At a competition with 5 runners, medals are awarded for first, second, and
third places. Each of the 3 medals is different. How many ways are there to
award the medals?
Decide if this is a permutation or a combination, and find the number of ways
to award the medals.

By completing the square,

[tex]x^2-16x+89=x^2-16x+64+25=(x-8)^2+25=0[/tex]

[tex](x-8)^2=-25[/tex]

[tex]x-8=\pm\sqrt{-25}[/tex]

[tex]x=8\pm5i[/tex]

[tex]\bf (\stackrel{x_1}{6}~,~\stackrel{y_1}{4})\qquad (\stackrel{x_2}{-6}~,~\stackrel{y_2}{4}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{4-4}{-6-6}\implies \cfrac{0}{-12}\implies 0 \\\\\\ \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-4=0(x-6)\implies y-4=0\implies y=4[/tex]

Answer: [tex]y-4=0[/tex]

Step-by-step explanation:

The equation of line passing through two points (a,b) and (c,d) is given by :-

[tex](y-b)=\dfrac{d-b}{c-a}(x-a)[/tex]

The standard form of equation of a line is given by :-

[tex]Ax+By+C=0[/tex], where A , B , and C are integers.

Then , the equation of line passing through two points  K(6,4) and L(-6,4) is given by :-

[tex](y-4)=\dfrac{4-4}{-6-6}(x-6)\\\\\Rightarrow\ y-4=(0)(x-6)\\\\\Rightarrow\ y-4=0[/tex]

Answer:

Rotation 90 degree counterclockwise then 2 units up.

Step-by-step explanation:

Given : Quadrilateral ABCD and A'B'C'D'.

To find: What transformation were applied to ABCD to obtain A’B’C’D.

Solution: We have given

A (3,6) →→ A'(-6,5)

B( 3,9)→→ B'(-9 ,5)

C(7,9)→→C'(-9 ,9)

D(7,6)→→D'(-6,9)

By the 90 degree rotational rule :   (x ,y) →→(-y ,x) and unit 2 unit up

A (3,6) →→ A'(-6,3) →→ A'(-6,3+2)

B( 3,9)→→ B'(-9 ,3)→→ B'(-9 ,3+2)

C(7,9)→→C'(-9 ,7) →→C'(-9 ,7+2)

D(7,6)→→D'(-6,7)→→D'(-6,7+2)

Therefore, Rotation 90 degree counterclockwise then 2 units up.

The transformation applied is Rotation 90 degree counterclockwise then 2 units up.

A coordinate system in geometry is a system that employs one or more integers, or coordinates, to define the position of points or other geometric components on a manifold such as Euclidean space.

The transformation which was applied to ABCD to obtain A’B’C’D be found by finding the change in the coordinates of the quadrilateral. Therefore,

As it is observed that the change in the coordinate is 90 degrees counterclockwise then 2 units up. Therefore, the transform of the coordinates can be done as (x ,y)⇒(-y ,x)⇒(-y, x+2).

Since the condition holds true, it can be concluded that the transformation applied is Rotation 90 degree counterclockwise then 2 units up.

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Answer:Pool B is being filled quicker

Step-by-step explanation: It is being filled quicker because 4 is a bigger number than 3 so there is your answer

By using the principles of similar triangles and setting up a ratio based on the given information, we can determine that the tree is approximately 22.22 feet tall.

We can solve this problem using similar triangles. In this case, the person (Mac) and his shadow form one triangle, while the tree and its shadow form another triangle. It's given that Mac is 5 feet tall and casts a 4 foot 6 inch (or 4.5 feet) shadow. At the same time, a nearby tree casts a 20 foot long shadow. The question asks, how tall is the tree?

What we need to do here is simple: set up a proportion using the ratios of the lengths of the sides of these two triangles. The ratio is height to shadow. So, let's use the information given:

Equating both ratios gives us: 5/4.5 = T/20. Solving this equation for 'T' will give us the height of the tree in feet.

Therefore, to compute 'T', cross multiply to get: T = (5*20)/4.5 = 22.22 feet approximately.

So, the tree is approximately 22.22 feet tall.

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bearing in mind that standard form for a linear equation means

• all coefficients must be integers, no fractions

• only the constant on the right-hand-side

• all variables on the left-hand-side, sorted

• "x" must not have a negative coefficient

[tex]\bf (\stackrel{x_1}{-3}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{2}~,~\stackrel{y_2}{1}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{1-2}{2-(-3)}\implies \cfrac{-1}{2+3}\implies -\cfrac{1}{5} \\\\\\ \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-2=-\cfrac{1}{5}[x-(-3)]\implies y-2=-\cfrac{1}{5}(x+3)[/tex]

[tex]\bf y-2=-\cfrac{1}{5}x-\cfrac{3}{5}\implies \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{5}}{5\left(y-2 \right)=5\left( -\cfrac{1}{5}x-\cfrac{3}{5} \right)}\implies 5y-10=-x-3 \\\\\\ 5y=-x+7\implies x+5y=7[/tex]

Answer:

(9,2)

Step-by-step explanation:

(1,2) is reflected over vertical line x=3 which means since (1,2) is 2 units over from x=3 that the reflection is going to be 2 units over in the other direction from x=3 so the first image is (5,2)

Now the last image is processed by a reflection through vertical line x=7

(5,2) is 2 units from x=7 so the reflection will also be 2 units from the vertical line x=7 so the final image is (9,2).

Answer (9,2)

Answer:

(9, 2 )

Step-by-step explanation:

Under a double reflection in 2 parallel vertical lines

The y- coordinate remains unchanged

The x- coordinate is twice the difference of the parallel lines added to the original x- coordinate, that is

image = (1 + 2(7 - 3), 2 ) = (1 + 8, 2) = (9, 2 )

Answer:

-2 = x

Step-by-step explanation:

4(1-x)+3x = -2(x+1)      Distribute

4 - 4x + 3x = -2x -2    Combine Like Terms (On the same sides of the equal sign)

4 -x = -2x -2               Combine Like Terms (On different sides of the equal sign)

4 = -3x - 2                   Solve

6 = -3x                        

-2 = x

Final answer:

The solution to the equation 4(1-x) + 3x = -2(x + 1) is found to be x = -6, after simplifying and checking by substitution back into the original equation, which confirms the solution by resulting in an identity.

Explanation:

To solve the equation 4(1-x) + 3x = -2(x + 1), we need to expand and simplify our equation:

4 - 4x + 3x = -2x - 2

4 - x = -2x - 2

Adding 2x to both sides we get:

4 + x = -2

Now subtracting 4 from both sides, we obtain:

x = -6

We then check this solution by plugging it back into the original equation, confirming that it simplifies to an identity. Therefore, x = -6 is the correct solution.

Answer:

yes it can

Step-by-step explanation:

1/2x-5=1/3x+6

Answer: third option.

Step-by-step explanation:

[tex]\frac{1}{2}x-5=\frac{1}{3}x+6[/tex] can be rewritten into two separate equationts:

[tex]\left \{ {{ y=\frac{1}{2}x-5} \atop {y=\frac{1}{3}x+6}} \right.\\\\[/tex]

You can observe that this linear equations are written in Slope-Intercept form:

[tex]y=mx+b[/tex]

  But the equations shown in options provided are written in Standard form:

[tex]Ax+By=C[/tex]

Therefore, you need to move the x term to the left side of the equation (In each equation):

- For the first equation:

[tex]y-\frac{1}{2}x=-5[/tex]

Simplifying:

[tex]\frac{2y-x}{2}=-5\\\\2y-x=-10[/tex]

 - For the second equation:

[tex]y-\frac{1}{3}x=6[/tex]

Simplifying:

[tex]\frac{3y-x}{3}=6\\\\3y-x=18[/tex]

Then the system of equations is:

[tex]\left \{ {{2y-x=-10} \atop {3y-x=18}} \right.[/tex]

Answer:

The width should be more than 5 m. So it can be  6,7 ...

Step-by-step explanation:

The length of a flower garden is = 9 m

The width of a flower garden is   = w

The area of a flower garden > 45 sq. m

The area of a flower garden  = l * b

l * b > 45

9 * b > 45

the width should be more than 5 m. So it can be  6,7 ...

Answer:

w > 36

Step-by-step explanation:

i got it right on edgeinuity

Answer:

15x+25y=975

x+y=55

Rearrange equation two so x is by itself.

x=-y+55

Plug the rearranged equation two into equation one.

15(-y+55)+25y=975

Evaluate the 'new' equation 1.

-15y+825+25y=975

10y+825=975

10y=150

y=15

Choose an equation to evaluate with y to get x. (i chose equation 2 because it was easier)

x+15=55

Evaluate the equation

x=55-15

x=40

So now we have x=40 and y=15

Evaluate those two terms with both equations to check the correctness.

15(40)+25(15)=975

600+375=975

975=975 (correct)

40+15=55

55=55 (correct)

Both equations are correct so the values of x and y are true.

Please mark as brainliest. :)

Check the picture below.

so the vertex at the bottom is exactly half-way of the opposite side, and equidistant from the other two vertices, meaning the two slanted sides are twins, and thus this is an isosceles triangle.

Answer:

[tex]2y=8[/tex]

Step-by-step explanation:

The given equations are:

[tex]4x+3y=17[/tex]

and

[tex]4x+y=9[/tex]

We subtract the second equation from the first equation to get:

[tex]4x-4x+3y-y=17-9[/tex]

This simplifies to:

[tex]2y=8[/tex]

When we subtract the second equation from the first one, we get:

[tex]2y=8[/tex]

the cheap answer is simply

(x-5)(x²+4x-2)

we can simply multiply the terms on one by the terms of the other and then add like-terms and simplify.

[tex]\bf (x-5)(x^2+4x-2)\implies \begin{array}{cllll} x^2+4x-2\\ \times x\\ \cline{1-1}\\ x^3+4x^2-2x \end{array}+ \begin{array}{cllll} x^2+4x-2\\ \times -5\\ \cline{1-1}\\ -5x^2-20x+10 \end{array} \\\\\\ x^3+4x^2-2x-5x^2-20x+10\implies x^3+4x^2-5x^2-2x-20x+10 \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill x^3-x^2-22x+10~\hfill[/tex]

Answer:

51x

Step-by-step explanation:

Answer:

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

Step-by-step explanation:

The exponential function is of the form

y = a [tex](b)^{x}[/tex]

To find a and b substitute the given points into the equation

Using (0, 2), then

2 = a [tex](b)^{0}[/tex] ⇒ a = 2

Using (1, 6), then

6 = 2 [tex](b)^{1}[/tex] ⇒ b = 3

Equation is y = 2 [tex](3)^{x}[/tex]

Answer:

translating it

Step-by-step explanation:

Shifting a function, in mathematics and economics, refers to the process through which the entire graph or model of a function is moved either up, down, left or right. A phase shift, commonly noted by φ, is an example of this, seen when aligning a cosine or sine function with initial conditions of data. In economics, factors like income, household preferences and taxes cause shifts in the consumption function.

When you shift a function in mathematics, you are essentially moving the entire graph of the function either up, down, left, or right. This is done by altering the function equation. For instance, the position of a block on a spring, modeled by a periodic function like a cosine function, can be shifted to the right. This rightward shift is termed a phase shift, typically denoted by the Greek letter phi (φ). The equation for the position as a function of time for the block becomes x(t) = Acos(wt + φ), where φ reflects the phase shift.

A shift in a function can also occur under economic contexts. Factors besides income can trigger the entire consumption function to shift, either parallelly up or down or make the slope of the consumption function steeper or flatter.

Shifting of functions is a crucial concept, whether we're handling a cosine or sine function to align the function with data's initial conditions or in economical situations where changes in income, taxes, or household preferences could cause significant shifts in the consumption function.

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Answer:

Third option: 80 people.

Step-by-step explanation:

Let be "x" the number of people that were surveyed.

You know that 48 people chose winter as their favorite season and these amount of people are the 60% of those questioned.

Knowing this,  you can calculate  the number of people that were surveyed with this procedure:

[tex]x=\frac{(48\ people)(100\%)}{60\%}\\\\x=(48\ people)(\frac{5}{3})\\\\x=80\ people[/tex]

Therefore, 80 people were surveyed. This matches with the third option.

Answer:

She can double the base length of the base so that the new volume is 120 cubic inches

Step-by-step explanation:

The method she can use to double the volume of triangular pyramid is she can double the base length of the base so that the new volume is 120 cubic inches.

A tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedral and the only one that has fewer than 5 faces.  

Volume of triangular pyramid :

Volume of triangular pyramid = 1/3 × Base Area × Height

According to the question

Naomi made a triangular pyramid from cardboard.  

The triangular base has a height = 4 inches

The triangular base has a base length = 15 inches

The height of the pyramid = 6 inches  

Therefore,

Base area of triangular pyramid

= [tex]\frac{1}{2} * Base\ of\ triangle * Height\ of\ triangle[/tex]

= [tex]\frac{1}{2} * 15 * 4[/tex]

= 30 [tex]inches ^{2}[/tex]

So,

Volume of triangular pyramid = 1/3 × Base Area × Height

                                                   = [tex]\frac{1}{3}[/tex] × 30 × 6

                                                   = 60  [tex]inches ^{3}[/tex]

Now,

Naomi wants to double the volume of the triangular pyramid

i.e

New Volume of triangular pyramid = 120 [tex]inches ^{3}[/tex]

Which can be obtain if she can double the base length of the base  

Hence, the method she can use to double the volume of triangular pyramid is she can double the base length of the base so that the new volume is 120 cubic inches.

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Answer:

what are the questions about it? cant answer if there are no questions

Step-by-step explanation:

Answer:

its domain isn (-10,∞) and its range is (0,∞)

Step-by-step explanation:

if this is the a.pex question regarding what is true about the function below, here you go

[tex]\bf \begin{array}{lcccl} &\stackrel{solution}{gallons}&\stackrel{\textit{\% of }}{antifreeze}&\stackrel{\textit{gallons of }}{antifreeze}\\ \cline{2-4}&\\ \textit{1st brand}&x&0.65&0.65x\\ \textit{2nd brand}&y&0.90&0.9y\\ \cline{2-4}&\\ mixture&40&0.8&32 \end{array}~\hfill \to \begin{cases} x+y&=40\\ \boxed{x}=40-y\\ \cline{1-2} 0.65x+0.9y&=32 \end{cases} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf \stackrel{\textit{substituting in the 2nd equation}}{0.65\left( \boxed{40-y} \right)+0.9y=32}\implies 26-0.65y+0.9y=32 \\\\\\ 26+0.25y=32\implies 0.25y=6\implies y=\cfrac{6}{0.25}\implies \blacktriangleright y=24 \blacktriangleleft \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{since we know that}}{x=40-y\implies }x=40-24\implies \blacktriangleright x=16 \blacktriangleleft[/tex]

Answer:

it should be used 16 gallons of 65% pue antifreeze and 24 gallons of 80% antifreeze.

Step-by-step explanation:

Let 'x' = amount of 65% antifreeze.

Let 'y' = amount of 90% antifreeze.

We need to obtain 40 gallons of mixture, then:

x + y = 40 gallons [1]

Also we know that the mixture should contain 80% pure antifreeze, then:

0.65x + 0.9y = 0.80(x+y) →  0.15x = 0.1y → y = 1.5x  [2]

Now, subtituting the value of 'y' into [1]:

x + y = 40 gallons → x + 1.5x = 40 gallons → 2.5x = 40 gallons

⇒ x = 16 gallons.

Then: y = 40 - 16 = 24 gallons.

Now, it should be used 16 gallons of 65% pue antifreeze and 24 gallons of 80% antifreeze.

Answer:

54

Step-by-step explanation:

Given:

[tex]3(a + b + c) 2[/tex]

Input the numbers provided into the equation

3(2 + 3 + 4)2

Add all of the numbers inside the parentheses

3(9)2

Multiply

3 * 2 = 6

6 * 9 = 54

Hey there! :)

3(a + b + c)2 ; when a = 2, b = 3, c = 4

In order to evaluate this, we simply need to plug everything in then simplify/

3(2 + 3 + 4)2

Simplify everything inside the parenthesis first.

3(9)2

Simplify.

27 × 2

Simplify.

54 is our final answer.

Hope this helped! :)

The correct option is Absolute value parent function. The absolute value parent function is a piecewise function because it is defined by different expressions depending on whether the input is positive or negative. In contrast, linear, quadratic, and exponential functions are defined by a single expression over their entire domains and are not piecewise.

The absolute value parent function is an example of a piecewise function. A piecewise function is a function that is defined by multiple sub-functions, each applied to a certain interval of the domain.

On the contrary, linear, quadratic, and exponential parent functions have a single expression defining them throughout their domain.

Linear functions, such as y = ax + b, have a constant rate of change and are not piecewise. Quadratic functions, like y = ax^2 + bx + c, create parabolas and are also defined by a single expression over their entire domain. Exponential functions, such as y = b^x, grow by a consistent percentage rate and are continuous and not piecewise.

Answer:

[tex]f(x)=7(b)^{x}-2[/tex]

[tex]f(x)=-5(b)^{x}+10[/tex]

Step-by-step explanation:

we know that

The y-intercept is the value of y when the value of x is equal to zero

Verify each case

case 1) we have

[tex]f(x)=7(b)^{x}-2[/tex]

so

For x=0

[tex]f(0)=7(b)^{0}-2[/tex]

[tex]f(0)=7(1)-2=5[/tex]

therefore

The function has a y-intercept of (0,5)

case 2) we have

[tex]f(x)=3(b)^{x}-5[/tex]

so

For x=0

[tex]f(0)=3(b)^{0}-5[/tex]

[tex]f(0)=3(1)-5=-2[/tex]

therefore

The function does not have a y-intercept of (0,5)

case 3) we have

[tex]f(x)=5(b)^{x}-1[/tex]

so

For x=0

[tex]f(0)=5(b)^{0}-1[/tex]

[tex]f(0)=5(1)-1=4[/tex]

therefore

The function does not have a y-intercept of (0,5)

case 4) we have

[tex]f(x)=-5(b)^{x}+10[/tex]

so

For x=0

[tex]f(0)=-5(b)^{0}+10[/tex]

[tex]f(0)=-5(1)+10=5[/tex]

therefore

The function has a y-intercept of (0,5)

case 5) we have

[tex]f(x)=2(b)^{x}+5[/tex]

so

For x=0

[tex]f(0)=2(b)^{0}+5[/tex]

[tex]f(0)=2(1)+5=7[/tex]

therefore

The function does not have a y-intercept of (0,5)

Answer:

for this question the correct answer is x= 10

Step-by-step explanation:

The true value of the equation In 20+ In 5 = 2 In x is x=10.

It is a function which is denoted through log or ln.

We have been given a log equation

log20+log5=2logx              (log m +log n =log(mn))

log(20*5)=2logx

log(100)=2logx

log[tex](10)^{2}[/tex]=2logx                (log[tex]m^{n}[/tex]=nlogm)

2 log 10=2 log x

x=10

Hence the required answer is x=10

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Answer:

-k (k+1) (k+2)

Step-by-step explanation:

-2k - k³ - 3k² (factor-k out)

-k (2 + k² + 3k) (rearrange to standard quadratic form)

-k (k² + 3k + 2) (factor expression inside parentheses using your favorite method)

-k (k+1) (k+2)

Answer:

Option c.

Step-by-step explanation:

The given expression is

[tex]-2k-k^3-3k^2[/tex]

We need to find the factor form of the given expression.

Taking out HCF.

[tex]-k(2+k^2+3k)[/tex]

Arrange the terms according to there degree.

[tex]-k(k^2+3k+2)[/tex]

Splitting the middle terms we get

[tex]-k(k^2+2k+k+2)[/tex]

[tex]-k((k^2+2k)+(k+2))[/tex]

[tex]-k(k(k+2)+(k+2))[/tex]

[tex]-k(k+1)(k+2)[/tex]

The factor form of given expression is -k(k+1)(k+2). Therefore, the correct option is c.

Answer:

The answer is the first table: (0, 0), (10, 50), (51, 255), (400, 2000)

Step-by-step explanation:

Let's check all of the tables:

Table 1:

x = 0, y = 0               ⇒        0 = 5 · 0         ⇒          0 = 0  

x = 10, y = 50           ⇒      50 = 5 · 10        ⇒        50 = 50

x = 51, y = 255         ⇒    255 = 5 · 51        ⇒      255 = 255

x = 400, y = 2000    ⇒ 2000 = 5 · 400     ⇒   2000 = 2000

Answer:

neither

Step-by-step explanation:

y would 1 but there is no x

Answer:(-1,-5)

Step-by-step explanation:

Because that's how math works. Its also what Khan Academy says soooo...

Answer:

So, the simplified version is

[tex]-5\sqrt{7}+7\sqrt{6}[/tex]

In words it can be written as minus 5 square root of 7 end root plus 7 square root of 6 end root

Step-by-step explanation:

[tex]5\sqrt{7}+12\sqrt{6}-10\sqrt{7}-5\sqrt{6}[/tex]

We need to simplify the above expression.

Combining the like terms of the above expression.

Like terms are those that have same variables.

[tex]=(5\sqrt{7}-10\sqrt{7})+(12\sqrt{6}-5\sqrt{6})[/tex]

[tex]=((5-10)(\sqrt{7})+((12-5)(\sqrt{6}))[/tex]

[tex]=((-5)(\sqrt{7})+((7)(\sqrt{6}))[/tex]

So, the simplified version is

[tex]-5\sqrt{7}+7\sqrt{6}[/tex]

In words it can be written as minus 5 square root of 7 end root plus 7 square root of 6

Answer:

f(x) = [tex]e^{x}[/tex]

Step-by-step explanation:

The only function that has it;s derivative equal to the function is the exponential function.

That is f(x) = f'(x) = [tex]e^{x}[/tex]

Ask: Which two numbers add up to -4 and multiply to -5?

-5 and 1

Rewrite the expression using the above

= (x - 5) and (x + 1)