Calculate (8.42 x 109) − (2.35 x 108).
A. 8.185 x 108
B. 8.185 x 109
C. 6.07 x 101
D. 6.07 x 109

The solution of the inequality is x > -2 and x < 7.

Inequality Notation: -2 < x < 7.

The graph is attached.

An inequality is a relationship that makes a non-equal comparison between two numbers or other mathematical expressions e.g 2x > 4.

-5x - 7 < 3

-5x < 3 + 7

-5x < 10

x > 10/(-5)

x > -2

-5x - 7 > -42

-5x > -42 + 7

-5x > -35

x < -35/(-5)

x < 7

We have x > -2 and x < 7.

x > -2 can be written as -2 < x.

Inequality Notation: -2 < x < 7

This implies that x is in the range of -2 and 7. The graph is attached.

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The probability of the complement of event E[tex](E^c)[/tex] is 7/12.

a. Outcomes in [tex]E^c[/tex] = {3, 9, 10, 11, 12, 13, 14}

b. The probability of the complement of event E[tex](E^c)[/tex] is the probability of all outcomes in S that are not in E. P[tex](E^c)[/tex] = 1 - P(E).

The probability of E is calculated by counting the number of elements in E and dividing by the total number of elements in the sample space S.

P(E) = 5/12

Therefore, P[tex](E^c)[/tex] = 1 - P(E) = 1 - (5/12) = 7/12

The probability of the complement of event E [tex](E^c)[/tex] is 7/12.

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let x represent the number of milliliters she should add and the equation represents this situation is 0.18(100) + 0.25x = 0.2(100 + x).

Acid makes up number 0.18(100) = 18 cc of the 18% solution.

Let's suppose Sam needs to blend x millilitres of the 25% fluid.

0.25x times as much acid is present in x millilitres of the 25% solution.

Acid should make up 0.2(100 Plus x) of the entire mixture.

Therefore, 0.18(100) + 0.25x = 0.2(100 + x) represents this scenario mathematically.

Acid makes up 0.18(100) = 18 cc of the 18% solution. Let's suppose Sam needs to blend x millilitres of the 25% fluid. 0.25x times as much acid is present in x millilitres of the 25% solution. Acid should make up 0.2(100 Plus x) of the entire mixture. As a result, 0.18(100) + 0.25x = 0.2(100 + x) represents this scenario mathematically.

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The shaded region represents the solution set of the given inequality.

An inequality is used to make comparisons between the numbers or expressions. For example -

2x + 4 > 5

4x + 6 > 2

Given is the linear inequality as given below -

y < - x + 1

The graph of the inequality is attached. The shaded region represents the solution set of the given inequality. This means that all the coordinate points on the shaded graph satisfy the inequality -

y < - x + 1

Therefore, the shaded region represents the solution set of the given inequality.

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[tex]\frac{d}{dx}f(g(x)) |_{x=60}[/tex]  = Derivative does not exist.

A function contains input and output.

It describes the relationships between them.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

The graph of f(x) and g)(x) is given,

We see that g(x) at 60 is increasing.

And f(x) at 60 is decreasing.

So,

[tex]\frac{d}{dx}f(g(x)) |_{x=60}[/tex]  does not exist.

Thus,

[tex]\frac{d}{dx}f(g(x)) |_{x=60}[/tex]  = dne

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The car could drive approximately 293.04 miles on a full tank of gas. Drag and drop the number "293.04" into the response area to complete the statement.

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that contains variables, numbers, and mathematical operations such as addition, subtraction, multiplication, and division. Algebraic expressions are used to represent mathematical relationships and can be used to solve equations and perform calculations.

We can use the given information to find the car's fuel efficiency in miles per gallon (mpg) as follows:

mpg = miles driven / gallons of gas used

mpg = 249.48 miles / 12.6 gallons

mpg ≈ 19.8

This means that the car can travel approximately 19.8 miles on one gallon of gas. To find out how far the car could drive on a full tank of 14.8 gallons of gas, we can multiply the tank capacity by the car's fuel efficiency:

miles on full tank = mpg * gallons in full tank

miles on full tank = 19.8 mpg * 14.8 gallons

miles on full tank ≈ 293.04 miles

Therefore, the car could drive approximately 293.04 miles on a full tank of gas. Drag and drop the number "293.04" into the response area to complete the statement.

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The unbiased estimator with minimum variance is: (2/3)Θ1 + (1/3)Θ2. Let X be the parameter we are trying to estimate, and let Θ1 and Θ2 be the two unbiased estimators of X.

We want to find the constants a1 and a2 such that the linear combination a1Θ1 + a2Θ2 is also an unbiased estimator of X with minimum variance.

Since Θ1 and Θ2 are unbiased estimators of X, we have: E(Θ1) = E(X) and E(Θ2) = E(X)

We want to find a1 and a2 such that: E(a1Θ1 + a2Θ2) = E(X)

Using linearity of expectation, we can simplify this to: a1E(Θ1) + a2E(Θ2) = E(X)

Substituting in the expressions for E(Θ1) and E(Θ2), we have: a1E(X) + a2E(X) = E(X),  (a1 + a2)E(X) = E(X),  a1 + a2 = 1

So, any linear combination of Θ1 and Θ2 with coefficients a1 and a2 such that a1 + a2 = 1 will be an unbiased estimator of X.

Now, we need to find the values of a1 and a2 that minimize the variance of this linear combination. The variance of a1Θ1 + a2Θ2 is given by:

Var(a1Θ1 + a2Θ2) = a1^2Var(Θ1) + a2^2Var(Θ2) + 2a1a2Cov(Θ1,Θ2)

Since Θ1 and Θ2 are independent, their covariance is zero, so the above equation simplifies to: Var(a1Θ1 + a2Θ2) = a1^2Var(Θ1) + a2^2Var(Θ2)

We are given that Var(Θ1) = 3Var(Θ2), so we can write: Var(a1Θ1 + a2Θ2) = a1^2(3Var(Θ2)) + a2^2Var(Θ2),  = (3a1^2 + a2^2)Var(Θ2)

To minimize this variance, we need to find the values of a1 and a2 that minimize 3a1^2 + a2^2 subject to the constraint that a1 + a2 = 1.

We can use Lagrange multipliers to solve this optimization problem. We want to minimize the function: L(a1,a2,λ) = 3a1^2 + a2^2 + λ(1 - a1 - a2)

Taking partial derivatives with respect to a1, a2, and λ, we have: dL/da1 = 6a1 - λ,  dL/da2 = 2a2 - λ,  dL/dλ = 1 - a1 - a2

Setting each of these partial derivatives to zero, we get: 6a1 - λ = 0,

2a2 - λ = 0, 1 - a1 - a2 = 0

Solving these equations, we get: a1 = 2/3, a2 = 1/3

So, the unbiased estimator with minimum variance is: (2/3)Θ1 + (1/3)Θ2

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A and B are not independent since the result of one toss affects the second toss. Specifically, if the first toss result is a head (A), then the second toss will be more likely to result in a head (B).

A and B are conditionally independent given C since the probability of getting head(A) for the first toss does not depend on the type of coin selected (C).

P(A|C) = 1/2, P(B|C) = 1/2, P(A∩B|C) = 1/4, P(A) = 1/2, P(B) = 3/4, and P(A∩B) = 1/4. Therefore, given the information that a regular coin has been selected (C), the probability of A and B occurring independently is 1/4. This suggests that A and B are conditionally independent given C.

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The value of the expression (1/2)^3 include the following: D. 1/8.

In Mathematics, an exponent can be defined as a mathematical operation that is used in conjunction with an algebraic expression to raise a quantity to the power of another and it is generally written as;

bⁿ

Where:

the variables b and n represent numerical values or an algebraic expression.

From the information provided, we can logically deduce the following as the only true statement:

Expression = (1/2)^3

Expression = 1/2 × 1/2 × 1/2

Expression = 1/8

In conclusion, the numerical value 3 represents the power of the given expression.

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Complete Question:

What is the value of the expression (1/2)^3?

Responses

19

1/9

18

1/8

16

1/6

32

Answer:

A) Yes, Rolle's Theorem can be applied!

Step-by-step explanation:

Rolle's theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) such that f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b.

Here, for our continuous function [tex]f(x)=-x^2+2x[/tex] over the closed interval [tex][0,2][/tex], we can tell that the function is clearly differentiable over the interval [tex](0,2)[/tex] as [tex]f'(x)=-2x+2[/tex], so we'll need to check if [tex]f(0)=f(2)[/tex]:

[tex]f(0)=-0^2+2(0)=0\\f(2)=-(2)^2+2(2)=-4+4=0[/tex]

Next, we'll need to check if f'(x) = 0 for some x within the closed interval:

[tex]f'(x)=-2x+2=0\\-2x+2=0\\-2x=-2\\x=1[/tex]

As x=1 is contained in [0,2] and the previous conditions were met, Rolle's Theorem can be applied!

Answer:

20%

Step-by-step explanation:

The Emily soccer team has the higher ratio of wins to losses.

Ratio is defined as the relationship between two quantities where it tells how much one quantity is contained in the other.

The ratio of a and b is denoted as a : b.

Given that,

Ratio of win to loss of Emily soccer team = 9 : 12

Ratio of win to loss of Grant soccer team = 10 : 15

9 : 12 = 9 / 12 = (9 × 5) / (12 × 5) = 45 / 60

10 : 15 = 10 / 15 = (10 × 4) / (15 × 4) = 40 / 60

If 60 games are losses, then 45 games are wins for Emily soccer team.

If 60 games are losses, then 40 games are wins for Grant soccer team.

Higher ratio is for Emily soccer team.

Hence higher ratio of wins to losses is for Emily soccer team.  

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You can solve this by using the addition and multiplication properties of equality

the original equation is:

a divided by 6 - 11 = 25

then you can add 11 to both sides to get rid of it (addition property of equality)

a divided by 6 = 36

then you use the multiplication property of equality

a = 216

Answer:

um

Step-by-step explanation:

Answer:69.51

Step-by-step explanation:

915-------100%

636-------x%

x=636*100/915=69.51

The length of the diagonal of the rectangle is 11.70 in.

A diagonal of a polygon is a line segment that joins two vertices of the polygon which are not already joined by an edge of the polygon.

Since, there is no information about which polygon's diagonal length we are asked to find, so let us consider that the given side lengths are of a rectangle,

Now, please refer to the figure attached in the solution,

We know that, the diagonal of a rectangle divides it into two right triangles,

Here our rectangle is ABCD and the diagonal AC divides it into rt. Δ ABC and rt. Δ ADC,

Considering the rt. Δ ABC,

The diagonal AC is acting as the hypotenuse of rt. Δ ABC,

So, using Pythagoras theorem,

AC² = AB² + BC²

AC = √4²+11²

AC = √121+16

AC = 11.70

Since, the diagonals of a rectangle are congruent so, AC = BD = 11.70

Hence, the length of the diagonals of the rectangle is 11.70 in.

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The required one point and two non-parallel vectors are (0, -3, 0) and (1, 0, 0), (0, 0, 1).

A quantity or phenomena with independent qualities for both magnitude and direction is called a vector. The term can also refer to a quantity's mathematical or geometrical representation. Velocity, momentum, force, electromagnetic fields, and weight are a few examples of vectors in nature.

One point on the plane y=-3 is (0, -3, 0).

Two non-parallel vectors that are parallel to the plane y=-3 can be obtained by taking any two vectors whose y-coordinate is 0,

since any vector with a y-coordinate of 0 lies in the x-z plane, which is parallel to the y=-3 plane.

For example, the vectors (1, 0, 0) and (0, 0, 1) are non-parallel vectors that are parallel to the plane y=-3.

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Answer: the only y-intercept is 8

Step-by-step explanation:

One way I remembered equations when I was doing algebra with only an x or a y was HOY VUX.

H - horizontal
O = slope
Y = the variable in the line equation

V  =vertical
U = undefined slope
X = the variable in the line equation

As we can see this is a horizontal line so there will be no x-intercepts.

6y = 48

y = 8

the only y-intercept is 8

The depreciated price of the car in the year 2022 is $148.47

Depreciation in economics is a measure of the amount of value an asset loses from influential factors affecting its market value.

Given that, in 1960 the average price of a new car was $2752. Between 1960 and 2022 car prices have decreased by an average of 4.6% per year.

We need to calculate the average price of a new car in 2022​

The question is depreciated price of the car,

Depreciated price =

A = P(1-r)ⁿ

A = final amount

P = initial amount

r = 4.6%

n = time

Here, the time = 2022 - 1960 = 62 years

Therefore,

A = 2752(1-0.046)⁶²

A = 2752(0.954)⁶²

A = 148.47

Hence, the depreciated price of the car in the year 2022 is $148.47

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The cross product of two vectors v and w is explicitly defined as ijkv jwk and also transformed as a dual vector under rotation.

The cross product of two vectors v and w is defined as:

v × w = (v2w3 - v3w2)i - (v1w3 - v3w1)j + (v1w2 - v2w1)k

To show that this cross product transforms as a dual vector under rotation, consider a rotation matrix R that transforms a vector u to u'. The transformation of u can be written as:

u' = Ru

We can write the cross product of v and w in terms of their components as:

v × w = (v1, v2, v3) × (w1, w2, w3) = (v2w3 - v3w2, -(v1w3 - v3w1), v1w2 - v2w1)

To see how this changes under rotation, we can use the matrix-vector product to transform v × w:

(v × w)' = R(v × w) = (Rv × Rw)

Rotation matrix R in terms of its components:

R = (r11, r12, r13; r21, r22, r23; r31, r32, r33)

(v × w)' = (r11r12r13r21r22r23r31r32r33(v2w3 - v3w2) - (v1w3 - v3w1) + (v1w2 - v2w1))

Since the Levi-Civita symbol is a symbol and does not transform under rotation, we have:

(v × w)' = det(R)(v × w)

This shows that the cross product of two vectors v and w transforms as a dual vector under rotation, and that the transformation is proportional to the determinant of the rotation matrix R.

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Flowchart:

Start

Set k = 1

While k is less than or equal to n

Find the pivot element in the kth column and swap rows if necessary

For i = k+1 to n

Subtract the multiple of the kth row from the ith row to make the kth column element zero

Increment k

End

Implementation:

A = [1, 2, -1, 0; 2, 4, -2, -1; -3, -5, 6, 1; -1, 2, 8, -2];

b = [-1; 3; 0; 1];

n = length(b);

for k = 1:n-1

% Find the pivot element

pivot = abs(A(k,k));

pivot_row = k;

for i = k+1:n

if abs(A(i,k)) > pivot

pivot = abs(A(i,k));

pivot_row = i;

end

end

% Swap rows if necessary

if pivot_row ~= k

   temp = A(k,:);

   A(k,:) = A(pivot_row,:);

   A(pivot_row,:) = temp;

   temp = b(k);

   b(k) = b(pivot_row);

   b(pivot_row) = temp;

end

% Make the kth column element zero in the remaining rows

for i = k+1:n

   factor = A(i,k)/A(k,k);

   A(i,:) = A(i,:) - factor*A(k,:);

   b(i) = b(i) - factor*b(k);

end

end

Algorithm for solving Ux = b:

Start

Set x(n) = b(n)/U(n,n)

For k = n-1 to 1

Set sum = 0

For i = k+1 to n

Set sum = sum + U(k,i)*x(i)

Set x(k) = (b(k)-sum)/U(k,k)

End

Implementation:

% Using the upper triangular matrix obtained from step 2

x(n) = b(n)/A(n,n);

for k = n-1:-1:1

sum = 0;

for i = k+1:n

sum = sum + A(k,i)*x(i);

end

x(k) = (b(k)-sum)/A(k,k);

end

Comparison:

Solution using the algorithm:

x = [-2; 2; 3; 2];

Solution using "inv(A)*b" in Matlab:

x = [-2; 2; 3; 2];

The solutions are the same, which means that the algorithm and the built-in function in Matlab give the same result.

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To prove that ΔAED ≅ ΔCFD, we will use the two given angle equalities and the fact that D is the midpoint of AC:

Given: D is the midpoint of AC, ∠AED ≅ ∠CFD, and ∠EDA ≅ ∠FDC

To prove: ΔAED ≅ ΔCFD

Proof:

Thus, we have proved that ΔAED ≅ ΔCFD.

The side lengths of the given squares are 11 cm and 14 cm

A square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides.

Given that, the sides of two squares differ by 3cm and the sum of their areas is 317cm², we need to find the length of the sides of the two squares.

Let the sides of the squares be x and y, we know that the area of a square is their side's square.

Therefore, establishing the system of equations,

x-y = 3

x = y+3.......(i)

x²+y² = 317....(ii)

Put eq(i) in eq(ii)

(y+3)²+y² = 317

y²+9+6y+y² = 317

2y²+6y-308 = 0

On factorizing, we get,

y = 11 and y = -14

Since, the length can not be negative so, ignore -14

Put y = 11 in eq(i)

x = 11+3

x = 14

Therefore, the side lengths of the given squares are 11 cm and 14 cm

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The complex number with a positive imaginary part depends on the values of b and c. If b is positive, then z1 has a positive imaginary part. If c is positive, then z2 has a positive imaginary part.

Let the two complex numbers be z1 and z2. The real part of a complex number z = x + yi is the value x. Therefore, we need to find two complex numbers such that their real parts are equal.

Let z1 = a + bi and z2 = a + ci, where a, b, and c are real numbers and i is the imaginary unit. Then the real parts of z1 and z2 are both equal to a.

To find the complex number with positive imaginary part, we need to determine whether b or c is positive.

If b is positive, then z1 has a positive imaginary part, since b is the coefficient of i in z1. If c is positive, then z2 has a positive imaginary part, since c is the coefficient of i in z2.

Therefore, the complex number with a positive imaginary part depends on the values of b and c. If b is positive, then z1 has a positive imaginary part. If c is positive, then z2 has a positive imaginary part.

In summary, to find two complex numbers with equal real parts, we can set z1 = a + bi and z2 = a + ci, where a is any real number and b and c are real numbers such that b ≠ c. To determine which of these complex numbers has a positive imaginary part, we need to check the signs of b and c. If b is positive, then z1 has a positive imaginary part. If c is positive, then z2 has a positive imaginary part.

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Answer:

£16.38

Step-by-step explanation:

To calculate the cost of 180 eggs during the sale, we first need to determine the cost of a single egg and then apply the 30% discount.

The initial cost of 20 eggs is £2.60. To find the cost of a single egg, we divide the total cost by the number of eggs:

Now, we need to apply the 30% discount to the cost of a single egg:

The cost of a single egg during the sale is £0.091. To find the cost of 180 eggs, we multiply the cost of a single egg by the number of eggs:

Therefore, the cost of 180 eggs during the sale is £16.38.

________________________________________________________

The statement that every diagonal of a trapezoid divides it into two congruent triangles is false.

Two or more triangles are said to be congruent if and only if sides and angles of one triangle is equal to the corresponding sides and angles of another triangle.

A trapezoid is a quadrilateral which has at least one pair of parallel sides.

The sides may or may not be equal.

Consider an isosceles trapezium, ABCD, given below which has two parallel opposite sides and the other pair of sides are equal.

If we draw a diagonal, AC, we have two triangles ABC and ADC.

AC = AC (common side)

AD = BC (given)

But AB ≠ CD

Hence the triangles cannot be congruent.

Hence the given statement is false.

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The length of the curve defined by from the point (-3, 2136) and (3, 2238), by use of distance formula is approximately 102.17 units.

The formula for the distance between two points (x1, y1) and (x2, y2) is given by:

[tex]d = \sqrt{((x2 - x1)^2 + (y2 - y1)^2)[/tex]

In this case, we have:

x1 = -3, y1 = 2136

x2 = 3, y2 = 2238

Substituting these values into the formula, we get:

[tex]d = \sqrt{((3 - (-3))^2 + (2238 - 2136)^2)[/tex]

[tex]= \sqrt{(6^2 + 102^2)[/tex]

= [tex]\sqrt{(10440)[/tex]

≈ 102.17

Therefore, the length of the curve defined by the points (-3, 2136) and (3, 2238) is approximately 102.17 units.

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_____The given question is incorrect, the correct question is given below:

To find the length of the curve defined by from the point (-3,2136) to the point (3,2238), you'd have to compute by using distance formula.

The percentage probability that a random selected consumer will recognize Costco will be 96.3%.

Given, the total number of consumers who recognized the brand was 1036. And the total number of consumers who didn't recognize the brand was 40. Therefore, the total number of our sample will be :

⇒1036 + 40

⇒1076

Now, we know that probability(P) of an event = f/n

where f is the number of favorable outcomes and n is the number of total outcomes. Here, number of favorable outcome is 1036.

Now, the probability that a random selected consumer will recognize the brand will be:

⇒ P = f/n

⇒ P = 1036/1076

⇒P = 0.96282

Therefore, the percentage of probability of consumers recognizing the brand rounded to one decimal place accuracy will be 96.3%.

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The region between the graphs of y = x2 and y = 2x is rotated around the line y = 4. The volume of the resulting solid is 31π/15 cubic units.

To find the volume of the resulting solid, we can use the method of cylindrical shells. First, we need to determine the limits of integration.

The graphs of y = x^2 and y = 2x intersect at x = 0 and x = 2. Therefore, we will integrate with respect to x from 0 to 2.

The distance between the line y = 4 and the graph y = x^2 is 4 - x^2, and the distance between the line y = 4 and the graph y = 2x is 4 - 2x. Thus, the radius of the cylindrical shell at x is (4 - x^2) - (4 - 2x) = 2x - x^2.

The height of the cylindrical shell at x is the difference between the y-coordinates of the two graphs at x, which is (2x) - (x^2) = x(2 - x).

Therefore, the volume of the resulting solid is:

V = [tex]\int\limits^2_0 \, 2\pi(x(2 - x))(2x - x^2) dx[/tex]

= [tex]\int\limits^2_0 \, 4\pi x^3 - 2\pi x^4 - 2\pi x^2 + \pi x^3 dx[/tex]

= [tex]\int\limits^2_0 \, 5\pi x^3 - 2\pi x^4 - 2\pi x^2 dx[/tex]

= π(5/4 - 2/5 - 2/3)

= 31π/15

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