In a school of 2000 students, the ratio of teachers to students is 3:80.
Some teachers join the school and the ratio changes to 1:20.
Find the number of teachers who joined the school.

1.  (a)The  vertex of the  function [tex]y=x^{2} +5x-7[/tex]  is [tex](-\frac{5}{2} ,-\frac{53}{4} )[/tex]

How is the vertex calculated?

In function [tex]y=x^{2} +5x-7[/tex],

a= 1 ,b= 5, c= -7

For a function   [tex]y=ax^{2} +bx+c[/tex]  , where (a≠0),

The vertex is [tex](-\frac{b}{2a} ,\frac{4ac-b^{2} }{4a} )[/tex]

[tex]x_v=-\frac{b}{2a}\\\\ =-\frac{5}{2*1} \\\\=-\frac{5}{2} \\\\\\y_v=\frac{4ac-b^{2} }{4a} \\\\=\frac{4*1*(-7)-(5^{2} )}{4*1}\\\\ =-\frac{53}{4}[/tex]

So, vertex  [tex](-\frac{5}{2} ,-\frac{53}{4} )[/tex] is minimum because a>0

(b)The solutions are [tex](\frac{\sqrt{53} -5}{2},0) (\frac{-5-\sqrt{53} }{2},0)[/tex]

How is the solution calculated?

When a function [tex]y=ax^{2} +bx+c =0[/tex]  where (a≠0),

[tex]x=\frac{-b\pm\sqrt{b^{2} -4ac} }{2a} \\\\x=\frac{-5\pm\sqrt{5^{2} -4(1)(-7)} }{2(1)} \\\\x=\frac{-5\pm\sqrt{53} }{2} \\\\x=\frac{\sqrt{53} -5}{2} , x=\frac{-5-\sqrt{53} }{2} \\\\\text{The solutions are } (\frac{\sqrt{53} -5}{2},0) (\frac{-5-\sqrt{53} }{2},0)[/tex]

2. (a)The  vertex of the  function [tex]y=-x^{2} +3x+8[/tex]  is [tex](\frac{3}{2} ,\frac{41}{4} )[/tex]

How is the vertex calculated?

In function [tex]y=-x^{2} +3x +8[/tex]

a= -1 ,b= 3, c= 8

For a function   [tex]y=ax^{2} +bx+c[/tex]  , where (a≠0),

The vertex is [tex](-\frac{b}{2a} ,\frac{4ac-b^{2} }{4a} )[/tex]

[tex]x_v=-\frac{b}{2a}\\\\ =-\frac{3}{2*(-1)} \\\\=\frac{3}{2} \\\\\\y_v=\frac{4ac-b^{2} }{4a} \\\\=\frac{4*(-1)*(8)-(3^{2} )}{4*(-1)}\\\\ =\frac{41}{4}[/tex]

So, vertex  [tex](\frac{3}{2} ,\frac{41}{4} )[/tex] is maximum because a<0

(b)The solutions are [tex](\frac{3+\sqrt{41} }{2},0) (\frac{3-\sqrt{41} }{2},0)[/tex]

How is the solution calculated?

When a function [tex]y=ax^{2} +bx+c =0[/tex]  where (a≠0),

[tex]x=\frac{-b\pm\sqrt{b^{2} -4ac} }{2a} \\\\x=\frac{-3\pm\sqrt{3^{2} -4(-1)(8)} }{2(-1)} \\\\x=\frac{-3\pm\sqrt{41} }{-2} \\\\x=\frac{3\pm\sqrt{41} }{2}\\\\x=\frac{3-\sqrt{41}}{2} , x=\frac{3+\sqrt{41} }{2} \\\\\text{The solutions are } (\frac{3+\sqrt{41} }{2},0) (\frac{3-\sqrt{41} }{2},0)[/tex]

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The dinner bill of Sally before the tip is $53.

Given that, Sally left an 18% tip. The tip was 9.50.

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

Let the dinner bill of Sally be x.

Now, 18% of x = 9.50

⇒ 18/100 × x = 9.50

⇒ 0.18x = 9.50

⇒ x = 9.50/0.18

⇒ x = 52.77

≈ 53

Therefore, the dinner bill of Sally before the tip is $53.

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Answer:

(x+3)(x–8)

Step-by-step explanation:

Normally you will always write factors in this form: (variable + smallest number)(variable – largest number)

Correct form: (x+2)(x–8)

Incorrect form: (x–2)(x+8)

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We will improve the argument by stating that two intersecting lines have vertically opposite angles equal in magnitude.

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The exponential equation [tex]14=5^x[/tex] is equivalent to the  given logarithmic equation.

The exponential function that is represented as y=[tex]m^x[/tex], where:

m=base, m>0

x= exponent

Knowing some of the main logarithm rules.

example: [tex]log(a*b)= log a + log b[/tex]

example: [tex]log\frac{a}{b} = log a - log b[/tex]

example: [tex]log\frac{a^b} = b*log a[/tex]

For solving this question, you should apply the logarithm rules and  rewrite the function as an exponential equation.

[tex]log_5(x)}-{log_5(25)=7[/tex]

[tex]\frac{log_5(x)}{log_5(25)} =7[/tex]

[tex]{log_5(x)}=7{log_5(25)}[/tex]

[tex]{log_5(x)}=7{log_5(5^2)}[/tex]

[tex]{log_5(x)}=7*2[/tex]

[tex]{log_5(x)}=14[/tex]

[tex]x=5^{14^}[/tex]

Therefore, the  exponential function is [tex]14=5^x[/tex]

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Rotate R and reflect 90

Step-by-step explanation:

Answer:

Step-by-step explanation:

To solve for x, isolate this equation from left to right.

4x+10=-26

First, subtract by 10 from both sides.

4x+10-10=-26-10

Solve.

-26-10=-36

4x=-36

Then, you divide by 4 from both sides.

4x/4=-36/4

Solve.

-36/4=-9

[tex]\Rightarrow \boxed{\sf{x=-9}}[/tex]

As a result, the solution is x=-9, which is the correct answer.

I hope this helps, let me know if you have any questions.

Answer:

-9

Step-by-step explanation:

First you subtract 10 from each side

4x + 10 = -26

-10 -10

then you divide by your x

4x = -36

4 4

the 4s cancel out leaving you with x only

x = -36/4

x = -9

Answer:

Step-by-step explanation:

The repeated part is 194805, six digits. Let the fraction be x.

Convert as follows:

Find prime factors of both numerator and denominator and simplify by cancelling common factors:

The common factors are:

When they cancel we are left with:

Answer:

Step-by-step explanation:

1) Find DGE

Bisecting an angle makes it half. A straight line is 180

180/2 = DGE

90 = DGE

2) Find FGE

Bisecting an angle makes it half. A straight line is 180

180/2 = FGE

90 =  FGE

Answer:

N<3

Step-by-step explanation:

[tex]m = \frac{y2 - y1}{x2 - x1} \\ m = \frac{7 - 6}{3 - 1} \\ m = \frac{1}{2} [/tex]

I ALSO PROVIDED YOU WITH THE FORMULA OF FINDING THE GRADIENT/SLOPE FOR FUTURE USE

HOPE THIS HELPS.

Answer: Y= 1/2x + 5.5

Answer:

-20-43w

Step-by-step explanation:

5(-4-8w)-3w

-20-40w-3w

-20-43w

Tanya yearly salary is

$35000 + 8% of her sales

Let her sales be x

Since, her sales is x

35000 + 0.08x > salary

To determine her necessary sales to earn over $50, 000

Let $50, 000 be her benchmark salary

35,000 + 0.08 * x > 50000

35000 + 0.08x > 50000

0.08x > 50000 - 350000

0.08x > 15000

Divide both sides by 0.08

0.08x/0.08 > 15000/0.08

x > 15000 / 0.08

x > 187,500

Therefore, she will need a sales of 188,000 and above to earn above $50,000

Answer:95

Step-by-step explanation:

This can simply be answered by adding the first digit of both numbers which would be 9 then add the secound number of both  numbers and that would be 5

The answer is 95

Answer:

if its 6^2 + 3^3 then its 63

Step-by-step explanation:

6^2 simplified is 36 and 3^3

6 x 6 = 36 + 27 = 63

                                        3 x 3 x 3 =27

                                        3 x 3= 9

                                        9 x 3= 27

Answer:

the answer is 8.

Step-by-step explanation:

Use the Pythagoras Theorem since it is a triangle.

[tex]a^{2}+b^{2}= c^{2}\\\\a^{2}+15^{2}= 17^{2}\\ a^{2}+225=289\\ a^{2}=289-225\\ a^{2}= 64\\ \sqrt{a}= \sqrt{64}\\ a=8[/tex]

Since you have the hypotenuse and base, just put those into the equation and find the missing side.

The correct answer is option B that is No, because two points with the same x-value have different y values.

A relation in mathematics may be defined as the term which describes the relationship between the input and output variables. A function may be defined as the expression in which for one value of input variable x there is only one output variable y. The input variable is called independent variable and output variable is called dependent variable. From the graph given in question it can be seen that at point x = 2 there are two points on the y axis that are y = 2 and y = 11. So, this violates the basic definition of function as for input x there are two outputs y and hence it cannot be regarded as function.

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Answer:

I believe you forgot to add the attachment

Step-by-step explanation:

There is no math problem I can see to help you

The two column proof is written as follows

Statement                                                    Reason

1. m ∠ d = 125°                                                 Given

2. m ∠ r = 125°                                                 Given  

3. m ∠ d = m ∠ r                                       Definition of equality

4. ∠ d ≅ ∠ r                                              Definition of congruent angles

This is a method of proof used in geometry. It helps to show how the required proof came to be possible. The major items in two column proof is:

The term equality as used in mathematics is defined to mean that the terms in comparison can be replace by the other

Using the given question as an instance:

m ∠ d = 125° and m ∠ r = 125°

This means that the two angles have same value and they can replace each other.

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The function is

The limit when x->+/- infinite are (analitically)

And, from the figure,

Therefore,

Therefore, the answer is option C, a>1.

Answer:

Step-by-step explanation:

According to the remainder theorem, the remainder when f(x) is divided by (x - n) is f(n).

As per question we are given:

Substitute and solve for c:

Note: I got a different value of c. This may be a result of a typo in the given expression. I provided a guide to solve such problems. Let me know is anything is unclear.

Option B. 2w + 11 = 1.5w + 14, w = 6 weeks is the correct answer

Miles run by Denasia = 11 miles

Miles run by Kenya = 14 miles

Let w represent the number of weeks after which both Kenya and Denasia will be running the same number of miles

Formulating the equation we get:

Miles run by Denasia + Increase in miles each week*Number of weeks = Miles run by Kenya + Increase in miles each week*Number of weeks

11 + 2w = 14 + 1.5w

0.5w = 3

w = 6

Hence, option B is the correct answer according to the given contraints and both Denasia and Kenya will run the same distance in the 6th week

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The functions that are equivalent are

What are roots of a number?

The root of a number is the inverse of the exponents. for instance

a squared has an inverse of square root of a.  

mathematically:

a^2 = (a)^1/2  has an inverse of √a

The data given:

[tex]\sqrt[4]{162^{x} }[/tex]

Solving the given data for the equivalence

[tex]\sqrt[4]{162^{x} }=162^{x/4}[/tex]

The above proves option A correct

[tex]\sqrt[4]{(3*3*3*3*2)^{x} }[/tex]

[tex]\sqrt[4]{(3^{4} *2)^{x} }[/tex]

[tex](3\sqrt[4]{(2) })^{x}[/tex]

The above solution makes option B correct

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

The above proves Option D correct

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complete question

Which functions are equivalent to f (x) = RootIndex 4 StartRoot 162 EndRoot Superscript x? Check all that apply.

f (x) = 162 Superscript StartFraction x Over 4

f (x) = (3 RootIndex 4 StartRoot 2 EndRoot) Superscript x

f (x) = 9 RootIndex 4 StartRoot 2 EndRoot Superscript x

f (x) = 126 Superscript StartFraction 4 Over x

f (x) = left-bracket 3 (2 Superscript one-fourth Baseline) right-bracket Superscript x

Answer:

[tex]\begin{aligned}&\phantom{=.} \dfrac{1}{8}x \cdot \dfrac{-4}{5}x\\\\&=\dfrac{1}{2} \cdot \dfrac{\boxed{-1}}{5} \cdot x \cdot \boxed{x}\\\\& = \dfrac{\boxed{-1}}{\boxed{10}} \;x\;^\boxed{2}\end{aligned}[/tex]

Step-by-step explanation:

According to the Commutative Property Law of Multiplication, changing the order or position of the numbers does not change the end result.

Therefore, collect like terms by moving the fractions to the left and the variables to the right.

As the denominator of the first fraction has been divided by 4, the numerator of the second fraction should also be divided by 4.

Therefore:

[tex]=\dfrac{1}{2} \cdot \dfrac{\boxed{-1}}{5} \cdot x \cdot \boxed{x}\\[/tex]

[tex]\textsf{Apply\;the\;fraction\;rule} \quad \dfrac{a}{c}\cdot \dfrac{b}{d}=\dfrac{ab}{cd}.[/tex]

[tex]\textsf{Apply\;the\;exponent\;rule}\quad \:aa=a^2.[/tex]

Therefore:

[tex]=\dfrac{\boxed{-1}}{\boxed{10}} \;x\;\boxed{^2}[/tex]

The distance between the person and the shuttle is changing at a rate of 640 feet/sec when the shuttle is 4000 ft high and is rising at a rate of 800 ft/sec.

Considering x to be the height of the triangle and y to be the hypotenuse of the triangle, a right-angled triangle will be formed according to the given information having a base of 3000 feet which is the distance of the person from the launch pad.

In this right-angle triangle x will be the height of the shuttle and y will be the distance between the person and the shuttle.

Now we apply the Pythagorean theorem to the triangle;

y² = x²+(3000)²

Now differentiating this equation with respect to time,t :

2y (dy/dt) = 2x(dx/dt) + 0

dy/dt = x(dx/dt)/y

As the shuttle is rising at a speed of 800 ft/sec, (dx/dt)=800

Substitute 4000 for x into the equation, y²=x²+(3000)², to find y;

y² = (4000)²+(3000)²

y = 5000

Now we substitute 4000 for x, 5000 for y, and 800 for dx/dt in the equation, dy/dt = x(dx/dt)/y

dy/dt = 4000(800) / 5000

dy/dt = 640

Therefore, the distance between the person and the shuttle is increasing at a rate of 640 feet/sec.

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Answer:[tex]3(5y - 7) - 2(9y - 11) = 4(8y - 13) - 17 \\➪ 15y - 21 - 18y + 22 = 32y - 52 - 17 \\ ➪15y - 18y - 32y = - 52 - 17 + 21 - 22\\➪-35y = - 70 \\➪35y = 70 \\ \: (cancel \: \: out \: "-" \: from \: both \: sides) \\ ➪y = \frac{ 70}{35} = 2 \\ \\ [/tex]

The two cars are 5 kilometers far apart from each other.

What is Pythagoras theorem?

The Pythagoras theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides.

Given that;

Two cars leave the same parking lot, with one heading north and the other heading east.

After several minutes, the northbound car has traveled 3 kilometers, and the eastbound car has traveled 4 kilometers.

Now,

The straight path is find by using the Pythagoras theorem as;

Straight path = √3² + 4²

                    = √9 + 16

                    = √25

                    = 5 km

Therefore,

The two cars are 5 kilometers far apart from each other.

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The smallest number from the given options is

[tex]5.48 × {10}^{ - 8} [/tex]

The correct answer option is option d

A.

[tex]1.28 × {10}^{ - 4} [/tex]

= 1.28 × 0.0001

= 0.000128

B.

[tex]2.85 × {10}^{ - 6} [/tex]

= 2.85 × 0.000001

= 0.00000285

C.

[tex]3.24 × {10}^{ - 5} [/tex]

= 3.24 × 0.00001

= 0.0000324

D.

[tex]5.48 × {10}^{ - 8} [/tex]

= 5 48 × 0.00000001

= 0.0000000548

Therefore, the smallest number is

[tex]5.48 × {10}^{ - 8} [/tex]

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Answer:

3.75 represents how much the flower grows every week.

Step-by-step explanation: