5. Your bill at a restaurant is $24.50. You tip 16%. What's
the final bill?

a) Contribution (objective value) = $132.14

b) The firm earns $132.14 at the optimal solution.

c) An additional hour of time available for the third machine is worth $0.14 to the firm.

d) An additional 5 hours of time available for the second machine will increase the objective value by $3.69.

The best result obtained from using computer software to solve the LP problem is: X1 = 11.43, X2 = 12.86, X3 = 5.71

b) The number of unused hours at the ideal solution is:

Machine 1 still has 8.57 hours of time left.

There are no hours left on Machine 2 at the moment.

There are still 94.29 hours left on Machine 3.

c) The shadow price of the third limitation is worth an extra hour of time available for the third machine. With the exception of increasing the right-hand side of the third constraint by one unit, we can solve the LP problem using the same constraints to determine the shadow price. Using LP to solve this issue, we discover that the shadow price for the third constraint is

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To find the conditions on a, b, c, and d such that the matrix B = [a b; c d] commutes with both [1 0; 0 1] and [0 0; 0 1], we need to determine when the product of B and each of these matrices is equal regardless of the order.

The necessary conditions for commutation are:

1. a = 1: This condition ensures that the first column of B remains unchanged when multiplied with [1 0; 0 1], ensuring commutation.

2. b = 0: This condition ensures that the second column of B is multiplied by the first column of [0 0; 0 1], which is a zero vector, resulting in a zero column.  

3. c = 0: This condition ensures that the first column of B is multiplied by the second column of [0 0; 0 1], which is a zero vector, resulting in a zero column.

4. d = 1: This condition ensures that the second column of B remains unchanged when multiplied with [0 0; 0 1], ensuring commutation.

In summary, the conditions for B to commute with both [1 0; 0 1] and [0 0; 0 1] are a = 1, b = 0, c = 0, and d = 1. These conditions ensure that the product of B with each of the given matrices is equal regardless of the order, resulting in commutation.

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Option D, rotation 180° about the origin, and then reflection across the y axis.

Given data :

Given vertex U(  4 ,2 )

U( 4 ,2 )→ U( -4 ,-2)

Next we will apply rule

Horizontal translation left '1' units and Vertical translation up 'd' units

U -4 ,-2) → U¹ ( -4 -1 ,-2 +3)

The new vertex is  U¹ ( -5 ,1)

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Answer:

  -8

Step-by-step explanation:

Put the given values where the corresponding variables are, and do the arithmetic.

__

  f(x) = 12 -3x

  f(5) = 12 -3(5) = -3

__

  j(x) = x² +4x -1 = (x +4)x -1

  j(-2) = (-2 +4)(-2) -1 = 2(-2) -1 = -5

__

  f(5) +j(-2) = -3 +(-5) = -8

So the required blanks for the series are filled with:

Blank 1: 4ke⁻ᵏ

Blank 2: ((k + 1)/k)e⁻¹

Blank 3: <

Blank 4: The series is convergent.

The given series is,

\(\sum_{k=1}^{\infty}\) 4ke⁻ᵏ

So the k th term of the series is given by,

aₖ = 4ke⁻ᵏ

Now,

aₖ₊₁/aₖ = (4(k+1)e⁻⁽ᵏ⁺¹⁾)/(4ke⁻ᵏ) = ((k + 1)/k)e⁻¹

Now the value of the limit is given by,

\(\lim_{k \to \infty}\) |aₖ₊₁/aₖ| = \(\lim_{k \to \infty}\) ((k + 1)/k)e⁻¹ = \(\lim_{k \to \infty}\) (1 + 1/k)e⁻¹ = (1 + 0)e⁻¹ = e⁻¹

since e > 2

then e⁻¹ < 1/2

So, e⁻¹ < 1

So, it is less than 1.

since  \(\lim_{k \to \infty}\) |aₖ₊₁/aₖ| = e⁻¹ < 1

Hence the given series \(\sum_{k=1}^{\infty}\) 4ke⁻ᵏ is convergent.

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The question is incomplete. The complete question will be -

The value of the surface integral ∫sf⋅ ds over the given surface S is 2√2.

To evaluate the surface integral ∫sf⋅ ds, we first need to parameterize the surface S which is the part of the sphere \(x^{2}\)+\(y^{2}\)+\(z^{2}\)=16 in the first octant.

One possible parameterization of S is:

x = r sinθ cosφ

y = r sinθ sinφ

z = r cosθ

where 0 ≤ θ ≤ π/2 and 0 ≤ φ ≤ π/2.

Next, we need to find the unit normal vector to the surface S. Since the surface is oriented toward the origin, the unit normal vector points in the opposite direction of the gradient vector of the function \(x^{2}\)+\(y^{2}\)+\(z^{2}\)=16 at each point on the surface S.

∇( \(x^{2}\)+\(y^{2}\)+\(z^{2}\)) = ⟨2x,2y,2z⟩

So, the unit normal vector to the surface S is

n = -⟨x,y,z⟩/4 = -⟨r sinθ cosφ, r sinθ sinφ, r cosθ⟩/4

Now, we can evaluate the surface integral using the parameterization and unit normal vector:

∫sf⋅ ds = ∫∫S f⋅n dS

= ∫0-π/2 ∫0-π/2 (-4r sinθ cosφ, -3r cosθ, 3r sinθ sinφ)⋅(-⟨r sinθ cosφ, r sinθ sinφ, r cosθ⟩/4) \(r^{2}\) sinθ dθ dφ

= ∫0-π/2 ∫0-π/2 (\(r^{3}\) \(sin^{2}\)θ/4)(12 \(sin^{2}\)θ) dθ dφ

= 3/4 ∫0-π/2 ∫0-π/2 \(r^{3}\)\(sin^{4}\)θ dθ dφ

= 3/4 ∫0-π/2 [\(r^{3/2}\)(2/3)] dφ

= 3/4 (2/3) \(2^{3/2}\)

= 2√2

Correct Question :

Evaluate the surface integral ∫sf⋅ ds where f=⟨−4x,−3z,3y⟩ and s is the part of the sphere \(x^{2}\)+\(y^{2}\)+\(z^{2}\)=16  in the first octant, with orientation toward the origin.∫∫SF⋅ dS=?

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The general solution of the differential equation is:

y = Ax^-5 + Bx^-5ln(x)

where A and B are constants determined by the initial or boundary conditions.

To solve the differential equation x^2y'' + 11xy' + 25y = 0, we can assume the solution to be of the form y = x^r, where r is a constant.

Then, we have:

y' = rx^(r-1)

y'' = r(r-1)x^(r-2)

Substituting these into the differential equation, we get:

x^2y'' + 11xy' + 25y = x^2r(r-1)x^(r-2) + 11xrx^(r-1) + 25x^r = 0

Dividing both sides by x^2, we have:

r(r-1) + 11r + 25 = 0

Simplifying, we get:

r^2 + 10r + 25 = (r+5)^2 = 0

Therefore, r = -5.

Thus, the general solution of the differential equation is:

y = Ax^-5 + Bx^-5ln(x)

where A and B are constants determined by the initial or boundary conditions.

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To determine the optimal order quantity for Rocky Mountain Tire Center, you must consider ordering costs, storage costs, and the purchase price of the tires. The order quantity should minimize the total cost including both ordering cost and storage cost.

The EOQ formula is given by: EOQ = √((2DS) / H)

Where: D = Annual demand (7,000 go-cart tires)

S = Ordering cost per order ($40) H = Holding cost - percentage of the purchase price (40% of the purchase price)

we need to determine the purchase price per tire based on the quantity ordered.

EOQ = √((2 * 7,000 * 40) / (0.4 * 15))

=118 tires

they should order approximately 118 tires.

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A part of a line that has 1 endpoint and extends indefinitely in only one direction is called a ray.

A ray is named using its endpoint first, and then any other point on the ray

Properties of ray:

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(a) By using the logistic equation P' = kP(1 - P/K), where P represents the fish population and K is the carrying capacity, we can determine the constant k. Given that the population tripled in the first year, we can set up the equation 3P = P(1 - kP/K) and solve for k. The value of k is approximately 0.0005833. We can then use this value to solve the logistic equation and find an expression for the population size after t years.

(b) To determine how long it will take for the population to increase to half of the carrying capacity, we set up the logistic equation P = 0.5K and solve for t. The solution to this equation gives us the time it takes for the population to reach half of the carrying capacity.

(a) To find the constant k, we set up the equation 3P = P(1 - kP/K) using the given information that the population tripled in the first year. By simplifying and solving for k, we find k ≈ 0.0005833. Now we can substitute this value of k into the logistic equation P' = 0.0005833P(1 - P/5100) and solve it to find an expression for the population size after t years.

(b) To determine the time it takes for the population to increase to half of the carrying capacity, we set up the equation P = 0.5(5100) using the logistic equation. By solving this equation, we can find the value of t that represents the time it takes for the population to reach half of the carrying capacity.

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Since the calculated test statistic (2.836) is greater than the critical value (1.645), we reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis that more than half of internet users have posted photos or videos online.

To test the hypothesis that more than half of internet users have posted photos or videos online, we can use a one-sample proportion test. Let p be the true proportion of internet users who have posted photos or videos online. The null and alternative hypotheses are:

H0: p <= 0.5

Ha: p > 0.5

We will use a significance level of 0.05.

Using the given information, we have:

n = 855

x = (56/100) * 855

= 479.6 (rounded to nearest whole number, 480)

The sample proportion is:

p-hat = x/n

= 480/855

= 0.561

The test statistic is:

z = (p-hat - p0) / √(p0 * (1 - p0) / n)

where p0 is the null proportion under the null hypothesis. We will use p0 = 0.5.

z = (0.561 - 0.5) / √(0.5 * (1 - 0.5) / 855)

= 2.836

Using a standard normal distribution table or calculator, the critical value for a one-tailed test at a 0.05 level of significance is approximately 1.645.

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Answer:

C. h=3V/a^2 is the correct answer

Step-by-step explanation:

Given:

V=1/3a^2h

It can be rewritten as

V=1/3 *( a^2) * h

Divide both sides by (1/3)*h

V / (1/3*h) = (1/3)* (a^2) * h / (1/3)*h

V÷1/3*h = a^2

V÷3/1*h=a^2

3V/h=a^2

Multiply both sides by h

3V/h*h=h*a^2

3V=h*a^2

h=3V/a^2

C. h=3V/a^2 is the correct answer

Answer:

C

Step-by-step explanation:

Answer:

4800 students

Step-by-step explanation:

120x40=4800students

4800 can sit in the class

Step-by-step explanation:

B represent the decay factor since it decays 0.2 every t year.

Answer:

A decay rate of 20% each year

Step-by-step explanation:

think of it as money to 1 in the parenthesis is 100 so whats 100-80 (i gt 80 because 80 turned into a decimal is .8) its 20 so its going to decay or go down 20% each year. hope this helps anyone :)

The correction made by the student is incorrect simplification of 30ab√2a+20a√2a

The perimeter of a polygon is the sum of its, all the side lengths.

Given that, the dimension of a rectangle is given by, √50a³b² and √200a³

The perimeter of the rectangle is = 2(length + width)

Solving for the perimeter,

2(√50a³b²+√200a³)

= 2(5ab√2a+10a√2a)

= 10ab√2a+20a√2a

The student solved it as 30ab√2a,

Which is incorrect, because 'b' is not there in both the expressions, hence, we cannot add them.

Hence, The correction made by the student is incorrect simplification of 30ab√2a+20a√2a

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The volume of ball is 32.49 cubic centimeter.

What is volume ?

 Every three-dimensional item takes up space in some way. The volume of this area is what is used to describe it. The area occupied within an object's three-dimensional bounds is referred to as its volume. The object's capacity is another name for it.

Here ,

The volume of a ball = 32490 cubic millimeters

To convert into cubic millimeter into cubic centimeter by dividing by 1000, Then,

=>  32490/1000

=> 32.49 cubic centimeter.

Hence the volume of ball in cubic centimeter is 32.49 .

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Answer:

yea i think so

Step-by-step explanation:

Answer:

i hope it helps

thank you

We are supposed to evaluate the integral:∫_0^1▒dx/(1+x^2 ).Using Romberg's method, we have to obtain an approximate value of x. The formula to calculate the integral by Romberg method is:

T_00 = h/2(f_0 + f_n)for i = 1, 2, …T_i0 = 1/2[T_{i-1,0} + h_i sum_(k=1)^(2^(i-1)-1) f(a + kh_i)]R(i,j) = (4^j T_(i,j-1) - T_(i-1,j-1))/(4^j-1)where h = (b-a)/n, h_i = h/2^(i-1).

The calculation is tabulated below: Thus, the approximate value of the integral ∫_0^1▒dx/(1+x^2 )using Romberg's method is:R(4,4) = 0.7854 ± 0.0007.

The question requires us to evaluate the integral ∫_0^1▒dx/(1+x^2 ) by using Romberg's method and then find an approximate value of x. Romberg's method is a numerical technique used to approximate definite integrals and it's known for producing highly accurate results.

The first step of the method is to apply the formula:T_00 = h/2(f_0 + f_n)which calculates the midpoint of the trapezoidal rule and returns an initial estimate of the integral.

We can use this initial estimate to calculate the next value of T_10, which is given by:T_10 = 1/2[T_00 + h_1(f_0 + f_1)]We can use the above formula to calculate the successive values of Tij, where i denotes the number of rows and j denotes the number of columns.

In the end, we can obtain the value of the integral by using the formula:

R(i,j) = (4^j T_(i,j-1) - T_(i-1,j-1))/(4^j-1)where i and j are the row and column indices, respectively.

After applying the above formula, we get R(4,4) = 0.7854 ± 0.0007Thus, the approximate value of the integral ∫_0^1▒dx/(1+x^2 )using Romberg's method is 0.7854 and the error is ± 0.0007. Hence, we can conclude that the value of x is 0.7854.

Romberg's method is a numerical technique used to approximate definite integrals and it's known for producing highly accurate results. The method involves calculating the midpoint of the trapezoidal rule and then using it to calculate the next value of Tij.

We can then obtain the value of the integral by using the formula R(i,j) = (4^j T_(i,j-1) - T_(i-1,j-1))/(4^j-1). The approximate value of the integral ∫_0^1▒dx/(1+x^2 )using Romberg's method is 0.7854 and the error is ± 0.0007.

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Answer:

  x = 10

Step-by-step explanation:

Use the Pythagorean theorem. The sum of the square of the sides is the square of the hypotenuse.

  x² +(√200)² = (√300)²

  x² = 300 -200

  x = √100 = 10

The length of the unknown side is 10 units.

Answer:

(-523) - 16 is -539

Step-by-step explanation:

mathmathmathmathmsthmatj

The number of stamps he has is 36.

What is a linear equation in math?

For this case we define the following variable:

We now write the equation that models the problem-

12 + (3/4x) = 39

3/4x = 39 - 12

(4/3)(3/4)x = 27 (4/3)

x = 36

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Answer: He has 36 stamps.

Step-by-step explanation: Just took test

The missing step in the two column proof is:

C. Statement: ∠1 ≅ ∠4, and ∠3 ≅ ∠5. Reason: Alternate Interior Angles Theorem

A two-column proof usually has numbered statements and corresponding reasons that show an argument in a logical order.

Statement 1: Points A, B, and C form a triangle.

Reason 1: Given

Statement 2: Let DE be a line passing through B and parallel to AC.

Reason 2: Definition of parallel lines

Statement 3: ∠3 ≅ ∠5 and ∠1 ≅ ∠4

Reason 3: Alternate interior angles theorem

Statement 4: m∠1 = m∠4 and m∠3 = m∠5

Reason 4: Definition of congruent angles

Statement 5: m∠4 + m∠2 + m∠5 = 180°

Reason 5: Angle addition and definition of a straight line

Statement 6: m∠1 + m∠2 + m∠3 = 180°

Reason 6: Substitution

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To make the augmented matrix correspond to a consistent system, we need to ensure that the third row is a linear combination of the first two rows. Let's denote the entries in the third row as a, b, and c, respectively. Then, the equation involving g, h, and k that makes the matrix consistent is:

-4g + 3h + 5k = a

7g - 5h - 9k = b

In a consistent system, all rows of the augmented matrix must be linearly dependent. This means that the third row should be a linear combination of the first two rows. By equating the corresponding entries in the third row to variables, we can find an equation involving g, h, and k that satisfies this condition.

In the given augmented matrix, the third row is represented by the variables g, h, and k. We denote the entries in the third row as a, b, and c, respectively. To create a consistent system, we need to find a relationship between these variables and the entries in the first two rows.

By comparing the entries in the first and second columns, we can set up the following equations:

-4g + 3h + 5k = a (equation 1)

7g - 5h - 9k = b (equation 2)

These equations ensure that the augmented matrix corresponds to a consistent system. If we substitute the values of g, h, and k into equations 1 and 2, the resulting values in the third row will satisfy the entries a and b, respectively.

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Answer:

8

Step-by-step explanation:

8/7  x 7 = 56/7 = 8

Answer:

8 days

Step-by-step explanation:

All you have to do is 8/7 multiplied by 7. Then you'll get your answer of 8 days.

Answer:

Answer:1.5*10^9 years

Step-by-step explanation:

The half life of the substance is given i.e 4.5B years. You can change it into seconds by multiplying it with 365 and 86400.

The current no of particles is given 78.88% which will be equal to 78.88 if you assume the substance had 100 no of particle initially.

Determine the decay constant by dividing 0.69 by half life

use these values in the equation for the answer.

[ln and e are inverse functions to eachother]

In physics lab, when you measure three quantities of x, y, and z, you can use these mathematical combinations to find meaningful relationships between these quantities:  

In this case, x=2.5 kg, y=0.8 m, and z=4.9 kg/m.

Based on the given quantities' scale, we know that x, y, and z have a relationship where:

z = x/y

Then, the mathematical combinations that might be meaningful are:

- (x/y) - z: This combination gives you the difference between the ratio of x and y, and z. This could be meaningful if you are trying to find the difference between two quantities that have different units.

- (x/z) - y: This combination gives you the difference between the ratio of x and z, and y. This could be meaningful if you are trying to find the difference between two quantities that have different units.

- x - (yz): This combination gives you the difference between the ratio of x and z, and y. This could be meaningful if you are trying to find the difference between two quantities that have different units.

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Answer:

Step-by-step explanation: