4 1/2 In radical form

The probability of the four events are: Event 1: 1/84Event 2: 3/14Event 3: 15/28 Event 4: 5/21

The total number of drinks = 9The number of sweet teas = 3The number of unsweetened teas = 6If you select 3 drinks at random, the following events can take place:

Event 1: All three drinks are sweet teas. The probability of event 1 = (Number of ways in which all three drinks can be sweet teas) / (Number of ways to select 3 drinks)The number of ways in which all three drinks can be sweet teas = 3C3 = 1 (because all three sweet teas are already fixed)The number of ways to select 3 drinks = 9C3 = (9 × 8 × 7)/(3 × 2 × 1) = 84Therefore, the probability of event 1 = 1/84 = 1/84

Event 2: Exactly two drinks are sweet teas. The probability of event 2 = (Number of ways in which two drinks are sweet teas and one is an unsweetened tea) / (Number of ways to select 3 drinks)The number of ways in which two drinks are sweet teas and one is an unsweetened tea = (3C2 × 6C1) = 18 (because you can choose 2 sweet teas from 3 and 1 unsweetened tea from 6)The number of ways to select 3 drinks = 9C3 = (9 × 8 × 7)/(3 × 2 × 1) = 84Therefore, the probability of event 2 = 18/84 = 3/14

Event 3: Exactly one drink is a sweet tea. The probability of event 3 = (Number of ways in which one drink is a sweet tea and the other two are unsweetened teas) / (Number of ways to select 3 drinks)The number of ways in which one drink is a sweet tea and the other two are unsweetened teas = (3C1 × 6C2) = 45 (because you can choose 1 sweet tea from 3 and 2 unsweetened teas from 6)The number of ways to select 3 drinks = 9C3 = (9 × 8 × 7)/(3 × 2 × 1) = 84Therefore, the probability of event 3 = 45/84 = 15/28

Event 4: All three drinks are unsweetened teas. The probability of event 4 = (Number of ways in which all three drinks can be unsweetened teas) / (Number of ways to select 3 drinks)The number of ways in which all three drinks can be unsweetened teas = 6C3 = 20 (because you can choose 3 unsweetened teas from 6)The number of ways to select 3 drinks = 9C3 = (9 × 8 × 7)/(3 × 2 × 1) = 84 Therefore, the probability of event 4 = 20/84 = 5/21

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Answer:

Step-by-step explanation:

First find the slope using m = (y2-y1)/(x2-x1)

m = (-1-7)/(-5-3)

   =  -8/-8

   =  1

y = mx + b

find the y-intercept

use 1 of the 2 points

Let's try (3,7)

7 = 1(3) + b

b = 4

Equation of the line is y = x + 4

The equation is:

Work/explanation:

First, we will use the slope formula and determine the slope:

[tex]\sf{m=\dfrac{y_2-y_1}{x_2-x_1}}[/tex]

where m = slope.

Plug in the data

[tex]\sf{m=\dfrac{-1-7}{-5-3}}\\\\\\\sf{m=\dfrac{-8}{-8}}\\\\\\\sf{m=1}[/tex]

The slope is 1; the equation so far is y = 1x + b or y = x + b.

Plug in the point:

[tex]\sf{7=3+b}[/tex]

[tex]\sf{3+b=7}[/tex]

Solve for b

[tex]\sf{b+3=7}[/tex]

[tex]\sf{b=4}[/tex]

So the y-intercept is 4; we plug that in and see that the equation is y = x + 4.

Answer:

Pamela is 30 years old.

Step-by-step explanation:

First equation:  

Since Pamela is 3 times older than Jakob, our first equation is given by:

P = 3J

Second Equation:

Since Pamela will be twice as old as Jakob in 10 years, our second equation is given by:

P = 2J + 10

Method to solve:  Substitution:

We can solve with substitution by isolating J in the second equation.  This will allow us to substitute it for J in the second equation and find P, Pamela's age:

Isolating J:

Step 1:  Divide both sides by 3

(P = 3J) / 3

P/3 = J

Substituting P/3 = J for J in P = 2J + 10:

P = 2(P/3) + 10

Step 1:  Distribute the 2 to P/3:

P = 2/3P + 10

Step 2:  Multiply both sides by 3 to clear the fraction:

(P = 2/3P + 10) * 3

3P = 2P + 30

Step 3:  Subtract 2P from both sides:

(3P = 2P + 30) - 2P

P = 30

Step 4:  Divide both sides by 2 to find P, Pamela's age:

(2P = 30) / 2

P = 30

Thus, Pamela is 30 years old.

Optional Steps to check the validity of our answer:

Plugging in 30 for P in P = 3J:

Step 1:  Divide both sides by 3:

(30 = 3J) / 3

10 = J

Thus, Jakob is 10 years old.

Checking the validity of answers with verbal statements:

Since 30 (i.e., Pamela's age) is indeed 3 times 10 (i.e., Jakob's age), this satisfies the first statement.

In 10 years, Pamela will be 40 as 30 + 10 = 40.

In 10 years, Jacob will be 20 as 10 + 10 = 20.

Since 40 (i.e., Pamela's age in 10 years) is indeed twice 20 (i.e., Jakob's age in 10 years), this satisfies the second statement.

Thus, our answer for Pamela's age is correct.

Answer:

a) [tex]\frac{x^2-4}{(x + 7)(x - 4)(x+2)}[/tex]

b)[tex]\frac{x^2-4x}{(x + 7)(x + 2)(x-4)}[/tex]

Step-by-step explanation:

x² + 3x - 28

= x² + 7x - 4x -28

= x(x + 7) - 4(x + 7)

=

= x² + 7x + 2x + 14

= x(x + 7) + 2(x + 7)

= (x + 7)(x + 2)

LCM of x² + 3x - 28 and x² + 9x + 14 is

(x + 7)(x - 4)(x + 2)

We can write:

[tex]\frac{x-2}{x^2 + 3x - 28}\\ \\= \frac{x-2}{(x + 7)(x - 4)}\\\\= \frac{x-2}{(x + 7)(x - 4)} *\frac{x+2}{x+2} \\\\=\frac{(x-2)(x+2)}{(x + 7)(x - 4)(x+2)} \\\\=\frac{x^2-2^2}{(x + 7)(x - 4)(x+2)} \\\\=\frac{x^2-4}{(x + 7)(x - 4)(x+2)}[/tex]

and

[tex]\frac{x}{x^2 + 9x + 14 }\\ \\= \frac{x}{(x + 7)(x + 2)}\\\\= \frac{x}{(x + 7)(x + 2)} *\frac{x-4}{x-4} \\\\= \frac{x(x-4)}{(x + 7)(x + 2)(x-4)} \\\\= \frac{x^2-4x}{(x + 7)(x + 2)(x-4)}[/tex]

Answer:

Area = 7.065 square miles

Step-by-step explanation:

The diameter of a circle is the distance across the circle passing through its centre, and it is equal to twice the radius. Therefore, if the diameter of a circle is 3 miles, then the radius is 1.5 miles.

The formula for the area of a circle is [tex]A = \pi r^2[/tex] ([tex]A = \pi \times r^{2}[/tex])

Substituting the value of the radius into this formula, we get:

Therefore, the area of the circle is approximately 7.065 square miles.

________________________________________________________

The answer is:

A = 7.069 miles²

Work/explanation:

We have the diameter, but we need to find the radius. We can do that by diving the diameter by 2:

radius = diameter ÷ 2 = 3 ÷ 2 = 1.5 miles

Now we move on to the next step - the formula.

The formula is:

[tex]\sf{A=\pi r^2}[/tex]

where:

diagram

[tex]\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\qbezier(2.3,0)(2.121,2.121)(0,2.3)\qbezier(-2.3,0)(-2.121,2.121)(0,2.3)\qbezier(-2.3,0)(-2.121,-2.121)(0,-2.3)\qbezier(2.3,0)(2.121,-2.121)(-0,-2.3)\put(0,0){\line(1,0){2.3}}\put(0.5,0.3){\bf\small 1.5\ miles}\end{picture}[/tex]

Plug in the data

[tex]\sf{A=\pi \times 1.5^2}[/tex]

[tex]\sf{A=\pi\times2.25}[/tex]

[tex]\sf{A=7.069~miles^2}[/tex]

Answer:

P = 9 + 9 + 4 + 4

Step-by-step explanation:

P=2(lxw) or l+l+w+w

Therefore, 9+9+4+4 is right.

Answer:

A

Step-by-step explanation:

the x- axis is a horizontal line. A line perpendicular to it will be a vertical line, parallel to the y- axis with equation

x = c ( c is the value of the x- coordinates the line passes through )

the only equation fitting this description from the list is

x = 3

f'(x) = 0 * x^(-2) + (-2/x^3) = -2/x^3.

So, the derivative of f(x) = 2/x^2 is f'(x) = -2/x^3.

To find f'(x) for the function f(x) = x + 2, we can use the power rule for derivatives.

The power rule states that if we have a function of the form f(x) = x^n, then the derivative is given by f'(x) = n*x^(n-1).

In this case, the function f(x) = x + 2 can be written as f(x) = x^1 + 2.

Applying the power rule, we differentiate each term separately:

f'(x) = d/dx (x^1) + d/dx (2)

The derivative of x^1 is 1x^(1-1) = 1x^0 = 1.

The derivative of a constant term like 2 is 0, as the derivative of a constant is always 0.

Therefore, f'(x) = 1 + 0 = 1.

So, the derivative of f(x) = x + 2 is f'(x) = 1.

To find f'(x) for the function f(x) = 2/x^2, we can use the power rule and the constant multiple rule.

The power rule states that if we have a function of the form f(x) = x^n, then the derivative is given by f'(x) = n*x^(n-1).

The constant multiple rule states that if we have a function of the form f(x) = cg(x), where c is a constant, then the derivative is given by f'(x) = cg'(x), where g'(x) is the derivative of g(x).

In this case, the function f(x) = 2/x^2 can be written as f(x) = 2 * x^(-2).

Applying the power rule and the constant multiple rule, we differentiate each term separately:

f'(x) = d/dx (2 * x^(-2))

Applying the constant multiple rule, the derivative of 2 is 0, as it is a constant term.

Applying the power rule, the derivative of x^(-2) is (-2) * x^(-2-1) = (-2) * x^(-3) = -2/x^3.

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Answer:

  (c)  31 units

Step-by-step explanation:

Given quadrilateral ABCD has AD║BC, with AD=3x+7 and BC=5x-9, you want to know the length of AD for the quadrilateral to be a parallelogram.

Opposite sides of a parallelogram are congruent, so for ABCD to be a parallelogram, we must have ...

  BC = AD

  5x -9 = 3x +7

  2x = 16

  x = 8

  AD = 3x +7 = 3(8) +7 = 31

The length of AD must be 31 units if ABCD is to be a parallelogram.

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Step-by-step explanation:

To convert all pulse rates of women to z-scores, we use the formula:

z = (x - μ) / σ

where x is the pulse rate, μ is the mean, and σ is the standard deviation.

Substituting the given values, we get:

z = (x - 77.5) / 11.6

Therefore, after converting all pulse rates of women to z-scores, the mean will be 0 and the standard deviation will be 1.

The probability that a randomly selected point within the square falls in the red-shaded square is 0.36

From the question, we have the following parameters that can be used in our computation:

The areas of the above shapes are

Red square = 3² = 9

White square = 5² = 25

The probability is then calculated as

P = Red square/White square

So, we have

P = 9/25

Evaluate

P = 0.36

Hence, the probability that a randomly selected point within the square falls in the red-shaded square is 0.36

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The height of the stump is 3 ft.

The given equation represents the height of the frog, h(t), as a function of time, t. To find the height of the stump, we need to determine the height when the time, t, is equal to 0.

In the equation h(t) = -16t² + 64t + 3, we substitute t = 0:

h(0) = -16(0)² + 64(0) + 3

Since any term multiplied by zero is zero, we can simplify further:

h(0) = 0 + 0 + 3

Therefore, the height of the stump, at time t = 0, is 3 ft. This means that when the frog initially leaps from the stump, the height of the stump itself is 3 ft.

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The value of tan 0 is -15/8.

The tangent is a periodic function that is defined by a unit circle. It is the ratio of the opposite side to the adjacent side of a right-angled triangle that contains the angle in question as one of its acute angles. The value of the tangent function can be positive, negative, or zero, depending on the quadrant in which the angle is located.

The given values are

cos0 = 8/17

sin0 = -15/17

We can use the trigonometric identity of tan = sin/cos to find the value of tan 0.

Substituting the given values, we get

tan 0 = sin 0 / cos 0

= (-15/17) / (8/17)

=-15/8

Therefore, the value of tan 0 is -15/8.

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Answer:

18

Step-by-step explanation:

Look at the values

2   184

7    94

Difference in volume: 184 liters - 94 liters = 90 liters

Difference in time: 7 minutes - 2 minutes = 5 minutes

rate in liters/minute: (90 liters) / (5 minutes) = 18 liters/minute

a) The solution to the function is t²(du/dt) + 2tu - u² = 0 and can be expressed in the form F(t, u, du/dt) = 0 which confirms it is homogeneous

b) The implicit solution is t/y + 1/t = 2ln|t| + C₁ and the explicit solution is y = (t + 1)/(2ln|t| + C₁)

a) To show that the given differential equation is homogeneous, we need to verify that it can be written in the form:

F(x, y, y') = 0

where F is a homogeneous function of degree zero.

Given:

(t² + 2ty)y' = y²

Let's rearrange the equation:

(t² + 2ty)y' - y² = 0

Now, let u = y/t. We can rewrite y' in terms of u:

y' = du/dt

Substituting these values into the equation, we get:

(t² + 2ty)(du/dt) - (y/t)² = 0

Expanding the equation:

t²(du/dt) + 2ty(du/dt) - (y²/t²) = 0

Now, let's substitute u = y/t into the equation:

t²(du/dt) + 2tu - u² = 0

We can see that this equation is of the form F(t, u, du/dt) = 0, which is homogeneous. Therefore, the given differential equation is homogeneous.

To find the degree of the equation, we need to determine the power of t in each term. Looking at the equation:

t²(du/dt) + 2tu - u² = 0

The highest power of t is 2, which means the degree of the equation is 2.

b) To find the implicit and explicit solutions to the initial value problem (IVP), we need to solve the homogeneous differential equation and apply the initial condition y(1) = 1.

Let's solve the homogeneous equation:

t²(du/dt) + 2tu - u² = 0

We can rewrite it as:

du/u² - dt/t² = -2dt/t

Integrating both sides:

∫(du/u²) - ∫(dt/t²) = -2∫(dt/t)

This simplifies to:

-1/u - (-1/t) = -2ln|t| + C₁

1/u + 1/t = 2ln|t| + C₁

Since u = y/t, we substitute u back:

1/(y/t) + 1/t = 2ln|t| + C₁

t/y + 1/t = 2ln|t| + C₁

This is the implicit solution to the given initial value problem.

To find the explicit solution, we need to solve for y in terms of t. Let's rearrange the equation:

t/y = 2ln|t| + C₁ - 1/t

Multiply both sides by y:

t = y(2ln|t| + C₁ - 1/t)

Now, let's simplify:

t = 2yln|t| + C₁y - 1

Rearranging the equation:

2yln|t| + C₁y = t + 1

Factoring out y:

y(2ln|t| + C₁) = t + 1

Dividing both sides by (2ln|t| + C₁), assuming C₁ ≠ -2ln|t|, we get:

y = (t + 1)/(2ln|t| + C₁)

This is the explicit solution to the given initial value problem.

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a) Using the standard divisors according to Jefferson's apportionment method, the number of salespeople assigned to work each shift is as follows:

Shift                              Morning     Midday     Afternoon     Evening   Total

Number of

salespeople assigned  1               3                  5                 7                16

b) The modified or adjusted divisor used represents the average number of customers for each shift divided by the standard divisor.

The standard divisor shows the ratio of the total population to the number of seats.

The standard divisor gives an insight about the number of people each seat represents.

Standard divisor = Total number of customers / Total number of salespeople

The total number of customers = 1,425

The number of salespeople = 16

a) Standard divisor = 1,425 ÷ 16

= 89.0625

Shift                              Morning     Midday     Afternoon     Evening   Total

Average number of

customers                     125            280             460             560       1,425

Standard divisor      89.0625     89.0625      89.0625      89.0625

b) Modified divisor     1.4035        3.1438          5.165           6.288

Number of

salespeople assigned  1               3                  5                 7                16

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Answer:

60°

Step-by-step explanation:

The sum of interior angles in a triangle is equal to 180°.

48° + 11x - 5 + 9x - 3 = 180°

40° + 20x = 180°

20x = 140°

x = 7

To find m∠A, replace x with 7:

m∠A = 9x - 3

9×7-3 = 60°

By finding the point where the graphs intercept, we will see that the solution of the system of equations is the point (0, 2).

The solutions of a system of equations are the points where the graphs of the different equations intercept when graphed.

Below you can see the graph of the system:

There you can see that we have only one intersection point at x = 0, y = 2, then we can conclude that our system has only one solution, and the solution is the point (0, 2).

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Answer:

136 [tex]ft^{2}[/tex]

Step-by-step explanation:

Find the area of the wall and subtract out the area of the window

16 x 9 = 144

4 x 2 = 8

144 - 8 = 136

Helping in the name of Jesus.

Answer:

1711

Step-by-step explanation:

[tex]4\cdot(14+8)^2-9\cdot(9-4)^2\\=4\cdot(22)^2-9\cdot(5)^2\\=4(484)-9(25)\\=1936-225\\=1711[/tex]

Make sure to follow order of operations!

Answer:

x = 90

Step-by-step explanation:

The given diagram shows a circle with intersecting chords, KM and JL.

To find the value of x, we can use the Angles of Intersecting Chords Theorem.

According to the Angles of Intersecting Chords Theorem, if two chords intersect within a circle, the angle formed at the intersection point is equal to half the sum of the measures of the arcs intercepted by the angle and its corresponding vertical angle.

Let the point of intersection of chords KM and JL be point P.

As the chords are straight lines, angle x° forms a linear pair with angle JPM.

Note: We cannot use the Angles of Intersecting Chords Theorem to find the value of x directly, since we have not been given the measures of the arcs KJ and ML. Therefore, we need to use the theorem to find m∠JPM first.

From inspection of the given diagram:

Using the Angles of Intersecting Chords Theorem, we can calculate the measure of angle JPM (shown in orange on the attached diagram):

[tex]\begin{aligned}m \angle JPM &=\dfrac{1}{2}\left(m\overset\frown{JM}+m\overset\frown{LK}\right)\\\\&=\dfrac{1}{2}\left(30^{\circ}+(2x-30)^{\circ}\right)\\\\&=\dfrac{1}{2}\left(30^{\circ}+2x^{\circ}-30^{\circ}\right)\\\\&=\dfrac{1}{2}\left(2x^{\circ}\right)\\\\&=x^{\circ}\end{aligned}[/tex]

As angle JPM forms a linear pair with angle x°, the sum of the two angles equals 180°:

[tex]\begin{aligned}m \angle JPM+x^{\circ}&=180^{\circ}\\\\x^{\circ}+x^{\circ}&=180^{\circ}\\\\2x^{\circ}&=180^{\circ}\\\\\dfrac{2x^{\circ}}{2}&=\dfrac{180^{\circ}}{2}\\\\x^{\circ}&=90^{\circ}\\\\x&=90\end{aligned}[/tex]

Therefore, the value of x is 90, which means that the two chords intersect at right angles.

The sum of the geometric sequence in this problem is given as follows:

5.77.

A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number, which is called the common ratio q.

The first term, the common ratio and the number of terms for this problem are given as follows:

[tex]a_1 = 10, q = -\frac{2}{3}, k = 8[/tex]

The formula for the sum of the first n terms is given as follows:

[tex]S_n = a_1\frac{1 - r^n}{1 - r}[/tex]

Hence the sum for this problem is given as follows:

[tex]S_8 = 10 \times \frac{1 - \left(-\frac{2}{3}\right)^8}{1 + \frac{2}{3}}[/tex]

[tex]S_8 = 5.77[/tex]

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Answer:

(2n - 3) + 5 is a correct expression.

Answer:

Step-by-step explanation:

[tex]\lim_{x\to 1} \frac{x^3-1}{x-1} \\= \lim_{x \to 1} \frac{(x-1)(x^2+x+1)}{(x-1)} \\= \lim_{x \to 1} (x^2+x+1)\\=(1)^1+1+1\\=1+1+1\\=3[/tex]

Tan (θ) = 13/9

θ = 55.305°

Answer:

A.

Step-by-step explanation:

Let's think of this as a clock.  We can see that the 2 lines start in the same place, around 3 o'clock.  Next, one of the line segments shifts down to around 6 o'clock.  Next, it shifts to about 9 o'clock.  Logically, the next step (in a clock) would be 12 o'clock, making A the correct choice.

We can also just use a regular circle, with one of the line segments moving 90 degrees each time.

Hope this helps! :)

Answer:

9, 9.5, 9.75, 9.875

Step-by-step explanation:

Notice the following pattern:

[tex]-22+16=-22+2^4=-6\\-6+8=-6+2^3=2\\2+4=2+2^2=6\\6+2=6+2^1=8[/tex]

Therefore, the next four terms will be:

[tex]8+2^0=8+1=9\\9+2^{-1}=9+0.5=9.5\\9.5+2^{-2}=9.5+0.25=9.75\\9.75+2^{-3}=9.75+0.125=9.875[/tex]

The required answer is: 9.61

Rounding the answer to two decimal places gives the best estimate for the mean percentage of underweight children as 9.61%.

To find the best estimate for the mean percentage of underweight children,

we need to calculate the midpoint of each interval and multiply it by the number of countries in that interval, then add the products and divide by the total number of countries.Below is the table that shows the midpoint of each interval and the corresponding calculations.

MidpointNumber of CountriesProducts16.73 (16 + 21.45) / 22383.29 (21.45 + 26.9) / 33253.13 (26.9 + 32.35) / 97194.08 (32.35 + 37.8) / 61411.025 (37.8 + 43.25) / 6 246.3 (43.25 + 48.7) / 2.

Total471Calculating the average of the above data giv

es:\[\frac{471}{23 + 3 + 9 + 6 + 6 + 2} = 471 / 49 = 9.6122\].

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A single 180-degree rotation about the origin is equivalent to the sequence of reflections mentioned, as it accomplishes the same changes to the shape's orientation and coordinates.

Yes, there is a single transformation that is equivalent to reflecting AABC across the y=x line and then reflecting the image across the y-axis. This single transformation is known as a 180-degree rotation about the origin.

When you reflect AABC across the y=x line, each point is transformed to its corresponding point on the opposite side of the line. This reflection effectively swaps the x-coordinates with the y-coordinates for each point, resulting in a new shape.

Now, when you reflect the newly formed image across the y-axis, you essentially negate the x-coordinates of each point. This is equivalent to rotating the shape 180 degrees about the origin.

A 180-degree rotation about the origin involves flipping the shape by exchanging the x and y coordinates and taking their negatives. This transformation achieves the same result as reflecting across the y=x line and then reflecting across the y-axis.

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