Find the markup and the cost of the following item. Round answers to the nearest cent.

A mirror selling for $98.00, marked up 30% on cost.

M=
C=


Find the markup and the cost of the following item. Round answers to the nearest cent.

A ream of paper selling for $2.19, marked up 11% on cost.

M=
C=

Answer:

(-10,-10)

Step-by-step explanation:

9x-9y=0

3x-4y=10

In elimination, we want both equations to have the same form and like terms to be lined up.  We have that.  We also need one of the columns with variables to contain opposites or same terms. Neither one of our columns with the variables contain this.  

We can do a multiplication to the second equation so that the first terms of each are either opposites or sames. It doesn't matter which.  I like opposites because you just add the equations together. So I'm going to multiply the second equation by -3.

I will rewrite the system with that manipulation:

9x-9y=0

-9x+12y=-30

----------------------Add them up!

0+3y=-30

     3y=-30

       y=-10

So now once you find a variable, plug into either equation to find the other one.

I'm going to use 9x-9y=0 where y=-10.

So we are going to solve for x now.

9x-9y=0 where y=-10.

9x-9(-10)=0  where I plugged in -10 for y.

9x+90=0  where I simplified -9(-10) as +90.

9x     =-90 where I subtracted 90 on both sides.

x=       -10 where I divided both sides by 9.

The solution is (x,y)=(-10,-10)

Answer:

cos(z) = .3846153846 and angle z = 67.38°

Step-by-step explanation:

Side UV is corresponding to side YX.  Side VW is corresponding to side YZ.  Side UW is corresponding to side XZ.

Starting with the first corresponding pair, we are told that side UV is 36, and that side YX is 3/5 of that.  So side YX is

[tex]\frac{3}{5}*36=21.6[/tex]

We are next told that side VW is 39, so side YZ is

[tex]\frac{3}{5}*39=23.4[/tex]

In order to find the cos of angle z, we need the adjacent side, which is side XZ.  Side XZ is 3/5 of side UW.  Right now we don't know the length of side UW, so we find it using Pythagorean's Theorem:

[tex]39^2-36^2=UW^2[/tex] and

[tex]UW^2=225[/tex] so

UW = 15

Now we can say that side XZ is

[tex]\frac{3}{5}*15=9[/tex]

The cos of an angle is the side adjacent to the angle (9) over the hypotenuse of the triangle (23.4) so our ratio is:

[tex]cos(z)=\frac{9}{23.4}[/tex]

which divides to

cos(z) = .3846153846

If you need the value of the angle, use the inverse cosine function on your calculator in degree mode to find that

angle z = 67.38°

3/6 and 7/14 can both be reduced to 1/2

Lets start with 3/6

3   divided by 3               1

--                                     ----

6  divided by 3                2

now, 7/14

7  divided by 7           1

--                                ----

14 divided by 7           2

Whatever you do to the numerator (top number) you have to do to the denominator (bottom number)

Answer:

The probability is [tex]\frac{5}{9}[/tex]

Step-by-step explanation:

Let A be the event of one red, zero white, and three blue chips,

And, B is the event of at least three blue chips,

Since, A ∩ B = A (because If A happens that it is obvious that B will happen )

Thus, the  conditional probability of A if B is given,

[tex]P(\frac{A}{B})=\frac{P(A\cap B)}{P(B)}=\frac{P(A)}{P(B)}[/tex]

Now, red chips = 5,

White chips = 3,

Blue chips = 7,

Total chips = 5 + 3 + 7 = 15

Since, the probability of one red, zero white, and three blue chips, when four chips are chosen,

[tex]P(A)=\frac{^5C_1\times ^3C_0\times ^7C_3}{^{15}C_4}[/tex]

[tex]=\frac{5\times 35}{1365}[/tex]

[tex]=\frac{175}{1365}[/tex]

[tex]=\frac{5}{39}[/tex]

While, the probability that of at least three blue chips,

[tex]P(B)=\frac{^8C_1\times ^7C_3+^8C_0\times ^7C_4}{^{15}C_4}[/tex]

[tex]=\frac{8\times 35+35}{1365}[/tex]

[tex]=\frac{315}{1365}[/tex]

[tex]=\frac{3}{13}[/tex]

Hence, the required conditional probability would be,

[tex]P(\frac{A}{B})=\frac{5/39}{3/13}[/tex]

[tex]=\frac{65}{117}[/tex]

[tex]=\frac{5}{9}[/tex]

Check the picture below.

where is the -16t² coming from?  that's Earth's gravity pull in feet.

[tex]\bf ~~~~~~\textit{initial velocity} \\\\ \begin{array}{llll} ~~~~~~\textit{in feet} \\\\ h(t) = -16t^2+v_ot+h_o \end{array} \quad \begin{cases} v_o=\stackrel{30}{\textit{initial velocity of the object}}\\\\ h_o=\stackrel{6}{\textit{initial height of the object}}\\\\ h=\stackrel{}{\textit{height of the object at "t" seconds}} \end{cases} \\\\\\ h(t)=-16t^2+30t+6 \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf \textit{vertex of a vertical parabola, using coefficients} \\\\ h(t)=\stackrel{\stackrel{a}{\downarrow }}{-16}t^2\stackrel{\stackrel{b}{\downarrow }}{+30}t\stackrel{\stackrel{c}{\downarrow }}{+6} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right)[/tex]

[tex]\bf \left(-\cfrac{30}{2(-16)}~~,~~6-\cfrac{30^2}{4(-16)} \right)\implies \left( \cfrac{30}{32}~,~6+\cfrac{225}{16} \right)\implies \left(\cfrac{15}{16}~,~\cfrac{321}{16} \right) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill (\stackrel{\stackrel{\textit{how many}}{\textit{seconds it took}}}{0.9375}~~,~~\stackrel{\stackrel{\textit{how many feet}}{\textit{up it went}}}{20.0625})~\hfill[/tex]

Answer:

Step-by-step explanation:

I'm not sure if this question is coming from a physics class or an algebra 2 or higher math class, but either way, the behavior of a parabola is the same in both subjects.  If a parabola crosses the x axis, those 2 x values are called zeros of the polynomial.  Those zeros translate to the time an object was initially launched and when it landed.  The midpoint is dead center of where those x values are located.  For example, if an object is launched at 0 seconds and lands on the ground 3 seconds later, it reached its max height at 2 seconds.  So what we need to do is find the zeros of this particular quadratic, and the midpoint of those 2 values is where the object was at a max height of 20.06.

I used the physics equation representing parabolic motion for this, since it has an easier explanation. This equation is

[tex]x-x_{0}=v_{0}+\frac{1}{2}at^2[/tex]

where x is the max height, x₀ is the initial height, v₀ is the initial upwards velocity, t is time (our unknown as of right now), and a is the acceleration due to gravity (here, -32 ft/sec^2).  Filling in our values gives us this quadratic equation:

[tex]20.06-6=30(t)+\frac{1}{2}(-32)t^2[/tex]

Simplifying that a bit gives us

[tex]14.06=30t-16t^2[/tex]

Rearranging into standard form looks like this:

[tex]0=-16t^2+30t-14.06[/tex]

If we factor that using the quadratic formula we find that the 2 times where the ball was launched and then where it came back down are

t = .925 and .95 (the ball wasn't in the air for very long!)

The midpoint occurs between those 2 t values, so we find the midpoint of those 2 values by adding them and dividing the sum in half:

[tex]\frac{.925+.95}{2}=.9375[/tex]

Therefore, the coordinates of the vertex (the max height) of this parabola are (.94, 20.06).  That translates to: at a time of .94 seconds, the ball was at its max height of 20.06 feet

Answer:

The ratio of the volume of one sphere to the volume of the right cylinder is 1 : 3 ⇒ answer B

Step-by-step explanation:

* Lets explain how to solve the problem

- The spheres touch the two bases of the cylinder

∴ The height of the cylinder = the diameters of the two spheres

∵ The diameter of the sphere = twice its radius

∴ The diameter of the sphere = 2r, where r is the radius of the sphere

∵ The height of the cylinder = 2 × diameter of the sphere

∴ The height of the cylinder = 2 × 2r = 4r

- The spheres touch the curved surface of the cylinder, that means

  the diameter of the sphere equal the diameter of the cylinder

∴ The maximum possible radius of the sphere is the radius of the

   cylinder

∵ The radius of the sphere is r

∴ The radius of the cylinder is r

- The volume of the cylinder is πr²h and the volume of the sphere is

  4/3 πr³

∵ The height of the cylinder = 4r ⇒ proved up

∵ The radius of the cylinder = r

∵ The volume of the cylinder = πr²h

∴ The volume of the cylinder = πr²(4r) = 4πr³

∵ The radius of the sphere = r

∵ The volume of the sphere = 4/3 πr³

∴ The ratio of the volume of one sphere to the volume of the right

   cylinder = 4/3 πr³ : 4 πr³

- Divide both terms of the ratio by 4 πr³

∴ The ratio = 1/3 : 1

- Multiply both terms of the ratio by 3

∴ The ratio = 1 : 3

∴ The ratio of the volume of one sphere to the volume of the right

  cylinder is 1 : 3

Answer:

81

Step-by-step explanation:

(g∘f)(-4) is another way of writing g(f(-4)).

First, find f(-4):

f(-4) = -2(-4) − 5

f(-4) = 3

Now plug into g(x):

g(f(-4)) = g(3)

g(f(-4)) = 3^4

g(f(-4)) = 81

Answer:

  see below

Step-by-step explanation:

Option 1: the two solution sets overlap at x=1.

Option 2: the two solution sets do not overlap; no solution.

Option 3: together, the two set describe all real numbers.

Answer:

The correct option is  Y = 1,000(1.15)6 2,313 bacteria

Step-by-step explanation:

According to the given statement a growth medium is  inoculated with 1,000 bacteria.

Bacteria grow at a rate of = 15% = 0.15

Add 1 to make it easier = 0.15+1 = 1.15

We have to find the population of the culture 6 days after inoculation.

= 1000(1.15)^6

=1000(2.313)

= 2313 bacteria

The population of the culture 6 days after inoculation = 2313 bacteria

Therefore the correct option is  Y = 1,000(1.15)6 2,313 bacteria....

Answer:AAAAAAAAAAAAAAAAAAAA

Answer:

Initial velocity is zero.

Step-by-step explanation:

According to second equation of motion

    [tex]s=ut+\frac{1}{2} at^2[/tex]

where s = distance traveled  

           t = time taken

          a = acceleration

          u = initial velocity

here in the question we have

      s = 10 m

     t = 1 second

    a = 20[tex]ms^{-2}[/tex]

plugging the known value in order to find the unknown which is u (initial velocity)

[tex]10=u(1)+\frac{1}{2} (20)(1)^2[/tex

10 = u +10

gives u =0

therefore initial velocity is zero.

If two lines l and m are parallel, then a reflection along line l followed by a reflection along line m is the same as a translation. Therefore, the correct answer is: A. translation.

If two lines l and m are parallel, a reflection along l followed by a reflection along m is equivalent to a translation. This can be understood geometrically: when a figure is reflected across parallel lines, it undergoes a displacement without changing its orientation.

The sequence of two reflections results in the figure being shifted along a path parallel to the original lines, akin to a translation. While reflections change orientation and translations shift position, the combination of two reflections across parallel lines maintains the figure's orientation while relocating it.

Answer:

  d = 8t

Step-by-step explanation:

A lot of math is about matching patterns. Here, the pattern you can match is given in the problem statement:

  d = 6t  . . . . . equation for distance traveled at 6 miles per hour

You are asked to write an equation for distance traveled at 8 miles per hour. You can see the number 6 in the above equation matches the "miles per hour" of the traveler. This should give you a clue that when the "miles per hour" changes from 6 to 8, the number in the equation will do likewise.

The equation you want is ...

  d = 8t . . . . . equation for distance traveled at 8 miles per hour

The distance traveled by a person biking at 8 miles per hour is represented by the equation d = 8t, where d is the distance, and t is the time in hours. For example, if they bike for 2 hours, they will have traveled 16 miles.

The question is about calculating the distance traveled by a person who can bike at a rate of 8 miles per hour. This can be represented by the equation d = rt, where d is the distance, r is the rate, and t is the time.

In this case, the rate (r) is 8 miles per hour. Therefore, the equation becomes: d = 8t, meaning the distance travelled is equal to 8 times the amount of time spent biking.

For example, if the person bikes for 2 hours, we would substitute 2 for t in the equation, which would look like this: d = 8 * 2. The resulting distance (d) is 16 miles.

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Answer:

First mechanic: 20 hours

Second mechanic: 15 hours

Step-by-step explanation:

First we create two equations where:

x - hours of first mechanic

y - hours of second mechanic

x+y=35

115x+45y=2975

Then, we multiply both sides of the first equation by 45, and then subtract it from the second equation:

45x+45y=1575

|115x+45y=2975

-|45x+45y = 1575

70x = 1400

x=20 hours

Now we know for how many hours the first mechanic worked. Now we just need to subtract that from the combined total to find the second mechanic's hours:

35-20=15 hours

Answer:

58 miles per hour

Step-by-step explanation:

First you need to divide 80 by 10 to see how many miles she was over the speed limit.

80/10 = 8 miles over the speed limit.

The speed limit was 50, so 50 + 8 = 58 miles.

So Cathy was driving at 58 miles per hour

Answer:

58mph

Step-by-step explanation:

Given:

Each mph over the speed limit gets fined $10

or mathematically, rate of fine = $10/mph

also, total fine was $80.

Number of mph over the speed limit,

= total fine ÷ rate of fine

= $80 ÷ $10/mph

= 8 mph.

Given that the speed limit was 50 mph, Cathy's final speed,

= speed limit + number of mph over speed limit

= 50 + 8

= 58 mph

Answer:

10

Step-by-step explanation:

First you do 8 × 4, this gives the total of the four numbers.

[tex]8 \times 4 = 32[/tex]

Then, subtract the total of 3 of the numbers from the total of 4 of the numbers.

[tex]32 - 22 = 10[/tex]

Answer:

  a)  73.52%

  b)  Regular membership is growing faster

Step-by-step explanation:

a)  r(t) = r0·1.047^(12t) . . . . regular members after t years, where r0 is the initial value of regular members at t=0.

Equivalently, this is ...

  r(t) = r0·(1.047^12)^t ≈ r0·1.7352^t

This shows the effective annual growth rate for regular members is 73.52%.

__

b) The 74% yearly growth rate of regular members is higher than the 65% yearly growth rate of VIP members. Regular membership is growing faster.

Answer:

The answer is 1.94

You need to have 1.94 hours a week to loose a monthly weight of 3 pounds

Answer:

The correct option is 2.

Step-by-step explanation:

Given information: Height of both prism are same.

Right rectangular prism has base dimensions of 3 inches by 12 inches.

Volume of a right rectangular prism:

[tex]V=Bh[/tex]

where, B is base area and h is height of the prism.

The volume of right rectangular prism is

[tex]V=(3\times 12)\times h=36h[/tex]

Therefore the volume of right rectangular prism is 36h cubic inches.

An oblique rectangular prism has base dimensions of 4 inches by 9 inches.

Volume of a oblique rectangular prism:

[tex]V=Bh[/tex]

where, B is base area and h is height of the prism.

The volume of right rectangular prism is

[tex]V=(4\times 9)\times h=36h[/tex]

Therefore the volume of oblique rectangular prism is 36h cubic inches.

The volumes are equal, because the heights are equal and the horizontal cross-sectional areas at every level are also equal.

Option 2 is correct .

The volumes are equal, because the heights are equal and the horizontal cross-sectional areas at every level are also equal.

The formula for the Volume of a right rectangular prism is:

V = B * h

where,

B is base area.

h is height of the prism.

Thus:

V = 3 * 12 * h

V = 36h

Similarly, the volume of the oblique rectangle is:

V = Bh

V = 4 * 9 * h

V = 36h

Thus, we can see that the volumes are equal, because the heights are equal and the horizontal cross-sectional areas at every level are also equal.

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Answer:

16

Step-by-step explanation:

i know because i did the test

Answer:

16 for people with ads i really need brainliest

Step-by-step explanation:

Answer:

x^2 -6x+44

Step-by-step explanation:

Develop form of (x-3)^2 is x^2 - 6x +9

Then y= x^2 -6x +9 + 35

So, y= X^2 -6x +44

Answer:

[tex]y=x^2 -6x+44[/tex]

Step-by-step explanation:

The standard form of a quadratic equation is:

[tex]y = ax ^ 2 + bx + c[/tex].

In this case we have the following quadratic equation in vertex form

[tex]y=(x-3)^2+35[/tex]

Now we must rewrite the equation in the standard form.

[tex]y=(x-3)(x-3)+35[/tex]

Apply the distributive property

[tex]y=x^2 -3x -3x +9+35[/tex]

[tex]y=x^2 -6x+9+35[/tex]

[tex]y=x^2 -6x+44[/tex]

the standard form of the equation is: [tex]y=x^2 -6x+44[/tex]

Answer:

The answer is 9

Answer:

Step-by-step explanation:

9

3x + y = -10. The standard form of the equation that is parallel to y = -3x + 3 and goes through point (-5, 5) is 3x + y = -10.

The equation is written in the slope-intercept form y = mx +b. So:

y = -3x + 3

The slope m = -3

Since the slopes of parallel lines are the same, we are looking for a slope line m = -3 and goes through point (-5, 5).

With the slope-intercept form:

y = mx + b

Introducing the slope m = -3:

y = -3x + b

Introducing the point (-5, 5):

5 = -3(-5) + b

5 = 15 + b

b= 5 - 15

b = -10

then

y = -3x -10

write the equation in standard form ax + by = c:

y = -3x -10

3x + y = -10

To find the standard form of the equation of a line that is parallel to y = -3x + 3 and passes through the point (-5, 5), let's follow these steps:
1. **Identify the slope**: Since parallel lines have the same slope, we can take the slope of the given line y = -3x + 3, which is -3.
2. **Use the point-slope form**: With the slope known and a point provided, we can use the point-slope form of the equation, which is:
  \[ y - y_1 = m(x - x_1) \]
  where \(m\) is the slope and \((x_1, y_1)\) is the point the line passes through.
3. **Apply the point and slope**: Insert the slope (-3) and the point (-5, 5):
  \[ y - 5 = -3(x - (-5)) \]
  \[ y - 5 = -3(x + 5) \]
4. **Distribute the slope**: Multiply -3 by both terms inside the parentheses:
  \[ y - 5 = -3x - 15 \]
5. **Isolate y:** Add 5 to both sides of the equation to isolate y:
  \[ y = -3x - 15 + 5 \]
  \[ y = -3x - 10 \]
6. **Convert to standard form**: The standard form of a linear equation is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) should be non-negative. Arrange the terms:
  \[ 3x + y = -10 \]
So the standard form of the equation that is parallel to y = -3x + 3 and passes through the point (-5, 5) is \(3x + y = -10\).

Answer:

Acute

Step-by-step explanation:

The Converse of the Pythagorean Theorem states that:

The side lengths 8, 14, and 15 are given. We can assume the hypotenuse (or c) is the longest side length, so it is 15.

It doesn't matter which order of the numbers are plugged in for a and b, so a and b will be 8 and 14.

Now we have to add [tex]a^2[/tex] and [tex]b^2[/tex] to see if the sum is greater than, less than, or equal to 15 (c).

Calculate the rest of the problem.

We have to find what [tex]15^2[/tex] is before we can make a decision using the Converse of the Pythagorean Theorem.

260 ([tex]a^2+b^2[/tex]) is greater than 225 ([tex]c^2[/tex]). This means that the triangle is acute because [tex]a^2+b^2>c^2[/tex].

Answer:

7 minutes

Step-by-step explanation:

g(x) = x^4 - 3x^2 + 4x - 5

g(2) = 2^4 - 3(2^2) + 4(2) - 5

g(2) = 16 - 3(4) + 4(2) - 5

g(2) = 16 - 12 + 8 - 5

16 - 12 = 4

4 + 8 = 12

12 - 5 = 7

input: 2

output: 7

ordered pair: (2,7)

By modeled function, the time taken to reach the school is 7 minutes.

A function which depicts the variation of a given dependent parameter represented by the variable in the function is known as a modeled function. For modeled function, we input the value of the dependent parameter in place of the given variable and the solution of function gives us the result of dependency.

Given that the time it takes to get to school, measured in minutes, is modeled using the function g(x) = [tex]x^{4} - 3x^{2} + 4x - 5[/tex] , where x is the number of stops the bus makes.

Also said that the bus makes 2 stops after we board.

Thus the dependent parameter is stop which is represented by the variable x in the modeled function g(x) and the solution of g(x) gives us the time period. We will get the time period by putting x = 2 in the modeled function.

Putting x = 2 in g(x), we get -

⇒ g(x) = [tex]2^{4} - 3*2^{2} + 4*2 - 5[/tex]

⇒ g(x) = 16 - 12 + 8 - 5

∴  g(x) = 7

Thus the time period is 7 units.

Therefore, by modeled function, the time taken to reach the school is 7 minutes.

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Answer:

Step-by-step explanation:

The temperature at a given altitude is

y = 36 - 3x

The temperature on the surface of the planet is the point (0,t) where t is the temperature for the given height.

y = 36 - 3*0

y = 36

So at the surface of the planet is 36 degrees C.

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

Effectively it is the slope of the equation which is - 3

So ever km going up will mean a loss of 3 degrees. I think they want you to write -3

Answer:

16 inches

Step-by-step explanation:

w = width of the rectangle

l = length of the rectangle

The perimeter of a rectangle is the length around it:

P = 2w + 2l

Given:

P = 92

l = w + 14

Substituting:

92 = 2w + 2(w + 14)

92 = 2w + 2w + 28

64 = 4w

w = 16

The width of the rectangle is 16 inches.

Final answer:

The width of the rectangle is 16 inches, deduced by using the perimeter formula for a rectangle and solving for width with given conditions.

Explanation:

To find the width of the rectangle given its perimeter and the fact that its length is 14 inches longer than its width, we start by listing what we know:

The perimeter of the rectangle is 92 inches.

The length (L) is 14 inches longer than the width (W).

Remember, the perimeter (P) of a rectangle is given by P = 2L + 2W. We can substitute L with W + 14, since the length is 14 inches more than the width. This gives us:

P = 2(W + 14) + 2W

Now we can plug in the value for P:

92 = 2(W + 14) + 2W

Then, distribute:

92 = 2W + 28 + 2W

Combine like terms:

92 = 4W + 28

Now, subtract 28 from both sides:

64 = 4W

Lastly, divide by 4 to solve for W:

16 = W

So, the width of the rectangle is 16 inches.

For this case we have the following expression:

[tex]\frac {11x-2-3 (1-7x) ^ 2} {(x + 1)}[/tex]

We must indicate the first step that allows to start the simplification of the expression.

It is observed that the first step to follow is to solve the square of the binomial that is in the numerator of the expression.

[tex](1-7x) ^ 2[/tex]

Answer:

Option A

Answer:

0

Step-by-step explanation:

11x-2-3(1-7x)^2/(1x+1)

11x-2-3-21x/1+1

-32x-2-3-1

-32x-1+1

-32=0

/-32 /-32

x=0

Answer:

revolutions per minute for 16 inches is 946 / minute

revolutions per minute for 14 inches is 1081 / minute

Step-by-step explanation:

Given data

radius = 16 inch

speed = 90 mph  = 90/60 = 1.5 miles/minute = 1.5 × 5280 feet /12 inches = 95040 inches /minute

radius 2 = 14 inches

to find out

revolutions per minute

solution

first we calculate the circumference of the wheel i.e. 2×[tex]\pi[/tex]×radius

circumference = 2×[tex]\pi[/tex]×16

circumference = 32[tex]\pi[/tex]

we know that revolution is speed / circumference

revolution = 95040/32[tex]\pi[/tex]

revolution = 945.38 / minute

we have given radius 14 inches than revolution will be i.e.

revolution = speed / circumference

circumference = 2×[tex]\pi[/tex]×14

circumference = 28[tex]\pi[/tex]

revolution =  95040/ 28[tex]\pi[/tex]

revolution = 1080.43 / minute

Answer:

Step-by-step explanation:

E

Answer:

B

Step-by-step explanation:

Edge

Answer: 0.0228

Step-by-step explanation:

Given : The  mean and the standard deviation of finish times (in minutes) for this event are respectively as :-

[tex]\mu=30\\\\\sigma=5.5[/tex]

If the distribution of finish times is approximately bell-shaped and symmetric, then it must be normally distributed.

Let X be the random variable that represents the finish times for this event.

z score : [tex]z=\dfrac{x-\mu}{\sigma}[/tex]

[tex]z=\dfrac{19-30}{5.5}=-2[/tex]

Now, the probability of runners who finish in under 19 minutes by using standard normal distribution table :-

[tex]P(X<19)=P(z<-2)=0.0227501\approx0.0228[/tex]

Hence, the approximate proportion of runners who finish in under 19 minutes = 0.0228

Final answer:

To find the proportion of runners finishing a 2-mile fun-run in under 19 minutes, we calculate the z-score for a finish time of 19 minutes given the mean of 30 and standard deviation of 5.5. The calculated z-score of -2 corresponds to about 2.28% of runners according to the standard normal distribution.

Explanation:

To find the approximate proportion of runners who finish in under 19 minutes, we will use the properties of the normal distribution because the distribution of finish times is approximately bell-shaped and symmetric. Given that the mean finish time (μ) is 30 minutes and the standard deviation (σ) is 5.5 minutes, we can calculate the z-score for a finish time of 19 minutes.

The z-score is calculated by the formula:
z = (X - μ) / σ
Where X is the finish time of interest (19 minutes), μ is the mean (30 minutes), and σ is the standard deviation (5.5 minutes).

Plugging in our values we get:
z = (19 - 30) / 5.5 = -11 / 5.5 = -2

Using a standard normal distribution table or calculator, we can find the area to the left of z = -2, which corresponds to the proportion of runners finishing in under 19 minutes. This value is approximately 0.0228 or 2.28% of the runners.