive $4 in change from the cashier.
- The change represents the difference between the amount paid and the total cost of the items purchased. In this situation, since Kay gave more money than the cost of the apples, she should receive the difference back as change.
- The calculation of the change is straightforward, as it involves subtracting the total cost from the amount given. The result represents the surplus amount that Kay should receive in return, ensuring a fair transaction.
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If f(x) = −2x² + 3x, select all the TRUE statements. a. f(0) = 5 b. f(a) = -2a² + 3a c. f (2x) = 8x² + 6x d. f(-2x) = 8x² + 6x
The true statements are b. f(a) = -2a² + 3a and d. f(-2x) = 8x² + 6x.
Statement b is true because it correctly represents the function f(x) with the variable replaced by 'a'. By substituting 'a' for 'x', we get f(a) = -2a² + 3a, which is the same form as the original function.
Statement d is true because it correctly represents the function f(-2x) with the negative sign distributed inside the parentheses. When we substitute '-2x' for 'x' in the original function f(x), we get f(-2x) = -2(-2x)² + 3(-2x). Simplifying this expression yields f(-2x) = 8x² - 6x.
Therefore, both statements b and d accurately represent the given function f(x) and its corresponding transformations.
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primo car rental agency charges $45per day plus $0.40 per mile. ultimo car rental agency charges $26 per day plus $0.85 per mile. find the daily mileage for
which the ultimo charge is twice the primo charge.
To find the daily mileage for which the Ultimo charge is twice the Primo charge, we can set up an equation and solve for the unknown value.
Let's start by defining some variables:
- Let x be the daily mileage.
- The Primo car rental agency charges $45 per day plus $0.40 per mile, so the Primo charge can be expressed as 45 + 0.40x.
- The Ultimo car rental agency charges $26 per day plus $0.85 per mile, so the Ultimo charge can be expressed as 26 + 0.85x.
According to the question, we need to find the value of x for which the Ultimo charge is twice the Primo charge. Mathematically, we can write this as:
26 + 0.85x = 2(45 + 0.40x)
Now, let's solve this equation step-by-step:
1. Distribute the 2 to the terms inside the parentheses on the right side of the equation:
26 + 0.85x = 90 + 0.80x
2. Move all the x terms to one side of the equation and all the constant terms to the other side:
0.85x - 0.80x = 90 - 26
3. Simplify and solve for x:
0.05x = 64
x = 64 / 0.05
x = 1280
Therefore, the daily mileage for which the Ultimo charge is twice the Primo charge is 1280 miles.
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For V = F3, let v1 = e1,v2 = e1 + e2,v3 = e1 + e2 + e3. Show that {v1,v2,v3} is a basis for V.
Hint : We know {e1,e2,e3} is a basis for F3, and hence a spanning set; show that {e1,e2,e3} ⊆ Span(v1,v2,v3), and
hence {v1,v2,v3} spans V . Use the fact that {e1,e2,e3} is also a linearly independent set to show that {v1,v2,v3} is a
linearly independent set, and hence a basis for V .
Since {v1, v2, v3} is linearly independent and spans V, it is a basis for V.
To show that {v1, v2, v3} is a basis for V, we need to demonstrate two things: linear independence and spanning.
Linear Independence: We need to show that the vectors v1, v2, and v3 are linearly independent, meaning that no vector in the set can be written as a linear combination of the others. In this case, we can observe that no vector in the set can be expressed as a linear combination of the others because they have distinct components. Each vector has a unique combination of 0s and 1s in its components.
Spanning: We need to show that every vector in V can be expressed as a linear combination of v1, v2, and v3. Since V = F3, every vector in V is a 3-dimensional vector. We can see that by choosing appropriate coefficients for v1, v2, and v3, we can express any 3-dimensional vector in V.
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please help!
Q2: Solve the given Differential Equation by Undetermined Coefficient-Annihilator
Approach. y" +16y=xsin4x
The general solution is the sum of the complementary and particular solutions: y(x) = y_c(x) + y_p(x) = c1 cos(4x) + c2 sin(4x) + ((1/16)x + 1/8) sin(4x) + (Cx + D) cos(4x).
y" + 16y = x sin(4x) using the method of undetermined coefficients-annihilator approach, we follow these steps:
Step 1: Find the complementary solution:
The characteristic equation for the homogeneous equation is r^2 + 16 = 0.
Solving this quadratic equation, we get the roots as r = ±4i.
Therefore, the complementary solution is y_c(x) = c1 cos(4x) + c2 sin(4x), where c1 and c2 are arbitrary constants.
Step 2: Find the particular solution:
y_p(x) = (Ax + B) sin(4x) + (Cx + D) cos(4x),
where A, B, C, and D are constants to be determined.
Step 3: Differentiate y_p(x) twice
y_p''(x) = -32A sin(4x) + 16B sin(4x) - 32C cos(4x) - 16D cos(4x).
Substituting y_p''(x) and y_p(x) into the original equation, we get:
(-32A sin(4x) + 16B sin(4x) - 32C cos(4x) - 16D cos(4x)) + 16((Ax + B) sin(4x) + (Cx + D) cos(4x)) = x sin(4x).
Step 4: Collect like terms and equate coefficients of sin(4x) and cos(4x) separately:
For the coefficient of sin(4x), we have: -32A + 16B + 16Ax = 0.
For the coefficient of cos(4x), we have: -32C - 16D + 16Cx = x.
Equating the coefficients, we get:
-32A + 16B = 0, and
16Ax = x.
From the first equation, we find A = B/2.
Substituting this into the second equation, we get 8Bx = x, which gives B = 1/8.
A = 1/16.
Step 5: Substitute the determined values of A and B into y_p(x) to get the particular solution:
y_p(x) = ((1/16)x + 1/8) sin(4x) + (Cx + D) cos(4x).
Step 6: The general solution is the sum of the complementary and particular solutions:
y(x) = y_c(x) + y_p(x) = c1 cos(4x) + c2 sin(4x) + ((1/16)x + 1/8) sin(4x) + (Cx + D) cos(4x).
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