Find the minimum of the objective
function, f, subject to the constraints
f = 7x + 4y
5x + y ≥
60
x+6y ≥ 60
4x+6y ≥ 204
x ≥ 0, y ≥0
Minimum =[?]

A polynomial function with rational coefficients and the given roots √3,-5 is :  x² + (5-√3)x -5√3

In mathematics, polynomial function can be defined as the equation which consist of coefficient, variables and exponents. A polynomial equation has positive numbers as their exponent which is also known as its degree. All the equations such as monomials, quadratic equation, cubic equation, linear equation of one degree, two degree and more  which has non  negative number as their powers all are part of polynomial equation and can be taken as types of the polynomial equations.

Some examples of polynomial equation are :

According to question,

roots are √3 , -5

Thus the factors are :

(x - √3 ) ( x - (-5) )

=> ( x - √3 )( x + 5 )

=> x²+ 5x - √3x - 5√3

=> x² + (5-√3)x -5√3      

Thus polynomial function for given roots is x² + (5-√3)x -5√3

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The lowest common multiple of 4 and 6 is 12

The lowest common multiple of 3 and 7 is 21.

The lowest common multiple of 5 and 10 is 10.

The lowest common multiple of 6 and 9 is 18.

The lowest common multiple of 4 and 6:

Multiple of 4 = 4 , 8, 12, 16, 20.

Multiple of 6 = 6, 12, 18, 24, 30

The lowest common multiple of 3 and 7:

Multiple of 3 =3, 6, 9, 12, 15, 18, ,21

Multiple of 7 = 7, 14, 21

The lowest common multiple of 5 and 10:

Multiple of 5 = 5, 10,

Multiple of 10 = 10, 20

The lowest common multiple of 6 and 9:

Multiple of 6 = 6, 12, 18, 24.

Multiple of 9 = 9, 18, 27

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keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the equation above

[tex]y = \stackrel{\stackrel{m}{\downarrow }}{\cfrac{2}{3}}x+4\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}}{\stackrel{slope}{\cfrac{2}{3}}~\hfill \stackrel{reciprocal}{\cfrac{3}{2}}~\hfill \stackrel{negative~reciprocal}{-\cfrac{3}{2}}}[/tex]

so, we're really looking for the equation of a line whose slope is -3/2 and that it passes through (2 , -6)

[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{-6})\hspace{10em} \stackrel{slope}{m} ~=~ - \cfrac{3}{2} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-6)}=\stackrel{m}{- \cfrac{3}{2}}(x-\stackrel{x_1}{2}) \implies y +6= -\cfrac{3}{2} (x -2) \\\\\\ y+6=-\cfrac{3}{2}x+3\implies y=-\cfrac{3}{2}x-3[/tex]

Answer to this question is:

Domain of given function is (-∞,-1)∪(-1,0)∪(0,∞),{x|x≠0,-1}

Range of given function is (-∞,0)∪(0,∞),{y|y≠0}

Inverse of given function  does not exist

The domain is all values of x that make the expression defined.The entire range of independent variable values is the domain of a function.

This definition can be stated simply as follows: The domain is the collection of all x-values that can be used to make the function "work" and produce actual y-values.

When choosing the domain, keep in mind that a fraction's denominator (bottom) cannot be 0.

Interval Notation:(-∞,-1)∪(-1,0)∪(0,∞)

Set-Builder Notation:{x|x≠0,-1}

The set of all valid y values constitutes the range..Graph may be used to find the values.

Interval Notation:(-∞,0)∪(0,∞)

Set-Builder Notation:{y|y≠0}

For calculation of inverse:

There is no inverse for any function that is not strictly monotonic. Whenever the first derivative of a function is ALWAYS zero, positive, or ALWAYS negative, the function is said to be monotonic. Basically, a function does not have an inverse function if there are two inputs for which the output is the same.

∴Domain of given function is (-∞,-1)∪(-1,0)∪(0,∞),{x|x≠0,-1}

Range of given function is (-∞,0)∪(0,∞),{y|y≠0}

Inverse of given function does not exist.

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Write 2 and 32 as a product of their prime factors then in index form. 2 = 2^1. 32 = 2x2x2x2x2 = 2^5. Use power rule (a^m)^n = a^mn to simplify:

32= 2^5

2= 2^1

(2^1)^1/2 x (2^5)^1/2 = 2^1/2 x (2^4/2x2^1/2)

2^1/2x4x2^1/2 = 4 x (2^1/2) ^2 = 4x2 = 8

=8

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Answer:

[tex]5 \dfrac{2}{3}\; \textsf {pounds}[/tex]

Step-by-step explanation:

Each dig gets 17/3 lbs of dog food

173 = 5 2/3

Answer:

(C) is correct.

Step-by-step explanation:

-12.8308

- as you can see this is a terminating decimal, with last digit = 8.

Answer:

Inverse property

Step-by-step explanation:

3/4 is the reciprocal of 4/3 which means they would cancel each other out. So you would only be left with x=x

the extraneous solution for the equation √ x - 3 = x - 5 is x = 4 or x = 7.

We are given an equation:

√ x - 3 = x - 5

Raise both sides to the power 2, we get that:

x - 3 = ( x - 5 )²

x - 3 = x² + 25 - 10 x

x² - 10 x - x + 25 + 3 = 0

x² - 11 x + 28 = 0

x² - 7 x - 4 x + 28 = 0

x( x - 7) - 4( x - 7) = 0

( x - 4 ) ( x - 7 ) = 0

x = 4 or x = 7.

Therefore, we get that, the extraneous solution for the equation √ x - 3 = x - 5 is x = 4 or x = 7.

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If the expression is log₃9x⁵,then the expanded form of this expression is [tex]log_{3}[/tex]9 +5[tex]log_{3}[/tex]x.

Given that the expression is log₃9x⁵.

We are required to find the expanded form of the expression.

Product property of logarithms says that the logarithm of a product is equal to a sum of logarithms and because logs are exponents, and we multiply like bases, we can add the exponents.

The expression is log₃9x⁵.

log₃9x⁵=is [tex]log_{3}[/tex]9 +[tex]log_{3}[/tex][tex]x^{5}[/tex]

= [tex]log_{3}[/tex]9 +5[tex]log_{3}[/tex]x

Hence if the expression is log₃9x⁵,then the expanded form of this expression is  [tex]log_{3}[/tex]9 +5[tex]log_{3}[/tex]x.

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Based on the number of cups of fresh cranberries needed for a recipe that serves 16 people, the number of cups of cranberries to make 28 servings of cranberry sauce is 7 cups

The number of cups of cranberries varies with the servings of cranberry sauce in a direct relationship.

If 4 cups of fresh cranberries can make 1 servings of cranberry sauce then 28 servings would need:

4       :        16

c       :         28

112 = 16c

c = 112 / 16

c = 7 cups

In conclusion, 7 cups of fresh cranberries are needed.

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Answer:

the value of 5 in 635,482 is 5,000

Step-by-step explanation:

to determine the place value of 5 you would have to start from the left to the right.

Answer:

Step-by-step explanation:

if f(x)=2x-4

then f(n+5)= 2(n+5)-4

                 =2n+10-4
                 =2n+6

Answer:

It would be 5^3 or 5 cubed.  Power a is 3

Step-by-step explanation:

Answer:

b) 16

c) 3

Step-by-step explanation:

b)

[tex](10 - 3 \times 2)^2\\\to (10 - 6)^2\\\to 4^2 \to 16[/tex]

c)

[tex]5^a = 125\\\to 5^a = 5^3\\\to a = 3[/tex]

The algebra property used on -9.97 + (4.68 + 13.08)= (-9.97 + 4.68) + (13.08) is; Associative property of addition.

We are given the number expression;

-9.97 + (4.68 + 13.08) = (-9.97 + 4.68) + (13.08)

There are different algebraic properties such as Associative Property, Commutative Property, distributive property, e.t.c

Now, Associative property of addition states that changing the grouping of addends does not change the sum.

For example, (4 + 5) + 6 = 4 + (5 + 6)

Similarly; (7 + 8) + 9= 7 + (8 + 9)

Looking at our question, we see that the only thing that was changed is the grouping of addends and it did not change the sum. This, we conclude that algebra property used is Associative property of addition.

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Answer:

166 degrees

Step-by-step explanation:

So first we need to find the angles in the triangle. One of the angles is 83 degrees since it's 180-97. Since the triangle is isosceles, the angle 'on top' of it is also 83 degrees. To find the last angle, you do 180-83-83, which is 14 degrees. To find x, you need to subtract 14 from 180, which is 166 degrees

Mr. Tan has a maximum of 76 students in his class, so he will have to give each one 36 pencils and 40 notebooks.

Arithmetic operations can also be specified by subtracting, dividing, and multiplying built-in functions.

We have been given that

Mr. Tan had 35 pencils and 42 notebooks

All of them to his students that he was short of 1 pencil and had an extra 2 notebooks.

Here the number of pencils = 35 and

the number of notebooks = 42 and

As per the given situation,

Therefore the required solution would be

maximum number of students = number of pencils + 1 + number of notebooks - 2

maximum number of students = 35+1+42-2 = 76

Therefore, Mr. Tan had 76 students.

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The concentration of hydrogen ions in seawater is 3.126 × 10⁻⁶ mol / dm³.

The pH level of seawater is 8.5.

If the pH is 8.5, and If the pH is 8.5, Assuming that this is measured under normal circumstances, we can apply the following relationship:

pH + pOH = pK_(w)

At 25 degrees Celcius, pK_(w) = 14

Where K_(w) is the dissociation constant for water = 1.0 × 10⁻¹⁴,

(At 25° C) but pK_(w) is the negative logarithm of K_(w).

pK_(w) = − log₁₀[ K_(w) ]

From this, we can convert pH, the measure of H₃O⁺ ions, into pOH, the measure of OH⁻ ions in the seawater:

pH + pOH = 14

8.5 + pOH = 14

pOH = 14 - 8.5

pOH = 5.5

Now,

pOH = − log₁₀[ OH⁻ ]

10^{−pOH} = [ OH⁻ ]

So,

10⁻⁵·⁵ = [ OH⁻ ] ≈ 3.126 × 10⁻⁶ mol / dm³

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since we know both triangles are similar then we can say that

[tex]\cfrac{TV}{QS}~~ = ~~\cfrac{TU}{QR}\implies \cfrac{x}{30}~~ = ~~\cfrac{13}{39}\implies 39x=390\implies x=\cfrac{390}{39}\implies \boxed{x=10}[/tex]

The percentage of the students who go to the movies twice a month or less and the percentage of the students that prefer comedies

40%

74%

This is further explained below.

Generally, the equation for is  mathematically given as

(a) Total number of students =200

Number of students who prefer action movies twice a month or less =60 Triceamonth or more =20

Therefore students prefer action makes =(60+20)/200

[tex]=\frac{\theta \phi}{200} 100\%[/tex]

=40%

(b) Number of students who go to movies twice a month er less,

For action =6o

For comely 88

Therefore % of students go to the movies twice a month or less -[tex]\left(\frac{60+80}{200}\right) \times 100 \%=(80+44)[/tex]$.

=74%

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Time taken by the copier to copy 60 color pages and 35 black and white pages is 3.7 minutes ( Option I ).

Given,

Time taken to copy pages ⋉ The number of pages being copied

Number of colour pages that can be copied in 1 minute = 21

Number of black and white pages that can be copied in 1 minute = 35

To find : Time taken by the copier to copy 60 color pages and 35 black and white pages

Given options :

(F) 0.9 minute (H) 2.9 minutes(G) 2.5 minutes (I) 3.7 minutes

Time taken by the copier to copy 60 colour pages

= 60 * 1 / 21 = 60 / 21 = 2.857 minutes

Time taken by the copier to copy 35 black and white pages

= 35 * 1 / 40 = 35 / 40 = 0.875 minutes

Total time taken by the copier to copy 60 color pages and 35 black and white pages

= 2.857 + 0.875 = 3.732 minutes

Therefore, time taken by the copier to copy 60 color pages and 35 black and white pages as 3.7 minutes.

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Answer:  [tex]y=-\frac{14}{5}[/tex]

Solving Linear Equations:

Solution with Steps:

[tex]14 =-5y[/tex]

[tex]\frac{14}{-5}=\frac{-5y}{-5}[/tex] - divide both sides by -5 to isolate for the variable

[tex]-\frac{14}{5}=y[/tex] - simplify to get your final answer

he spends 86 dollars, thus 100-86=14

your answer is 14

The measures of the angles are

1. Angle TLN, MZQ and NLM

The angles TLN and NLM are supplementary angles

So, we have

TLN + NLM = 180

This gives

3x - 18 + y = 180

Also, we have

TLN + MZQ = 180

So, we have

3x - 18 + 5x + 14 = 180

Evaluate the like terms

8x -4 = 180

This gives

8x = 184

Divide

x = 23

Substitute x = 23 in 3x - 18 + y = 180

3 x 23 - 18 + y = 180

Evaluate

51 + y = 180

So, we have

y = 129

The above computations imply that:

TLN = 3 x 23- 18 = 51 degrees

NLM = y = 129 degrees

MZQ = 5 x 23 + 14 = 129 degrees

2. Angle JZQ and TLN

The above means that

JZQ = TLN

So, we have

9x + 9 = 8x + 19

Evaluate

x = 10

So, we have

JZQ = TLN = 9 * 10 + 9

JZQ = TLN = 99 degrees

3. Angle ZMR, LMO and LMR

The angles ZMR and LMO are congruent angles

So, we have

ZMR = LMO

This gives

x + 14 = 2x - 6

Evaluate

x = 20

Also, we have

LMR + LMO = 180

So, we have

y + 2x - 6 = 180

y + 2 x 10 - 6 = 180

y + 14 = 180

So, we have

y = 166

The above computations imply that:

LMO= ZMR = 20 + 14 = 34 degrees

LMR = y = 166 degrees

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The result of simplifying each number 10¹/₂ . 10¹/₂ using the exponent rules is 10

To solve this exercise we have to resolve algebraic operations following the exponent rules.

10¹/₂ . 10¹/₂

Using the product rule that indicates that: the exponent result will be the addition of these exponents, we have:

10(¹/₂⁺¹/₂)

10(²/₂)

10¹

10

In mathematics an exponent is the number of time that a number, called (base) is multiplied by itself. It is also called, power or index.

Example: 3² = 3*3 = 9

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Answer:

x=5+4

x=9

hope this helps

Answer:

Step-by-step explanation:

a = g - b - c

x = (n - p)7n

x = 4 is not an extraneous solution.

Given equation [tex]2(2x)^{1/3}[/tex] + 1 = 5

Subtracting 1 on both the sides of the equation, we get

[tex]2(2x)^{1/3}[/tex] + 1 - 1 = 5 - 1

[tex]2(2x)^{1/3}[/tex] = 4

Dividing the equation by 2, we get

[tex]2(2x)^{1/3}[/tex] = 2

Taking cube of both the sides of the equation, we get

2x = 8

x = 4

Check for extraneous solution by keeping the value of x in original equation

[tex]2(2x)^{1/3}[/tex] + 1 = 5

[tex]2(2(4))^{1/3}[/tex] + 1 = 5

[tex]2(8)^{1/3}[/tex] + 1 = 5

2(2) + 1 = 5

5 = 5

LHS = RHS

Thus, x = 4 is not an extraneous solution.

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