Determine the domain and range for the relation I

The number of ways to choose 6 letters from a set of 13 is given as follows:

a) Order does not matter: 1716.

b) Order matters: 1,235,520.

A combination is used when the order does not matter, hence, for item a, the number of ways is obtained as follows:

C(13,6) = 13!/(6!7!) = 1716.

A permutation is used when the order matters, hence, for item b, the number of ways is obtained as follows:

P(13,6) = 13!/7! = 1,235,520.

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The given statement is false. Because If the r-value, or correlation coefficient, of a data set, is negative, the coefficient of determination is positive.

Two variables with a negative correlation tend to move in opposing directions.

While a correlation value of -0.3 or below shows a very weak association, one of -0.8 or lower suggests a significant negative relationship.

The complete question is;

"The given statement is true or false

If the r-value, or correlation coefficient, of a data set, is negative, the coefficient of determination is negative."

Because If a data set's r-value, or correlation coefficient, is negative, the coefficient of determination is positive.

Hence, the given assertion is incorrect.

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The graph of the feasible region is attached

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

\(\left\{ \begin{array}{lr} y + 7x \ge 10 \\ 8y + 2x \ge 20 \\ y + x \ge 4 \\ y + x\le 10 \\ x \ge 0 \\ y \ge 0\end{array}\)

To plot the graph of the feasible region, we plot each inequality in the domain x ≥ 0 and y ≥ 0

Using the above as a guide, the graph is attached

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Answer:

10.7

Step-by-step explanation:

Since this is a right triangle, we can use trig functions to calculate the hypotenuse.

We know the opposite side from the angle labeled 59 degrees.

sin 59 = opp side / hypotenuse

sin 59 = 9.2/x

x = 9.2/ sin 59

x = 10.733

Answer:

Step-by-step explanation:

a). By applying Pythagoras theorem in right triangle ABC,

AC² = AB² + BC²

20² = 16² + BC²

BC = \(\sqrt{400-256}\)

     = 12

Therefore, another leg of the triangular base = 12 cm

Area of the triangular base ABC = \(\frac{1}{2}(\text{Base})(\text{Height})\)

                                                      = \(\frac{1}{2}(16)(12)\)

                                                      = 96 cm²

Volume of a triangular prism = Area of the triangular base × height

                                       960 = 96 × b

                                          b = 10 cm

b). Area of the triangle ABC = 96 cm²

Answer:

f(x) = x³ - 4x + 6

f'(x) = 3x² - 4

a) f'(2) = 3(2²) - 4 = 12 - 4 = 8

6 = 8(2) + b

6 = 16 + b

b = -10

y = 8x - 10

b) 3x² - 4 = 0

3x² = 4, so x = ±2/√3 = ±(2/3)√3

= ±1.1547

f(-(2/3)√3) = 9.0792

f((2/3)√3) = 2.9208

c) 3x² - 4 = 8

3x² = 12

x² = 4, so x = ±2

f(-2) = (-2)³ - 4(-2) + 6 = -8 + 8 + 6 = 6

6 = -2(8) + b

6 = -16 + b

b = 22

y = 8x + 22

f(2) = 6

y = 8x - 10

The equation perpendicular to the tangent is y = -1/8x + 25/4

-The smallest slope on the curve is 2.92

The curve has the smallest slope at the point (1.15, 2.92)

The equations at tangent points are y = 8x + 16 and  y = 8x - 16

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

y = x³ - 4x + 6

Differentiate

So, we have

f'(x) = 3x² - 4

The point is (2, 6)

So, we have

f'(2) = 3(2)² - 4

f'(2) = 8

The slope of the perpendicular line is

Slope = -1/8

So, we have

y = -1/8(x - 2) + 6

y = -1/8x + 25/4

We have

f'(x) = 3x² - 4

Set to 0

3x² - 4 = 0

Solve for x

x = √[4/3]

x = 1.15

So, we have

Smallest slope = (√[4/3])³ - 4(√[4/3]) + 6

Smallest slope = 2.92

So, the smallest slope is 2.92 at (1.15, 2.92)

Here, we set f'(x) to 8

3x² - 4 = 8

Solve for x

x = ±2

Calculate y at x = ±2

y = (-2)³ - 4(-2) + 6 = 6: (-2, 0)

y = (2)³ - 4(2) + 6 = 6: (2, 0)

The equations at these points are

y = 8x + 16

y = 8x - 16

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Answer:

The answers are=

The x-coordinate of the vertex is (38, 0) and the y-coordinate of the vertex is (19, 1768.9).

It is defined as the graph of a quadratic function that has something bowl-shaped.

(x - h)² = 4a(y - k)

(h, k) is the vertex of the parabola:

a = √[(c-h)² + (d-k²]

(c, d) is the focus of the parabola:

It is given that:

The graph of the parabolic path is shown in the picture.

From the graph:

The x-coordinate of the vertex is (38, 0)

The x-coordinate represents time in seconds

The y-coordinate of the vertex is (19, 1768.9)

The y-coordinate represents the height in meters

Thus, the x-coordinate of the vertex is (38, 0) and the y-coordinate of the vertex is (19, 1768.9).

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Option(A) is the correct answer is A. 1,335 birds.

To calculate the number of birds that will be left after 20 years, we need to consider the annual decline rate of 2%.

We can use the formula for exponential decay:

N = N₀ * (1 - r/100)^t

Where:

N is the final number of birds after t years

N₀ is the initial number of birds (2,000 in this case)

r is the annual decline rate (2% or 0.02)

t is the number of years (20 in this case)

Plugging in the values, we get:

N = 2,000 * (1 - 0.02)^20

N = 2,000 * (0.98)^20

N ≈ 2,000 * 0.672749

N ≈ 1,345.498

Rounded to the nearest whole number, the number of birds that will be left after 20 years is 1,345.

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Answer:

10*0=1, according to the law of indices which states that anything to the zero power is equal to one

1*4=1×1×1×1=1

1×1.0=1.0=1

From the given parallelogram:

a) x equals 10

b) The length of side DA is approximately 51.64.

a) We first solve for x, using the fact that opposite sides of a parallelogram are equal in length. Thus:

AB = CD, which gives us:

5x - 18 = 2x + 12

3x = 30

x = 10

b) To solve for DA, we need to use the formula for the area of a parallelogram:

Area = base × height

Since opposite sides of a parallelogram are parallel, its height is the perpendicular distance between AB and CD.

Let E be the point on CD such that AE is perpendicular to CD. Then, we have:

AE = CD sin(∠C)

AE = (2x + 12) sin(∠C)

Also, we have:

BE = AB sin(∠C)

BE = (5x - 18) sin(∠C)

Since AE = BE, we have:

(2x + 12) sin(∠C) = (5x - 18) sin(∠C)

2x + 12 = 5x - 18

3x = 30

x = 30/3

x = 10

Therefore, AE = BE = 2x + 12 =  (2*10) + 12 = 32

The base of the parallelogram is AB = 5x - 18 = 5*10 - 18 = 32

So, the area of the parallelogram is:

Area = base × height

Area = AB × AE

Area = 32 × 32

Area = 1024

Finally, we can use the formula for the length of a diagonal of a parallelogram:

DA² = AB² + AD² - 2AB × AD cos(∠A)

The opposite angles of a parallelogram are equal, so, we have:

∠A = ∠C

cos(∠A) = cos(∠C)

cos(∠A) = (2x + 12) / AE

cos(∠A) = (2x + 12) / 32

Substituting x = 10, we get:

cos(∠A) = 11/16

Thus, we get:

DA² = (5x - 18)² + (2x + 12)² - 2(5x - 18)(2x + 12) cos(∠A)

DA² = (32)² + (52)² - 2(32)(52)(11/16)

DA² ≈ 2668.06

DA = √(2668.06)

Therefore, the length of DA is ≈ 51.64

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4.65 kg the mass of each item (assuming the bag's mass is negligible). (Sand density is 3.00 g/cm3, while the density of gold is 19.3 g/cm3)

Given that,

A gold statue that is perched on a pedestal with a weight-sensitive alarm is replaced by a thief using a bag of sand. The statue and the bag of sand have the exact same volume, 1.55 L.

We have to calculate the mass of each item (assuming the bag's mass is negligible). (Sand density is 3.00 g/cm3, while the density of gold is 19.3 g/cm3)

We know that,

Mass of gold = Volume × Density

= 1.55×1000 cc × 19.3

= 29915 grams

= 29.915 Kg

Mass of sand = 1.55×1000×3

= 4650g

= 4.65 Kg

Volume = 1.55 L

= 1.55 × 1000 cc

= 1550 cc

Mass = Volume × Density

Mass of gold = 1550 × 19.3

= 29915 grams

= 29.915 kg

Mass of sand = 1550 × 3

= 4650 grams

= 4.65 kg

Therefore, 4.65 kg the mass of each item (assuming the bag's mass is negligible). (Sand density is 3.00 g/cm3, while the density of gold is 19.3 g/cm3)

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The property you are referring to is called the associative property of multiplication. According to this property, when multiplying three numbers, the grouping of the numbers does not affect the result. In other words, you can change the grouping of the factors without changing the product.

In the equation you provided: 3x(5x7) = (3x5)7

The associative property allows us to group the factors in different ways without changing the result. So, whether we multiply 5 and 7 first, or multiply 3 and 5 first, the final product will be the same.

Answer:

12

Step-by-step explanation:

Take the equation where the interval < or = 4

g(4)= 0.5(4)+10

=2+10

=12

When x is equal to 1, the Function f(x) = -4x - 5 yields a value of -9.

The find ƒ(1) for the function f(x) = -4x - 5, we need to substitute x = 1 into the function and evaluate the expression.

Replacing x with 1, we have:

ƒ(1) = -4(1) - 5

Simplifying further:

ƒ(1) = -4 - 5

ƒ(1) = -9

Therefore, when x is equal to 1, the value of the function f(x) = -4x - 5 is ƒ(1) = -9.

Let's break down the steps taken to arrive at the solution:

1. Start with the function f(x) = -4x - 5.

2. Replace x with 1 in the function.

3. Evaluate the expression by performing the necessary operations.

4. Simplify the expression to obtain the final result.

In this case, substituting x = 1 into the function f(x) = -4x - 5 gives us ƒ(1) = -9 as the output.

It is essential to note that the notation ƒ(1) represents the value of the function ƒ(x) when x is equal to 1. It signifies evaluating the function at a specific input value, which, in this case, is 1.

Thus, when x is equal to 1, the function f(x) = -4x - 5 yields a value of -9.

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Answer:

Step-by-step explanation:

The thousand mark is the 4th number when going from right to left. So it would be the {6},976. When it comes to rounding, you go "5 and above, give it a shove, 4 and below, let it go. 6,976 rounded to the nearest thousand is 7,000,  3,983 rounded to the nearest thousand is 4,000,  13,560 rounded is 14,000.

7,000 + 4,000+ 14,000 = 25,000

There are 7 pencils measuring more than 5 1/2 inches but less than 7 2/4 inches.

An inch is a unit of length in the imperial and US customary systems of measurement. Equal to 1/12 of a foot, it is usually denoted by the symbol "in". It is also equal to 2.54 centimeters. The inch is commonly used in the construction industry, engineering, and other fields.

To answer this question, we must first convert all of the measurements into the same unit of length. We will convert 5 1/2 inches and 7 2/4 inches into decimals by dividing the fractions by their denominators. 5 1/2 inches is equal to 5.5 inches and 7 2/4 inches is equal to 7.25 inches.

Now, we can calculate how many pencils are measuring more than 5.5 inches but less than 7.25 inches. To do this, we will subtract the smaller number (5.5) from the larger number (7.25).

7.25 - 5.5 = 1.75

This means that there are 1.75 inches between the two measurements. Since each pencil takes up about 0.25 inches, we can divide 1.75 by 0.25 to determine how many pencils would fit in that space.

1.75 / 0.25 = 7

Therefore, there are 7 pencils measuring more than 5 1/2 inches but less than 7 2/4 inches.

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Complete questions as follows-

how many pencils measure more than 5 1/2 inches but less than 7 2/4 inches?

The logarithmic function f(x) = a·log₃(x - 4), passing through the points (13, 7), has the values;

5 a. The value of a is 3.5

b. Please find attached the graph of the function, f(x) = 3.5·log₃(x - 4), created with MS Excel

A logarithmic function is a function that contain and involves logarithm operation and it is the inverse of an exponential  function

The function is f(x) = a·log₃(x - 4),

x > 4 and a > 0

The coordinates of a point on the graph of the function, f is A(13, 7)

5 a. The value of a can be found by plugging in the value of (13, 7) = (x, f(x)), as follows

f(13) = 7 = a·log₃(13 - 4) = a·log₃9 = a·log₃3²

7 = a·log₃3²

7 = 2·a·log₃3 = 2·a·1 = 2·a

2·a = 7

a = 7 ÷ 2 = 3.5

a = 3.5

5 b. The coordinates of the x-intercept of the graph = (5, 0)

The equation of the function is;

f(x) = 3.5·log₃(x - 4)

A third point on the graph is given when f(x) = 14 as follows;

f(x) = 14 = 3.5·log₃(x - 4)

log₃(x - 4) = 14 ÷ 3.5 = 4

3⁴ = x - 4

x = 3⁴ + 4 = 85

Which gives the point, (85, 14)

Similarly, we have the point (31, 10.5), (7, 3.5)

Please find attached the graph of f(x) created with MS Excel

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The inequality on the graph is the third option:

(2/5)*x - 3/2 ≥ y

Let's analyse the graph if the inequality.

We can see that there is a solid linear equation with a positive slope, and the shaded area is below that line, then the inequality is of the form:

y ≤ linear equation.

We know that the symbol "≤" must be used because of the solid line.

We also can see that when x = 0, y takes a velue between -1 and -2.

With that in mind the correct option is the third one:

(4/5)*x - 2y ≥ 3

Isolating y we get:

(4/5)*x - 3 ≥ 2y

(2/5)*x - 3/2 ≥ y

Changing the order:

y ≤ (2/5)*x - 3/2

That is the graphed inequality.

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Answer: $179.70

Step-by-step explanation:

The simple interest will be:

= (Principal × Rate × Time) / 100

= ($6000 × 5.99% × 5) / 10

= ($6000 × 0.0599 × 5) / 10

= $1797 / 10

= $179.70

Therefore, the total interest on the loan is $179.70

Answer:

96

Step-by-step explanation:

First divide 360 by 2.

Then divide 180 by 5 and that is how many 5 pound bags he will use.

Then divide 180 again by 3 and that is how many 3 pound bags he will use.

Answer:

\(\huge\boxed{\sf Opposite = 125.86 \ ft}\)

Step-by-step explanation:

This question will be solved using trigonometric ratios.

Angle = θ = 40°

Adjacent = 150 ft

Opposite = ?

Using trigonometric ratio, tan θ.

\(\displaystyle tan \theta = \frac{opposite}{adjacent} \\\\Put \ the \ given.\\\\tan \ 40 = \frac{opposite}{150} \\\\0.839 = \frac{opposite}{150} \\\\Multiply \ both \ sides\ by \ 150\\\\0.839 \times 150 = opposite\\\\125.86 \ ft = Opposite\\\\Opposite = 125.86 \ ft\\\\\rule[225]{225}{2}\)

Answer:

C) -2/9

Step-by-step explanation:

\(\displaystyle -\frac{1}{6}\biggr[3-15\biggr(\frac{1}{3}\biggr)^2\biggr]\\\\=-\frac{1}{6}\biggr[3-15\biggr(\frac{1}{9}\biggr)\biggr]\\\\=-\frac{1}{6}\biggr[3-\frac{15}{9}\biggr]\\\\=-\frac{1}{6}\biggr[\frac{27}{9}-\frac{15}{9}\biggr]\\\\=-\frac{1}{6}\biggr[\frac{12}{9}\biggr]\\\\=-\frac{1}{6}\biggr[\frac{4}{3}\biggr]\\\\=-\frac{4}{18}\\\\=-\frac{2}{9}\)

Answer:

Hence, Option (C) - 2/9 is the Answer:

Step-by-step explanation:

-1/6 [3 -15(1/3)^2]

-1/6(3 -15)(1/9))

-1/6(3 - 5/3)

-1/6 (4/3)

Hence, Option (C) - 2/9 is the Answer:

I hope it helps!

During the exponential phase, e.coli bacteria in a culture increase in number at a rate proportional to the current population. If growth rate is 1.9% per minute and the current population is 172.0 million, the population 7.2 minutes from now can be calculated using the following formula:

P(t) = P ₀e^(rt)where ,P₀ = initial population r = growth rate (as a decimal) andt = time (in minutes)Substituting the given values, P₀ = 172.0 million r = 1.9% per minute = 0.019 per minute (as a decimal)t = 7.2 minutes

The population after 7.2 minutes will be:P(7.2) = 172.0 million * e^(0.019*7.2)≈ 234.0 million (rounded to the nearest tenth)Therefore, the population of e.coli bacteria 7.2 minutes from now will be approximately 234.0 million.

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The equation 2sin(x)tan(x) + 3 = 0 has been proved to equal 2cos²(x) - 3cos(x) - 2 = 0

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

2sinxtanx+3=0

Rewrite as

2sin(x)tan(x) + 3 = 0

Express tan(x) as tan(x) = sin(x)/cos(x)

So, we have

2sin²(x)/cos(x) + 3 = 0

Multiply through by cos(x)

2sin²(x) + 3cos(x) = 0

Express sin²(x) as 1 - cos²(x)

So, we have

2(1 - cos²(x)) + 3cos(x) = 0

Expand

2 - 2cos²(x) + 3cos(x) = 0

Multiply through by -1

-2 + 2cos²(x) - 3cos(x) = 0

Rearrange the terms

2cos²(x) - 3cos(x) - 2 = 0

Hence, the equation has been proved

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Answer:

141

Step-by-step explanation:

calculator

Answer:

141

Step-by-step explanation:

22+125=147

147-6=141

therefore

the answer is 141

Perpendicular lines intersect at 90° - definition of perpendicular.

Here is the two column proof:

     Statement                                        Reason                                              

Answer:

∠ GJK = 102°

Step-by-step explanation:

(2x + 92) and (3x + 63) are same- side interior angles and sum to 180° , so

2x + 92 + 3x + 63 = 180

5x + 155 = 180 ( subtract 155 from both sides )

5x = 25 ( divide both sides by 5 )

x = 5

Then

∠ GJK = 2x + 92 = 2(5) + 92 = 10 + 92 = 102°

Answer:

√20 miles

Step-by-step explanation:

a² + b² = c²

                                           |

                     ?                    |   2 miles ↓

                                           |

          ------------------------------

                     ← 4 miles  

2² + 4² = c²

4 + 16 = c²

20 = c²

√20 = c

I hope this helps!

The length of the straight path they must run to get back to the starting line is 4.47 miles.

Displacement [x] of an object is the length of the straight line joining the initial and final position of the object. Mathematically, displacement can be written as-

Δx = x[2] - x[1]

Now -

Δx = vΔt              {[v] is velocity}

So we can write -

vΔt = x[2] - x[1]

Given are the runners at a cross-country meet such that they ran 2 miles to south and then 4 miles to west from the starting line.

We can find the straight path they must run to get back to the starting line using the Pythagoras theorem. The base will be equal to distance ran towards south and height will be equal to the distance ran towards west. The length of the hypotenuse will be equal to that of displacement or the straight path they must run to get back to the starting line. Using Pythagoras theorem -

[h]² = [b]² + [p]²

[h]² = (2 x 2) + (4 x 4)

[h]² = 4 + 16

[h]² = 20

[h] = 4.47 miles

Therefore, the length of the straight path they must run to get back to the starting line is 4.47 miles.

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