Instructions: Identify the vertices of the feasible region and use them to find the maximum and/or minimum value for the given linear programming constraints.

System of Linear Programming:

z=−3x+5y

x+y≥−22

x−y≥−4

x−y≤2

Minimum value of z:

4. Compute the derivative.

[tex]y = 2x^2 - x - 1 \implies \dfrac{dy}{dx} = 4x - 1[/tex]

Find when the gradient is 7.

[tex]4x - 1 = 7 \implies 4x = 8 \implies x = 2[/tex]

Evaluate [tex]y[/tex] at this point.

[tex]y = 2\cdot2^2-2-1 = 5[/tex]

The point we want is then (2, 5).

5. The curve crosses the [tex]x[/tex]-axis when [tex]y=0[/tex]. We have

[tex]y = \dfrac{x - 4}x = 1 - \dfrac4x = 0 \implies \dfrac4x = 1 \implies x = 4[/tex]

Compute the derivative.

[tex]y = 1 - \dfrac4x \implies \dfrac{dy}{dx} = -\dfrac4{x^2}[/tex]

At the point we want, the gradient is

[tex]\dfrac{dy}{dx}\bigg|_{x=4} = -\dfrac4{4^2} = \boxed{-\dfrac14}[/tex]

6. The curve crosses the [tex]y[/tex]-axis when [tex]x=0[/tex]. Compute the derivative.

[tex]\dfrac{dy}{dx} = 3x^2 - 4x + 5[/tex]

When [tex]x=0[/tex], the gradient is

[tex]\dfrac{dy}{dx}\bigg|_{x=0} = 3\cdot0^2 - 4\cdot0 + 5 = \boxed{5}[/tex]

7. Set [tex]y=5[/tex] and solve for [tex]x[/tex]. The curve and line meet when

[tex]5 = 2x^2 + 7x - 4 \implies 2x^2 + 7x - 9 = (x - 1)(2x+9) = 0 \implies x=1 \text{ or } x = -\dfrac92[/tex]

Compute the derivative (for the curve) and evaluate it at these [tex]x[/tex] values.

[tex]\dfrac{dy}{dx} = 4x + 7[/tex]

[tex]\dfrac{dy}{dx}\bigg|_{x=1} = 4\cdot1+7 = \boxed{11}[/tex]

[tex]\dfrac{dy}{dx}\bigg|_{x=-9/2} = 4\cdot\left(-\dfrac92\right)+7=\boxed{-11}[/tex]

8. Compute the derivative.

[tex]y = ax^2 + bx \implies \dfrac{dy}{dx} = 2ax + b[/tex]

The gradient is 8 when [tex]x=2[/tex], so

[tex]2a\cdot2 + b = 8 \implies 4a + b = 8[/tex]

and the gradient is -10 when [tex]x=-1[/tex], so

[tex]2a\cdot(-1) + b = -10 \implies -2a + b = -10[/tex]

Solve for [tex]a[/tex] and [tex]b[/tex]. Eliminating [tex]b[/tex], we have

[tex](4a + b) - (-2a + b) = 8 - (-10) \implies 6a = 18 \implies \boxed{a=3}[/tex]

so that

[tex]4\cdot3+b = 8 \implies 12 + b = 8 \implies \boxed{b = -4}[/tex].

If a data set contains three points, and two of the residuals are -10 and 20, the third residual is 10 (option B).

A residual is the difference between the observed value and the estimated value of the quantity of interest.

The residual of a data points should normally sum up to zero (0). This means the following applies:

-10 + 20 + x = 0

x = 10

Therefore, if data set contains three points, and two of the residuals are -10 and 20, the third residual is 10.

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Answer:

A

Step-by-step explanation:

x - 4 / (x + 5)(x - 1)

let's expand:

x - 4 / x² + 4x - 5

4 - 4 / 16 + 16 - 5 = 0 so answer is 4

Answer:

[tex]\frac{x^2}{9}+\frac{y^2}{4}=1[/tex]

Step-by-step explanation:

So an ellipse can be expressed in two different but very similar forms:

Horizontal Major Axis:

   [tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]

Vertical Major Axis:
   [tex]\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1[/tex]

In both equations the length of the major axis is "2a" and the length of the minor axis is "2b"

In the equation you provided, the major axis is horizontal, and the minor axis is vertical.

So looking at the horizontal length, you can see that it's 6, and since this ellipse has a horizontal major axis, that means 2a=6, which means a=3

Looking at the vertical length, you can see that it's 4, and since the ellipse has a vertical minor axis, that means 2b=4, which means b=2

The last thing to note is that the center of an ellipse is (h, k) in the equation, and since here the center is (0, 0) it's (x-0)^2 and (y-0)^2 in the denominator which is just x^2 and y^2

So let's plug in the values into the equation:

[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]

(h, k) = (0, 0)

a = 3

b = 2

[tex]\frac{x^2}{3^2}+\frac{y^2}{2^2}=1\\\\\frac{x^2}{9}+\frac{y^2}{4}=1[/tex]

Here we go ~

The given figure is of a horizontal ellipse with :

so, a = 3

hence, b = 2

And as it's shown in the figure, the ellipse has its centre at origin, so we can write it's equation as :

[tex]\qquad \sf  \dashrightarrow \: \cfrac{ {x}^{2} }{ {a}^{2} } + \cfrac{ {y}^{2} }{b {}^{2} } = 1[/tex]

[ now, plug in the values ]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{ {x}^{2} }{ {3}^{2} } + \cfrac{ {y}^{2} }{2 {}^{2} } = 1[/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{ {x}^{2} }{ {9}^{} } + \cfrac{ {y}^{2} }{4 {}^{} } = 1[/tex]

That's all, ask me if you have any questions ~

a.i) The number of inches per minute that the Galapagos tortoise walks are 216 inches (6 x 36).

a.ii) The time it would take a Galapagos tortoise to walk across the U.S. in hours is 2,933 hours (1,056,000/(6 x 60).

b.i) The Hare runs 31,680 inches per minute (1,900,800/60).

b.ii) The time it would take the Hare to complete the race across the U.S. in hours is 20 hours (600/30).

c.) For the Tortoise to pass the Hare, the Hare needs to sleep for 2,913 hours (2,933 - 20) or 121.4 days (2,933/24).

In this exercise, the following equivalent values are used for computing the distance and speed:

Speed of Galapagos Tortoise per minute = 6 yards/ min (60 yards in 10 minutes)

Speed of Hare = 30mph

30 miles = 52,800 yards (30 x 1,760)

52,800 yards = 1,900,800 inches (52,800 x 36)

= 880 yards per minute (52,800/60)

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[tex]f(x) = \sqrt[3]{11x} \\ y = \sqrt[3]{11x} \\ x = \sqrt[3]{11y} \\ x {}^{3} = 11y \\ y = \frac{x {}^{3} }{11} [/tex]

[tex]f(x) = \frac{x}{7} + 10 \\ y = \frac{x}{7} + 10 \\ x = \frac{y}{7} + 10 \\ x - 10 = \frac{y}{7} \\ y = \frac{x - 10}{7} [/tex]

[tex]f(x) = \frac{7}{x} - 2 \\ y = \frac{7}{x} - 2 \\ x = \frac{7}{y} + 2 \\ x - 2 = \frac{7}{y} \\ y = \frac{7}{x - 2} [/tex]

[tex]f(x) = 9x - 6 \\ y = 9x - 6 \\ x = 9y - 6 \\ x + 6 = 9y \\ y = \frac{x + 6}{9} = g(x)[/tex]

What is ordinary number?

1 : a number designating the place (such as first, second, or third) occupied by an item in an ordered sequence — see Table of Numbers. 2 : a number assigned to an ordered set that designates both the order of its elements and its cardinal number.

Answer:

200

Step-by-step explanation:

If 40% are roses than 60% must be sunflowers (40% + 60% = 100%)

We are trying to find 60% of some number is 120.

Let n = the total number of flowers

.6n = 120  Divide both sides by .6

n = 200

Based on the mean of the sample and the proportion who read below grade level, the probability that a second sample would have a proportion greater than 0.54 is 0.9772.

The probability that the second sample would be selected with a proportion greater than 0.54 can be found as:

P (x > 0.54) = P ( z > (0.54 - 0.30) / 0.12))

Solving gives:

P (x > 0.54) = P (z > 2)

P (x > 0.54) = 0.9772

In conclusion, the probability that the second sample would be selected with a proportion greater than 0.54 is 0.9772.

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[tex]g(x)=f(4x)=(4x)^2 = 16x^2[/tex]

The graph is shown in the attached image.

Answer:

480

Step-by-step explanation:

1/5 of 400 =   1/5 * 400 = 80

  increase 400 by 80   = 480

From the standard normal table, the respective probabilities are; 0.908 and 0.841

Formula for calculating the standard score or z score is:

z = (x - μ)/σ

where:

z is the standard score

x is the raw score

μ is the population mean

σ is the population standard deviation

A) We are given;

x = 95

μ = 110

σ = 15

Thus;

z = (95 - 110)/15

z = 1.33

From the normal standard distribution table we can find the probability from the z-score as;

P(x > 95) = 1 - 0.091759.

P(x > 95) = 0.908

B) We are given;

x = 125

μ = 110

σ = 15

Thus;

z = (125 - 110)/15

z = 1

From the normal standard distribution table we can find the probability from the z-score as;

P(x > 95) = 1 - 0.158655.

P(x > 95) = 0.841

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The domain of the function is [-4, 4) and the range of the function is [-5, 2)

The domain

As a general rule, it should be noted that the domain of a function is the set of input values or independent values the function can take.

This means that the domain is the set of x values

From the graph, we have the following intervals on the x-axis

x = -4 (closed circle)

x =4 (open circle)

This means that the domain of the function is [-4, 4)

The range

As a general rule, it should be noted that the range of a function is the set of output values or dependent values the function can produce.

This means that the range is the set of y values

From the graph, we have the following intervals on the y-axis

y = -5 (closed circle)

y = 2 (open circle)

This means that the range of the function is [-5, 2)

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Answer: B. All real numbers

Step-by-step explanation:

See attached image.

So, the equation is sin(x + y)/sin(x - y) = (1 + cotxstany)/(1 + cotxtany)

The question has to to with trigonometric identities?

Trigonometric identities are equations that show the relationship between the trigonometric ratios.

Given the equation sin(x + y)/sin(x - y)

Using the trigonometric identities.

So, sin(x + y)/sin(x - y) = (sinxcosy + cosxsiny)/(sinxcosy + cosxsiny)

Dividing the rnumerator and denominator of ight hand side by sinx, we have

sin(x + y)/sin(x - y) = (sinxcosy + cosxsiny)/sinx/(sinxcosy + cosxsiny)/sinx

sin(x + y)/sin(x - y) = (sinxcosy/sinx + cosxsiny/sinx)/(sinxcosy/sinx + cosxsiny/sinx)

= (cosy + cotxsiny)/(cosy + cotxsiny) (since cosx/sinx = cotx)

Dividing the numerator and denominator of the right hand side by cosy, we have

= (cosy + cotxsiny)/cosy/(cosy + cotxsiny)/cosy

= (cosy/cosy + cotxsiny/cosy)/(cosy/cosy + cotxsiny/cosy)

= (1 + cotxstany)/(1 + cotxtany)   [since siny/cosy = tany]

So,  sin(x + y)/sin(x - y) = (1 + cotxstany)/(1 + cotxtany)

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Answer: [tex]\large\boxed{12x^3-30x+3=0}[/tex]

Step-by-step explanation:

[tex]Solve:\\\ 6(2x^2-5)x=-3\\\\Use\ distributive\ property\ first\\\\(12x^2-30)x=-3\\\\Now\ multiply\ the\ terms\ in\ ()\ by\ 'x'\\\\12x^3-30x=-3\\\\\large\boxed{12x^3-30x+3=0}[/tex]

[tex]l = sec {}^{2} x - 1 \\ l = \frac{1}{cos {}^{2} x} - \frac{cos {}^{2} x}{cos {}^{2} x} \\ l = \frac{1 - cos {}^{2} x}{cos {}^{2}x } \\ l = \frac{sin {}^{2} x}{cos {}^{2} x} = ( \frac{sinx}{cosx} ) {}^{2} = tan {}^{2} x[/tex]

Answer:

A

Step-by-step explanation:

If digits can be repeated, that means there are 10 options for each place in the pin code. 10*10*10*10 = 10,000

If digits can not be repeated, there are 10 options for the first digit, 9 options for the second digit, 8 options for the third digit, and 7 options for the fourth digit. 10*9*8*7 = 5040

Answer:

See Below

Step-by-step explanation:

y = a (x-7)^2 -3       where 'a'  is any number except 0

[tex] \frac{3a}{4} > - 16 \\ 3a > - 64 \\ a > \frac{ - 64}{3} \\ a > 21 \frac{1}{3} [/tex]

[tex]\huge\text{Hey there!}[/tex]

[tex]\huge\textbf{Equation:}[/tex]

[tex]\mathsf{\dfrac{3}{4}a > -16}[/tex]

[tex]\huge\textbf{Simplifying it:}[/tex]

[tex]\mathsf{\dfrac{3}{4}a > -16}[/tex]

[tex]\mathsf{\dfrac{3}{4}a > - \dfrac{16}{1}}[/tex]

[tex]\huge\textbf{Divide \boxed{\dfrac{4}{3}} to both sides:}[/tex]

[tex]\mathsf{\dfrac{4}{3}\times\dfrac{3}{4}a > -16\times\dfrac{4}{3}}[/tex]

[tex]\huge\textbf{Simplify it:}[/tex]

[tex]\mathsf{a > \dfrac{4}{3}\times -16}[/tex]

[tex]\mathsf{a > - \dfrac{16}{1} \times\dfrac{4}{3}}[/tex]

[tex]\mathsf{a > \dfrac{-16\times4}{1\times3}}[/tex]

[tex]\mathsf{a > \dfrac{-64}{3}}[/tex]

[tex]\mathsf{a > -\dfrac{64}{3}}[/tex]

[tex]\mathsf{a > -21 \dfrac{1}{3}}[/tex]

[tex]\huge\textbf{Therefore, your answer should be:}[/tex]

[tex]\huge\boxed{- \frak{21\dfrac{1}{3}}}\huge\checkmark[/tex]

[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]

The sequence of angles from smallest to largest is m∠T,m∠S,m∠R.

Given three angles be m∠T=4x-52°,m∠S=x+38°,m∠R=2x+47°.

We are required to arrange the angles in ascending order.

Ascending order means values that are written in a way that smallest values come first and larger values comes last.

First we have to find the angles at one value of x.

The values will differ as the values of x differ so we have to choose point or value of x.

Suppose x=20.

Then the angles are:

m∠T=4x-52°

=4*20-52

=80-52

=28°

m∠S=x+38°

=20+38

=58°

m∠R=2x+47°

=2*20+47

=40+47

=87°

Hence the sequence of angles from smallest to largest is m∠T,m∠S,m∠R.

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Answer:

$15666.83 (2dp)

Step-by-step explanation:

Mean = Total of all values / Number of Values

= [tex]\frac{17484+14978+13521+14500+18540+14978}{6}[/tex]

=[tex]\frac{94001}{6}[/tex]

= $15666.83 (2dp)

Answer:

k ≈ 50.77

Step-by-step explanation:

using the cosine ratio in the right triangle

cos19° = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{48}{k}[/tex] ( multiply both sides by k )

k × cos19° = 48 ( divide both sides by cos19° )

k = [tex]\frac{48}{cos19}[/tex] ≈ 50.77 ( to 2 dec. places )

Hi :)

            We'll use sohcahtoa to solve this problem

[tex]\Large\boxed{\begin{tabular}{c|1} \sf{Sohcahtoa} ~&~~~~~Formula~~~~~~~ \\ \cline{1-2} \ \sf{Soh} & Opp~\div \text{hyp}\\\sf{Cah} & Adj \div \text{hyp}\\\sf{Toa} & Opp \div \text{adj} \end{tabular}}[/tex]

Looking at our triangle, we can clearly see that we have :

Set up the ratio

[tex]\longrightarrow\darkblue\sf{cos(19)=\dfrac{48}{k}}[/tex]

solve for k

[tex]\longrightarrow\darkblue\sf{k\cos(19)=48}[/tex] > multiply both sides by k to clear the fraction

[tex]\longrightarrow\darkblue\sf{k=\dfrac{48}{\cos(19)}}[/tex]  > divide both sides by cos (19)

[tex]\star\longrightarrow\darkblue\sf{k\approx50.77}\star[/tex]

[tex]\tt{Learn~More ; Work\ Harder}[/tex]

:)

DE ≅ CE given that side AD and side BC are equal and angle ∠BCD and angle ∠ADC are equal. This can be obtained by using triangle congruency theorems.

Triangle congruency theorems required in the question,

In the question it is given that,

⇒ Side DE and side CE are equal ⇒ AD ≅ BC

⇒ angle ∠BCD and angle ∠ADC are equal ⇒ ∠BCD ≅ ∠ADC

Therefore we can say that,

ΔADC ≅ ΔBCD according to the SAS triangle congruency theorem

ΔAED ≅ ΔBEC according to the AAS triangle congruency theorem

Thus DE ≅ CE since corresponding parts of congruent triangles are congruent (CPCTC).

Hence DE ≅ CE given that side AD and side BC are equal and angle ∠BCD and angle ∠ADC are equal.

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Answer:

mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size.

Answer:

A total of 204 textbooks were sold with 153 math textbooks and 51 psychology textbooks being sold, as the number of math textbooks was three times the number of psychology textbooks.

Given that,

The combined total of math and psychology textbooks sold in a week is 204.

The number of math textbooks sold is three times the number of psychology textbooks sold.

To solve this problem,

Assign variables to represent the number of math and psychology textbooks sold.

Let's say "M" represents the number of math textbooks and "P" represents the number of psychology textbooks.

We know that the combined total of math and psychology textbooks sold is 204,

So the equation be:

M + P = 204  .....(i)

We are also given that the number of math textbooks sold was three times the number of psychology textbooks sold.

In equation form, this can be expressed as:

M = 3P

Now, we can substitute the value of M in terms of P into the equation (i):

3P + P = 204

Combining like terms, we get:

4P = 204

Dividing both sides by 4, we find:

P = 51

So, the number of psychology textbooks sold is 51.

To find the number of math textbooks sold,

Substitute this value back into the equation:

M = 3P

M = 3(51)

M = 153

Therefore, the number of math textbooks sold is 153.

Hence,

153 math textbooks and 51 psychology textbooks were sold.

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Answer:

below

Step-by-step explanation:

Answer:

A. [tex]\frac{17}{52}[/tex]

B. [tex]\frac{17}{52}[/tex]

C. [tex]\frac{2}{13}[/tex]

Step-by-step explanation:

A.

There are 52/4 diamonds in the deck and 4 '5's in the dech of cards

52/4 = 13 + 4 = 17

Therefore, you have a  [tex]\frac{17}{52}[/tex] chance of drawing one of those cards.

B.

There are 13 hearts in the deck and 4 jacks. Therefore, your odds are the same : [tex]\frac{17}{52}[/tex]

C.

There are 4 jacks in a deck of cards and 4 '8's in a deck of cards

Therefore your probability is [tex]\frac{8}{52}[/tex] which simplifies to [tex]=\frac{2}{13}[/tex]

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