the park is 6 miles due west of your house, and the library is 11 miles north of your house. How far is the shortest distance from the park to the library? Round to the nearest half mile.

Answer:

x^2 + y^2 = 5^2

or x^2 +y^2 = 25

Step-by-step explanation:

The equation of a circle is usually written in the form

(x-h)^2 + (y-k)^2 = r^2

Where (h,k) is the center and r is the radius

The center is at the origin so (h,k) = (0,0) and  the radius is 5 so r=5

x^2 + y^2 = 5^2

or x^2 +y^2 = 25

Answer:

x² + y² = 25

Step-by-step explanation:

(x - h)² + (y - k)² = r²

Center ( h⁰, k⁰)  

radius = 5/r

(x - 0)² + (y - 0)² = 5² ⇒ x² + y² = 25

This will be your answer : x² + y² = 25

Answer:

Option A is correct.

Step-by-step explanation:

We need to find the quotient of the polynomials

[tex](6x^3+8x^2+16)\div(2x+4)[/tex]

The quotient is: 3x^2-2x+4

The remainder is: 0

The division is shown in the figure attached.

Option A is correct.

Answer:

15 days

Step-by-step explanation:

10 men : 6 days

4 men : ? days

? = 15

Answer:

The area of DBE = 27 square units

Step-by-step explanation:

Area of ABC = 3 square units

In the figure, we can see that ABC was enlarged so that BDE is formed where side BD = 6 and AB was = 2

Hence ABC was enlarged 3 times its size.

We know by formula that:

Area of ABC = 1/2(base x perpendicular)

3 = 1/2(2 x p)

=> p = 3

As ABC was enlarged 3 times its size, the perpendicular of BDE must be 3*p

= 3*3 = 9

AREA OF DBE = 1/2(base*perp)

= 1/2(6*9)

= 27 square units

Answer:

The area of DBE = 27 square units

Step-by-step explanation:

Area of ABC = 3 square units

In the figure, we can see that ABC was enlarged so that BDE is formed where side BD = 6 and AB was = 2

Hence ABC was enlarged 3 times its size.

We know by formula that:

Area of ABC = 1/2(base x perpendicular)

3 = 1/2(2 x p)

=> p = 3

As ABC was enlarged 3 times its size, the perpendicular of BDE must be 3*p

= 3*3 = 9

AREA OF DBE = 1/2(base*perp)

= 1/2(6*9)

= 27 square units

Step-by-step explanation:

Answer:

8 times

Step-by-step explanation:

We know that the radius of smaller sphere is r,

The volume of sphere is given by:

[tex]V_1=\frac{4}{3} \pi r^{3}[/tex]

where V_1 is the volume of the small sphere.

As we know that the radius of large sphere is double of the smaller sphere, the radius of large sphere will be 2r

Let V_2 be the volume of large sphere

[tex]V_2=\frac{4}{3}\pi (2r)^{3} \\ =\frac{4}{3}\pi *8r^3[/tex]

Separating 8 aside

[tex]V_2=8(\frac{4}{3}\pi r^{3})\\V_2=8V_1[/tex]

We can see that the volume of large sphere is eight times the volume of small sphere ..

Answer:

8 times

Step-by-step explanation:

Given

ratio of radii = a : b, then

ratio of volumes = a³ : b³

Here ratio of radii = 1 : 2, hence

ratio of volumes = 1³ : 2³ = 1 : 8

Thus the volume of the large sphere is 8 times the volume of the small sphere

Answer:3.4

Step-by-step explanation:

Answer:

(x + 5 y)^2

Step-by-step explanation:

Simplify the following:

4 (x - y)^2 - 12 (x - y) (x + y) + 9 (x + y)^2

(x - y) (x - y) = (x) (x) + (x) (-y) + (-y) (x) + (-y) (-y) = x^2 - x y - x y + y^2 = x^2 - 2 x y + y^2:

4 x^2 - 2 x y + y^2 - 12 (x - y) (x + y) + 9 (x + y)^2

4 (x^2 - 2 x y + y^2) = 4 x^2 - 8 x y + 4 y^2:

4 x^2 - 8 x y + 4 y^2 - 12 (x - y) (x + y) + 9 (x + y)^2

(x + y) (x - y) = (x) (x) + (x) (-y) + (y) (x) + (y) (-y) = x^2 - x y + x y - y^2 = x^2 - y^2:

4 x^2 - 8 x y + 4 y^2 - 12 x^2 - y^2 + 9 (x + y)^2

-12 (x^2 - y^2) = 12 y^2 - 12 x^2:

4 x^2 - 8 x y + 4 y^2 + 12 y^2 - 12 x^2 + 9 (x + y)^2

(x + y) (x + y) = (x) (x) + (x) (y) + (y) (x) + (y) (y) = x^2 + x y + x y + y^2 = x^2 + 2 x y + y^2:

4 x^2 - 8 x y + 4 y^2 - 12 x^2 + 12 y^2 + 9 x^2 + 2 x y + y^2

Grouping like terms, 4 x^2 - 8 x y + 4 y^2 - 12 x^2 + 12 y^2 + 9 (x^2 + 2 x y + y^2) = 9 (x^2 + 2 x y + y^2) + (4 y^2 + 12 y^2) - 8 x y + (4 x^2 - 12 x^2):

9 (x^2 + 2 x y + y^2) + (4 y^2 + 12 y^2) - 8 x y + (4 x^2 - 12 x^2)

4 y^2 + 12 y^2 = 16 y^2:

9 (x^2 + 2 x y + y^2) + 16 y^2 - 8 x y + (4 x^2 - 12 x^2)

4 x^2 - 12 x^2 = -8 x^2:

9 (x^2 + 2 x y + y^2) + 16 y^2 - 8 x y + -8 x^2

9 (x^2 + 2 x y + y^2) = 9 x^2 + 18 x y + 9 y^2:

9 x^2 + 18 x y + 9 y^2 + 16 y^2 - 8 x y - 8 x^2

Grouping like terms, 9 x^2 + 18 x y + 9 y^2 + 16 y^2 - 8 x y - 8 x^2 = (9 y^2 + 16 y^2) + (18 x y - 8 x y) + (9 x^2 - 8 x^2):

(9 y^2 + 16 y^2) + (18 x y - 8 x y) + (9 x^2 - 8 x^2)

9 y^2 + 16 y^2 = 25 y^2:

25 y^2 + (18 x y - 8 x y) + (9 x^2 - 8 x^2)

x y 18 + x y (-8) = 10 x y:

25 y^2 + 10 x y + (9 x^2 - 8 x^2)

9 x^2 - 8 x^2 = x^2:

25 y^2 + 10 x y + x^2

The factors of 25 that sum to 10 are 5 and 5. So, 25 y^2 + 10 x y + x^2 = (x + 5 y) (x + 5 y):

(x + 5 y) (x + 5 y)

(x + 5 y) (x + 5 y) = (x + 5 y)^2:

Answer:  (x + 5 y)^2

[tex]4(x-y)^2-12(x-y)(x+y)+9(x+y)^2 =\\4(x^2-2xy+y^2)-12(x^2-y^2)+9(x^2+2xy+y^2)=\\4x^2-8xy+4y^2-12x^2+12y^2+9x^2+18xy+9y^2=\\x^2+10xy+25y^2[/tex]

which can factorised into [tex](x+5y)^2[/tex]

[tex]g(f(x))=5(4x^2+5x+3)-7=20x^2+25x+15-7=20x^2+25x+8[/tex]

Answer:

value of n = 2

Step-by-step explanation:

[tex]\frac{4}{5n}-\frac{1}{5}=\frac{2}{5n}[/tex]

We need to solve the above equation.

Find Least Common multiplier from 5n, 5 = 5n

Multiply both sides of the equation by 5n

[tex]\frac{4}{5n}*5n-\frac{1}{5}*5n=\frac{2}{5n}*5n\\Solving:\\4 -n = 2\\[/tex]

Now finding the value of n.

Adding -4 on both sides

[tex]4 -n-4 = 2-4\\-n= -2[/tex]

Multiplying both sides by -1

n = 2

The value of n is 2.

We can verify by putting value of n in the given question.

[tex]\frac{4}{5n}-\frac{1}{5}=\frac{2}{5n}\\n=2\\\frac{4}{5*2}-\frac{1}{5}=\frac{2}{5*2}\\\frac{4}{10}-\frac{1}{5}=\frac{2}{10}\\\frac{2}{5}-\frac{1}{5}=\frac{1}{5}\\\frac{2-1}{5}=\frac{1}{5}\\\frac{1}{5}=\frac{1}{5}[/tex]

So, value of n = 2

Answer:

It's C : n=1/2

Have a great day :)

The equation for [tex]\( G(F(x)) = 4x^2 - 20x + 26 \)[/tex].

To find [tex]\( G(F(x)) \)[/tex], we first need to substitute the expression for [tex]\( F(x) \)[/tex] into the function [tex]\( G(x) \)[/tex].

Given:

[tex]\[ F(x) = 2x - 5 \]\[ G(x) = x^2 + 1 \][/tex]

Replace x in [tex]\( G(x) \)[/tex] with [tex]\( F(x) \):[/tex]

[tex]\[ G(F(x)) = (2x - 5)^2 + 1 \][/tex]

Now, expand [tex]\( (2x - 5)^2 \)[/tex] using the binomial theorem:

[tex]\[ (2x - 5)^2 = (2x - 5)(2x - 5) \]\[ = 4x^2 - 10x - 10x + 25 \]\[ = 4x^2 - 20x + 25 \][/tex]

Now, substitute this expression back into [tex]\( G(F(x)) \)[/tex]:

[tex]\[ G(F(x)) = 4x^2 - 20x + 25 + 1 \]\[ G(F(x)) = 4x^2 - 20x + 26 \][/tex]

Answer:

C) the value of a person's wealth or income

Step-by-step explanation:

If an economy doesn't have money nobody would be wealthy because nobody would have money.

The statement which is rue about the effects of the transformations on the graph of function f to obtain the graph of function g is D. The graph of function f(x) is shifted right 3 units and up 4 units.

Transformation of geometrical figures or points is the manipulation of a given figure to some other way.

Different types of transformations are Rotation, Reflection, Glide reflection, Translation and Dilation.

Given is a function f(x) and the transformed function g(x) = f(x - 3) + 4.

After translation, the original figure is shifted from a place to another place without affecting it's size.

Here the transformation is both horizontal and vertical translation.

f(x) is first changed to f(x - 3).

When f(x) is changed to f(x - d), the function is shifted right d units.

So here the function is shifted right to 3 units.

Similarly when f(x) changed to f((x) + d, then the function is shifted up d units.

So the function is also shifted up 4 units.

Hence the transformation is the graph is shifted right to 3 units and up 4 units.

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Answer: I think problem 2

Step-by-step explanation: hope this helps

Answer:

To find the area of a square with only the diagonal known,

Square the diagonal and divide by 2:

8^2 = 64

64/2 = 32

The area is 32 cm^2

Answer:

32

Step-by-step explanation:

I’m correct

The exact value of sin(11π/12) is (√6 - √2)/4.

To find the exact value of sin(11π/12), we can use trigonometric identities to determine the value of an equivalent angle in the first quadrant. Since π/12 is the reference angle of 11π/12, which is in the second quadrant, we can find the sine of the equivalent angle in the first quadrant. The reference angle in the first quadrant is π/12. Therefore, the exact value of sin(11π/12) is the same as the exact value of the sine of π/12. Using a trigonometric identity, we find that sin(π/12) = (√6 - √2)/4.

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[tex]\bf \textit{Logarithm of exponentials} \\\\ \log_a\left( x^b \right)\implies b\cdot \log_a(x) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ 4\log_9(3)=\log_9(x)\implies \log_9(3^4)=\log_9(x)\implies 3^4=x\implies 81=x[/tex]

The value of x will be equal to 81.

Expression in maths is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication and division

4log₉(3)   =  log₉x

By using the logarithmic property:-

Log[tex]_a[/tex]( x[tex].^b[/tex])  =  b Log[tex]_a[/tex](x)

4log₉(3)   =  log₉x

log₉ ( 3 )⁴  =  log₉ ( x )

3⁴   =  x

x   =  81

Therefore the value of x will be equal to 81.

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Answer:

The 4 pack is the better deal.

Step-by-step explanation:

You need to divide the amount of cost by the number of rolls per pack to get a per roll price.

4 pack = $2.04

$2.04/4 rolls = cost $.51/roll

for the 9 pack $4.68.

$4.68/4 rolls = $.52/1 roll

The better deal is the 4 pack because $.51 is less than $.52 per roll for the 9 pack.

Answer:

A

Step-by-step explanation:

U can already eliminate C and D since the z needs to be with 2. A and B almost look the same but in the radicals the second number goes on top resulting in 5/6 which leads to A.

Answer:

The correct answer is the third option

Step-by-step explanation:

Have different slope but have the same y-intercept, so they have no solution.

y = 1/2x - 3

To graph this, the slope 1/2 which means go up once and to the right twice and b = -3, which means the y-intercept.

second equation:

y =  -1/2x -3

To graph this, the slope -1/2 which means go down once and to the right twice and b = -3, which means the y-intercept.

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Answer:

y=-12

Step-by-step explanation:

For the slope of line given two points I like to line up and subtract. Then put 2nd diff/1st diff.

Let's do that:

(-6  , -12 )

(0 ,  -12)

-----------

-6      0

2nd/1st=0/-6=0  slope should have been obvious as 0 since there is no rise in the points (you can see this from their y-coordinate)

Anyways the y-intercept is when it crosses the y-axis. The y-axis is crossed when the x-coordinate is 0. You have that given here which is -12.

So the equation is y=0x-12 or jusy y=-12

If you already knew that horizontal lines were of the form y=a number

then you could have just skip to y=-12

y=-12 just means all the ordered pairs on this line have the y-coordinate being -12 which is what we have in the set of points you gave

[tex](g\circ f)(x)=(2x-1)^2-2=4x^2-4x+1-2=4x^2-4x-1[/tex]

Answer:

[tex]4x^2 -4x -1[/tex]

Step-by-step explanation:

Given functions,

[tex]f(x) = 2x - 1-----(1)[/tex]

[tex]g(x) = x^2 -2-----(2)[/tex]

∵ (gof)(x) = g[f(x)]    ( Composition of functions )

[tex]\implies (gof)(x) = g(2x-1)[/tex] ( From equation (1) )

[tex]=(2x-1)^2 - 2[/tex]  ( From equation (2) )

[tex]=4x^2 + 1 - 4x - 2[/tex]

[tex]=4x^2 -4x -1[/tex]

Answer:

The answer to this question is D- 15.

Step-by-step explanation:

This is how to solve it

30 students and 2 at random this are the key words

you do 30/2

your answer is equal to 15

Answer: B. 435

Step-by-step explanation: This is a combination problem. The teacher is going to choose two students at random, but the order in which the two students are chosen doesn't matter. (meaning, student A being chosen 1st and student B being chosen 2nd is the same result as Student B being chosen 1st and student A being chosen 2nd) Since this is a combination's problem, we use the combination function nCr.

The nCr function is written as [tex]\frac{n!}{r!(n-r)!}[/tex], where n represents the total number of things to choose from while r represents the number of objects  that will be taken from the set. In this case, n=30 since there are 30 total students to choose from while r=2 since the teacher is picking two students from the group of 30.  The exclamation marks next to the variables represent a factorial.

A factorial is the product of all positive integers less than or equal to the integer next to the factorial. For example, 6! indicates 6 factorial, which is equal to 6 x 5 x 4 x 3 x 2 x 1 = 720. Therefore, 30! equals 30 x 29 x 28 x 27 x 26......... and so on until its been multiplied by every positive integer less than 30.

Using the nCr function, we plug the values in to get [tex]\frac{30!}{2!(30-2)!}[/tex]. After doing some simplification and factoring, we get the equation [tex]\frac{30 * 29}{2}[/tex], which yields 435 possible combinations. This can be done because 30 factorial and 28 factorial share 28 factors due to the nature of factorials, simplifying 30 factorial to simply 30 multiplied by 29. The equation yields 435 possible combinations, thus meaning that there are 435 possible ways to choose 2 students from 30 students.

Answer:

10

Step-by-step explanation:

Leave the 5 , change division to multiplication and turn the fraction upside down.

5 × [tex]\frac{2}{1}[/tex] = 5 × 2 = 10

Answer:

x = 1 or x = -7

Step-by-step explanation:

Step-by-step explanation:

Do you mean by completing the square?

x² + 6x - 7 = 0

x² + 6x = 7

Take half of 6, square it, and add to both sides.  (6/2)² = 9.

x² + 6x + 9 = 7 + 9

(x + 3)² = 16

x + 3 = ±4

x = -3 ± 4

x = -7, 1

So the roots are x=-7 and x=1.

For this case, we must build a quotient that, when multiplied by the divisor, eliminates the terms of the dividend until it reaches the rest.

The attached figure shows the quotient given by:

[tex]x ^ 3 + x ^ 2 + x + 1[/tex]

Answer:

Quotient: [tex]x ^ 3 + x ^ 2 + x + 1[/tex]

See attached image

Answer:

Quotient is: x^3+x^2+x+1  

Step-by-step explanation:

We need to solve (x^4-1) ÷  (x-1) using synthetic division.

In synthetic division we write the coefficients in decreasing order of their powers. We have x^4-1 that can be written as: 1x^4 + 0x^3 + 0x^2 + 0x -1

so our coefficients will be

1 0 0 0 -1

and for synthetic division, we take the constant term of divisor and change its sign.

We have x-1, constant term -1 so, our value will be 1.

The division is attached in the figure below.

Quotient is: x^3+x^2+x+1  

Answer:

[tex]\frac{1}{(m-4)(m-3)}[/tex]

Step-by-step explanation:

The question requires you to simplify using quadratic identities

Re-write the numerator.

[tex]\frac{m+3}{m^2-16} \\\\\\\\=\frac{m+3}{(m+4)+(m-3) }[/tex]

Re-write the denominator as;

[tex]\frac{(m+3) +(m-3)}{m+4}[/tex]

Re-arrange expression

[tex]\frac{m+3}{(m+4)+(m-4)} * \frac{m+4}{(m+3)+(m-3)}[/tex]

cancel the terms that are alike to remain with

[tex]\frac{1}{(m-4)(m-3)}[/tex]

Answer:

The expression which equivalent is 1/[(m - 4)(m - 3)] ⇒ 2nd answer

Step-by-step explanation:

* Lets revise how to divide two fractions

- To divide a/b and c/d change the division operation to multiplication

  operation and reciprocal the fraction after the division sign

# a/b ÷ c/d = a/b × d/c

* Lets solve the problem

∵ (m + 3)/(m² - 16) ÷ (m² - 9)/(m + 4)

- Use the factorization to simplify the fractions

∵ m² - 16 is a different of two squares

- The factorization of the different of two squares a² - b² is

∵ a² = a × a , -b² = b × -b

∴ a² - b² = (a + b)(a - b)

- Use this way with m² - 16

∵ m² = m × m

∵ -16 = 4 × -4

∴ m² - 16 = (m + 4)(m - 4)

- Similar factorize m² - 9

∵ m² = m × m

∵ -9 = 3 × -3

∴ m² - 9 = (m + 3)(m - 3)

- Now lets write the fraction and simplify it

∵ (m + 3)/(m² - 16) ÷ (m² - 9)/(m + 4)

∴ (m + 3)/[(m + 4)(m - 4)] ÷ [(m + 3)(m - 3)]/(m + 4)

- Change the division operation to multiplication operation and

  reciprocal the fraction after the division sign

∴ (m + 3)/[(m + 4)(m - 4)] × (m + 4)/[(m + 3)(m - 3)]

- We can cancel (m + 4) in the denominator of the first fraction with

 (m + 4) in the numerator of the second fraction and cancel (m + 3)

 in the numerator of the first fraction with (m + 3) in the denominator

 of the second fraction

∴ 1/(m - 4) × 1/(m - 3) ⇒ multiply the two fractions

∴ 1/[(m - 4)(m - 3)]

* The expression which equivalent is 1/[(m - 4)(m - 3)]

Answer:

40 it is a trick question

Step-by-step explanation:

Answer:

The correct option is:   40

Step-by-step explanation:

Measure of an arc is actually the measure of the central angle corresponding with that same arc.

Here, the corresponding central angle for arc [tex]\widehat{AB}[/tex] is [tex]\angle1[/tex].

Given that,  [tex]m\angle 1= 40[/tex]°

So.....

[tex]m\widehat{AB}=m\angle 1\\ \\ m\widehat{AB}=40\°[/tex]

Answer:

D = 9.4868

Step-by-step explanation:

The expression is the following

D = √((x2-x1)^2+(y2-y1)^2)

Where

(x1,y1) = (4,6)

(x2,y2) = (7,-3)

D = √((7-4)^2+(-3-6)^2)

D = √((3)^2+(-9)^2)

D = √(9+81)

D = √(90)

D = 9.4868

Answer:

[tex]d = \sqrt{(7-4)^{2}+(-3-6)^{2}}[/tex]

Step-by-step explanation:

The expression used for calculating distance between two points involves the square root of sum of squares of differences of x-intercepts and y-intercepts.

The formula is given by:

[tex]d = \sqrt{(x_{2} -x_{1} )^{2}+(y_{2}- y_{1} )^{2} }[/tex]

Here,

[tex](x_{1},y_{1}) = (4,6)\\ (x_{2},y_{2}) = (7,-3)\\Putting\ the\ values\\d = \sqrt{(7-4)^{2}+(-3-6)^{2}}[/tex]

Hence, the following expression will give the distance between given points

[tex]d = \sqrt{(7-4)^{2}+(-3-6)^{2}}[/tex]

Solving it will give:

[tex]d = \sqrt{(3)^{2}+(-9)^{2}}\\= \sqrt{9+81}\\=\sqrt{90}[/tex]

Answer:

[tex]\boxed{\text{A. }\math{\left \{ x \, | \, x \in \mathbb{R}, x < -2 \right \}}}}[/tex]

Step-by-step explanation:

The open circle means that the point is not included in the solution set, and the arrow pointing left means that all numbers less than -2 are members.

In set-builder notation, each term has a special meaning. The braces enclose the members of the set.

Here's how you translate the notation,

[tex]\begin{array}{rcl}\\\left \{ & = & \text{The set of}\\x & = & \text{all x values}\\| & = &\text{such that}\\x & = & x\\\in & = &\text{is a member of}\\\mathbb{R}, & = &\text{all real numbers, and}\\x < -2 & = & \text{x is less than -2}\\\end{array}\\\text{The answer is }\boxed{\textbf{A. }\mathbf{\left \{ x \, | \, x \in \mathbb{R}, x < -2 \right \} }}}[/tex]