The normal line to a graph of a function f at a point (c, f(c)) is the line through (c, f(c)) perpendicular to the tangent line to the graph of f at (c, f(c)). See the figure. If f is a function whose derivative at c is f
′
(
c
)
≠
0
,
the slope of the normal line to the graph of f at (c, f(c)) is −
1
f
′
(
c
)
.
Then an equation of the normal line to the graph of f at (c, f(c)) is y
−
f
(
c
)
=
−
1
f
′
(
c
)
(
x
−
c
)
.
Find the slope of the normal line to the graph of the function at the indicated point.
f
(
x
)
=
4
x
2
+
2
a
t
(
1
,
6
)

The image of point A(-3, -5) after a translation right 4 units and up 2 units is the point B(1,−3)

Transformation is the movement of a point from its initial location to a new location. Types of transformations are reflection, rotation, translation and dilation.

Translation is the movement of a point either up, left, right or down in the coordinate plane.

The point A = (1 - 4, -3 - 2) = (-3, -5)

The image of point A(-3, -5) after a translation right 4 units and up 2 units is the point B(1,−3)

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The volume of the space filled with tissue paper in the packaging box is 198.264 cubic centimeters

Constant = k

= 2.1³

= 9.261

Volume of large : volume of small = k

Vl / Vs = k

Vl = Vs × k

= 24 × 9.261

= 222.264 cubic inches

Therefore,

Volume of tissue paper = volume of large - volume of small

= 222.264 cubic inches - 24 cubic inches

= 198.261 cubic inches

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Answer:426.6

Step-by-step explanation:

.138 x =.088(x - 69.7) + .394(69.7)

.138 x -  .088x = 69.7(.394 - .088)

.05x = 21.3282

/

x = 426.564

This rounds to x = 426.6

Step-by-step explanation:

M=2G

M=J+10

G=J-8

2G=J+10

G=J-8

2(J-8)=J+10

2J-16=J+10

J=26

An equation which is equivalent to the equation bsin(A) = asin(B) is: B. [tex]\frac{sinA}{a} =\frac{sinB}{b}[/tex]

The law of sines is also referred to as sine law or sine rule and it can be defined as an equation that relates the side lengths of a triangle to the sines of its angles.

Mathematically, the law of sines is given by this equation:

[tex]\frac{sinA}{a} =\frac{sinB}{b}[/tex]

In this context, we can infer and logically deduce that an equation which is equivalent to the equation bsin(A) = asin(B) is [tex]\frac{sinA}{a} =\frac{sinB}{b}[/tex].

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Answer: B

Explanation: sin(A)/a=sin(B)/b

The solution to the given expression [98/(82 + 283)] * (69 - 29) is; 784/73

We want to solve the expression;

[98/(82 + 283)] * (69 - 29)

Let us first simplify the first part which is;

98/(82 + 283) = 98/365

Then let us simplify the second part which is;

69 - 29 = 40

Thus, we have the expression;

(98/365) * 40

This can be simplified further to get;

(98/73) * 8 = 784/73

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The number of students with heights less than 163 cm should be expected is 12.

According to the statement

The mean of height is 175 cm

and  the standard deviation of height is 6 cm.

We use normal distribution here with formula

Z= X - μ /σ

Here X is 167.5 and μ is 175 and σ is 6 cm.

Substitute the values in it then

Z = 167.5 - 175 / 6

Z = -1.25

-1.25 have a p value 0.012

Out of 1000 students:

0.012 x 1000 = 12.

So, The number of students with heights less than 163 cm should be expected is 12.

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Jamie purchases 4 chocolate (3.57), 7 strawberry/vanilla, and 7 peanut butter cupcakes for an office party.

The steps involved are:

Given that,

Jamie is ordering cupcakes for an office party.

There are three flavors - Chocolate, Strawberry/vanilla, and peanut flavors.

Jamie purchased a total of 18 cupcakes for $55.75

The cost for a single cupcake of each flavor is as follows:

Chocolate - $3

Strawberry  - $2.75

Peanut - $3.50

Consider the number of cupcakes of each flavor as x, y, and z respectively.

So, the equations we can write from this data are,

x + y + z = 18 and

3x + 2.75y + 3. 50z = 55.75

But it is given that,

There are twice as many chocolate cupcakes purchased as peanut butter cupcakes.

So, z = 2x

Then the above two equations become,

3x + y = 18 and

10x + 2.75y = 55.75

By solving these two equations we get,

x = 3.57 ≅ 4

y = 7.28 ≅ 7

Since we know that z = 2x,

⇒ z = 2 × 3.57 = 7.14 ≅ 7

So,

There are 4 chocolate, 7 strawberry, and 7 peanut cupcakes.

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The distance between two points (4,-2) and (-6,4) exists [tex]2\sqrt{34}.[/tex]

Distance between two points exists the length of the line segment that joins the two given points. Distance between two points in coordinate geometry can be estimated by finding the length of the line segment merging the provided coordinates.

Let the points be (4,-2) and (-6,4).

The distance between two points [tex](x_{1} ,y_{1})[/tex] and [tex](x_{2} , y_{2} )[/tex] exists provided by the distance between the two points

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

substituting the points in the equation, we get

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

simplifying the equation, we get

[tex]$=\sqrt{136}[/tex]

[tex]$=2\sqrt{34}[/tex]

The distance between two points (4,-2) and (-6,4) exists [tex]2\sqrt{34}.[/tex]

Therefore, the correct answer is option c. [tex]$2\sqrt{34}.[/tex]

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Answer:

a)  Time until both vehicles meet is 1.5 hours after starting at noon.  That makes it 1:30PM.

b)  Average speed of car is 84 km/h

Step-by-step explanation:

A -----------------------z------------B

          Left      Speed(km/h)      Time

Car:   12PM            X  

Van:   12PM           60

Car/Van

DistanceCar        AB + z

DistanceVan       Az

Ratio:                  (AB+z)/Az  = 7/5

Time until both meet = T (in hours)

Distance Car:            xT

Distance Van:           60T

====

  xT = AB + z

  60T = Az

---

(xT/60T)= (7/5)

x = 60(7/5)

x = 84 km/h

=====

Time for car to reach B is:    time (hr) = 108 km/(84 km/h)

                                                 time = 1.286 hours

Distance for at 1.289 hours is:    distance (km) = (60 km/h)*(1.286 h)

                                                   distance = 77.14 km

At 1.286 hours, the car reverses direction.  The van is (108 km - 77.14 km) or 30.86 km away.

Add the distances travelled by both vehicles after the car reverses direction at 1.286 hours.  The sum will be 30.86 km when they meet, at time of T.

Car Distance + Van Distance = 30.86 km

T(84 km/h) + t(60 km)

They meet when they are 0 km apart, which can be modeled with the following equation:

Van travel Distance - Car Travel Distance = 0 starting at 1.286 hr.

Let t be the time after 1.286 hours that both vehicles meet/collide.

t*(60 km/h)  +  t(84 km/h) = 30.86 km

t(60+84) = 30.86 km

t(144 km/h) = 30.86 km

t = 0.2143 hr

Total time until the car and van meet is 1.286 hr + 0.2143 hr for a total of 1.50 hours.

=================

a)  Time until both vehicles meet is 1.5 hours after starting at noon.  That makes it 1:30PM.

b)  Average speed of car is 84 km/h

==============

CHECK

Is the ratio of the distance travelled by the car and the van until they meet in the ratio of 7/5?

Car distance is (1.5 hr)(84 km/h) = 126 km

Van distance is (1.5 hr)(60 km/h) = 90.0 km

Ratio is 126/90 or 1.4

Ratio of 7/5 is 1.4

YES

Answer:

  A.  The reflection preserves the side lengths and angles of triangle . The dilation preserves angles but not side lengths.

Step-by-step explanation:

Reflection is a rigid transformation. It preserves both angles and side lengths. Dilation preserves angles, but changes all lengths by the same scale factor.

The described triangle was subject to reflection, which preserves angles and lengths. It was also subject to dilation, which preserves angles, but not lengths.

The appropriate description is that of choice A.

Answer:

-3

Step-by-step explanation:

look at the hint: slope (most call it gradient) = change in y/change in x

so you take 2 x and y variables and subtract

15 - 9 = 6   ← change in y

0 - 2 = -2   ← change in x

slope (gradient) = 6/-2

slope (gradient) = -3

The exact value of sin 5π/4 is 0. 0677

It is important to note that π = 3. 142

Let's find sin 5π

Substitute the value

sin 5π = sin (5 × 3. 142) = sin 15. 71

= [tex]\frac{sin 15. 71}{4}[/tex]

Find the sine of the number

= [tex]\frac{0. 27076}{4}[/tex]

= [tex]0. 0677[/tex]

Thus, the exact value of sin 5π/4 is 0. 0677

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Answer:

-  sqrt2/2

Step-by-step explanation:

Answer:

Step-by-step explanation:

The function with the greatest rate of change over the interval [0, 2] is: D. a

The expansion of the expression (x + 2y)² is x² + 4xy + 4y².

The given expression is (x + 2y)². The expansion of the expression will be:

(x + 2y)²

= (x + 2y)(x + 2y)

= x² + 2xy + 2xy + 4y².

= x² + 4xy + 4y²

The expansion is x² + 4xy + 4y².

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Considering the definition of an equation and the way to solve it, there are 10 apples in the bag.

An equation is the equality existing between two algebraic expressions connected through the equals sign in which one or more unknown values, called unknowns, appear in addition to certain known data.

The members of an equation are each of the expressions that appear on both sides of the equal sign while the terms of an equation are the addends that form the members of an equation.

The solution of a equation means determining the value that satisfies it. In this way, by changing the unknown to the solution, the equality must be true.

To solve an equation, keep in mind:

Being "or" the number of oranges in the bag, and knowing that:

the equation in this case is:

or + apples= 15

or + 2or= 15

Solving:

3or= 15

or= 15 ÷ 3

or= 5

As a bag of apples and oranges contains twice as many apples as oranges, then 2or= 2×5= 10 apples

Finally, there are 10 apples in the bag.

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One sample Z Test is used by cate.

According to the statement

The average tuition in Illinois is $28,950 with a standard deviation of $3,470.

Assuming we have a large enough sample, since we are testing one group's mean relative to a hypothesized value, this would be a one-sample z-test.

So, One sample Z Test is used by cate.

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Answer:

9cm

Step-by-step explanation:

If the triangles are similar then their lengths are in a ratio to each other, Ex: PR is 3 times LN, 2*3 = 6cm

This gives us a ratio of 1:3

PQ = LM*3 = 3cm*3 = 9cm

The x-values are the heartbeat(s) and the y-values are the time(s)

The variables are given as:

Time(s) and Heartbeat(s)

As a general rule, the action that is being done is the independent variable.

The action here is the heartbeat.

So, the x-values are the heartbeat(s) and the y-values are the time(s)

A possible value of this is;

x = 100, y = 60 seconds

So, the ordered pair is (100,60)

When represented as ratio, we have;

Ratio = 60/100

Simplify

Ratio = 0.6 heartbeat per second

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Answer:

Triangle DEF is a right, scalene triangle. It is not isosceles, obtuse, acute, or equilateral

Step-by-step explanation:

All we know about m and n are that they are not equal to each other and they are positive. This was given in the problem. See image. Once it is graphed you can see on the graph the lengths of DE and EF. Use Pythagorean theorem to calculate DF. see image.

DE is horizontal and EF is vertical, so you can see their slopes or calculate using a formula. Calculate the slope of DF. Slope is y-y on top of a fraction and x-x on the bottom of the fraction.

Lastly, use midpoint formula to find the midpoints. Average the x's and average the y's to find the x- and y-coordinates of the midpoints. See image.

Finally, DEF is a right triangle. The graph as well as the slopes show us that DE and EF form a right angle. So DEF must be a right triangle (and not obtuse nor acute) We were told that m doesNOT equal n, so the triangle cannot have two equal sides, so it cannot be isosceles (2 equal sides) nor equilateral (3 equal sides) It has 3 different lengths of sides; that is called scalene.

Answer:

41 yards by 118 yards

Step-by-step explanation:

Let W and L be the Width and Length of the playing field:

The perimeter, P, would be P = 2W + 2L

We are told that P = 318 yards.

We learn that L = 3W - 5

Combining this information, we can say:

P = 2W + 2L                         [Oriinal equation]

318 = 2W + 2L                      [P is 318]

318 = 2W + 2(3W-5)             [ Use the definition of L = 3W - 5]

318 = 2W + 6W - 10

328 = 8W

W = 41 yards

Use this value of W to find L:

L = 3W - 5

L = 3(41) - 5

L = 123 - 5

L = 118

CHECK

Does 2*(118) + 2*(41) = 318?  YES

Is the length 5 yards less than triple the width?

3*(41) - 5 = 118  YES

Domain refers to the term which best describes the set of all possible input values for a function and is therefore denoted as option C.

These are all the values which go into a function which is defined and is denoted as the sign below:

dom · ( f ).

They are also among the values which form the independent variables in this type of context while on the other hand, the output is referred to as the range.

These set of values are usually inputted into the function to check its acceptability and also retrieve the output gotten from the type of operation given. It is also used to identify the independent variable which are present thereby making it the most appropriate choice.

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Answer:-----------

Step-by-step explanation:------------------

Answer:

???????

Step-by-step explanation:

Please tell the question so i can answer it

There are 120 ways in which 5  riders and 5 horses can be arranged.

We have,

5 riders and 5 horses,

Now,

We know that,

Now,

Using the arrangement formula of Permutation,

i.e.

The total number of ways [tex]^nN_r = \frac{n!}{(n-r)!}[/tex],

So,

For n = 5,

And,

r = 5

As we have,

n = r,

So,

Now,

Using the above-mentioned formula of arrangement,

i.e.

The total number of ways [tex]^nN_r = \frac{n!}{(n-r)!}[/tex],

Now,

Substituting values,

We get,

[tex]^5N_5 = \frac{5!}{(5-5)!}[/tex]

We get,

The total number of ways of arrangement = 5! = 5 × 4 × 3 × 2 × 1 = 120,

So,

There are 120 ways to arrange horses for riders.

Hence we can say that there are 120 ways in which 5  riders and 5 horses can be arranged.

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Solving a system of equations we can see that:

First, let's define the variables:

With the given information we can write 3 equations:

A = (3/4)*V

V = K - $18

A + V + K = $386.4

Then we need to solve a system of equations:

A = (3/4)*V

V = K - $18

A + V + K = $386.4

We can first replace the first equation into the third to get:

(3/4)*V + V + K = $386.4

And rewrite the second to get:

k = V + $18

And replace that in the other equation:

(3/4)*V + V + V + $18 = $386.4

Now we can solve this for V:

V*(1 + 1 + 3/4) = $386.4 - $18

V*(11/4) = $368.40

V = (4/11)*$368.40 = $133.96

Now that we know the value of V, we can replace it in:

K = V + $18 = $133.96 + $18 = $151.96

A = (3/4)*V = (3/4)*$133.96 = $100.47

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A) Sample size of compound event = 25

B) New Sample size = 36

What is Compound event?

A compound event is one that has more than one sample point. Simple events are less complicated than compound events. These occurrences entail the possibility of multiple occurrences. One is the probability of all possible outcomes of a compound event.

An event that consists of two or more simple events is called a compound event. Simple events are those that can only have one result, whereas complex events can have a variety of distinct results. Compound events can consist of several independent or dependent occurrences, where the result of one event has no bearing on the likelihood of the other.

Given,

Number of buttons in a container = 5

Lucas picks one button, replaces it and then picks another.

Therefore,

Sample size of the compound event

                 n = 5[tex]\times[/tex]5

                 n  = 25

Now, one more button is added in the container and in total there are 6 buttons.

Lucas repeats the same process again and hence,

Sample size of the new compound event will be,

                  n = 6[tex]\times[/tex]6

                  n = 36

Full Question is - Lucas has five buttons in a container. Each button is a different shape. There is a star, an oval, a hexagon, a circle, and a heart. He picks one button, replaces it, and then picks another button. The sample size for this compound event is  ___  Suppose one square-shaped button is added to the container. If Lucas repeats the same picking process, then the sample size would be ____

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Answer:

2.5

Step-by-step explanation:

v = u + at is one of the kinematics equations.

Given v = 10, t = 2, u = 5, we will substitute these values into the equation to find a.

[tex]v = u + at\\10=5+2a\\2a=10-5\\2a = 5\\a = \frac{5}{2} \\= 2.5[/tex]

[tex]f(x) = (x + 3)(x + 6) \\ f(x) = x {}^{2} + 6x + 3x + 18 \\ f(x) = x {}^{2} + 9x + 18[/tex]

Answer:

C.) y = x² + 9x + 18

Step-by-step explanation:

C.) is correct because the function has the accurate x-intercepts. X-intercepts are the x-values in which the line passes through the x-axis. We can prove this by setting the function equal to 0, factoring, then solving for "x". You can factor the function by asking yourself the question, "which two numbers multiply to 18 and add to 9?"

y = x² + 9x + 18                             <----- Original function

0 =  x² + 9x + 18                           <----- Set the function equal to 0

0 = (x + 6)(x + 3)                           <----- Factor

0 = x + 6             0 = x + 3

-6 = x                  -3 = x

So, at (-3, 0) and (-6, 0), the function crosses the x-axis. This matches the given graph.

The order of the numbers which satisfies commutative property exists 1/7 + (-1)+ 2/7.

The commutative property exists a math rule that states that the order in which we multiply numbers does not change the product. The commutative property uses for addition and multiplication. The property states that phrases can “commute,” or transfer locations, and the outcome will not be affected. This exists described as a + b = b + a for addition, and a × b = b × a for multiplication.

Arranging the order of the numbers the fractions

[tex]$& \frac{1}{7}+(-1)+\frac{2}{7} \\[/tex]

[tex]$=& \frac{1}{7}+\frac{2}{7}+(-1) \\[/tex]

Simplifying the equation, we get

[tex]$= \frac{1}{7}+(-1) \\[/tex]

[tex]$&=\frac{4}{7}[/tex]

Therefore, the correct answer is option A. 1/7 + (-1)+ 2/7

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The formula is [tex]\frac{n(n+1)(2n+1)}{6}[/tex]

What are series?

In mathematics, we can describe a series as adding infinitely many numbers or quantities to a given starting number or amount.

We will find the formula as shown as below:

Let [tex]S=1+2^2+3^2+4^2+................+n^2[/tex]

We know [tex](n+1)^3=n^3+3n^2+3n+1[/tex]

[tex](1+1)^3=1^3+3(1)^2+3(1)+1[/tex]

[tex](2+1)^3=2^3+3(2)^2+3(2)+1[/tex]

[tex](3+1)^3=3^3+3(3)^2+3(3)+1[/tex]

.

.

[tex](n+1)^3=n^3+3(n)^2+3(n)+1[/tex]

On adding

[tex]2^3+3^3+4^3......(n+1)^3=(1^3+2^3+3^3+.....+n^3)+3(1^2+2^2+.....+n^2)+3(1+2+3....n)+(1+1+1+....+1)[/tex]

[tex]2^3+3^3+4^3......(n+1)^3-(1^3+2^3+3^3+.....+n^3)=3S+\frac{3n(n+1)}{2}+(1+1+1+....+1)[/tex]

[tex](n+1)^3-1^3=3S+\frac{3n(n+1)}{2} +n[/tex]

[tex]n^3+3(n)^2+3(n)+1-1=3S+\frac{3n(n+1)}{2} +n[/tex]

[tex]2n^3+6n^2+6n=6S+3n^2+3n+2n[/tex]

[tex]6S=2n^3+3n^2+n[/tex]

[tex]6S=2n^2(n+1)+n(n+1)[/tex]

[tex]6S=(n+1)(2n^2+n)[/tex]

[tex]6S=n(n+1)(2n+1)[/tex]

[tex]S=\frac{n(n+1)(2n+1)}{6}[/tex]

Hence, the formula is [tex]\frac{n(n+1)(2n+1)}{6}[/tex]

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