Exercise 21 1. A girl moves from a point P on a bearing of 060° to a point Q, 40m away. She then moves from the point Q, on a bearing of 120° to a point R. The bearing of P from R is 255, Calculate, correct to three significant figures, the distance between P and R.​

The first table with the following values of x and y, shows a linear function.

x = -4, y = 8

x = -1, y = 2

x = 1, y = 2

x = 2, y = 4

x = 3, y = 6

Definition of a Linear Function:

A polynomial function of degree zero or one that has a straight line as its graph is referred to as a linear function in calculus and related fields.

For the above chosen option, the value of y corresponding to a value of x is twice that of its x value. This linear function can be represented as follows,

y = | 2x | .......... (1)

In this linear function, mod represents, that y is always maintained positive. Besides, the value of y is always twice that of the x. Hence, the linear function can also be written as, y ± 2x = 0.

Since, equation (1) is of the form, y = mx + c, it produces a straight line. Here, m = 2 and c = 0. Thus, it is a linear function.

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Triangle A"B"C" are similar triangles to triangle ABC and all corresponding angles are congruent. Also, triangle A"B"C" is twice the size of triangle ABC.

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.

Triangle ABC is translated 3 units to the left and downward 10 units to form triangle A'B'C', then dilated by a factor of 2 to form triangle A''B''C''.

Hence:

Triangle A"B"C" are similar triangles to triangle ABC and all corresponding angles are congruent. Also, triangle A"B"C" is twice the size of triangle ABC.

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The equation of the exponential function is [tex]y = 3^{-x -5} - 3[/tex]

An exponential equation is represented as:

y = b^x

Where b represents the base.

The base is 3.

So, we have:

[tex]y = 3^x[/tex]

When reflected over the y-axis, we have:

[tex]y = 3^{-x[/tex]

When shifted right by 5 units, we have:

[tex]y = 3^{-x -5[/tex]

Lastly, the function has an asymptote of y=-3

So, we have:

[tex]y = 3^{-x -5} - 3[/tex]

Hence, the equation of the exponential function is [tex]y = 3^{-x -5} - 3[/tex]

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Add both

Put in first one

Find

Considering the domain of the function, it is restricted for:

B. [tex]x \neq -3, x \neq -2, x \neq 0[/tex].

The domain of a function is the set that contains all possible input values for the function.

A fraction cannot have a denominator of zero, hence:

We solve these two inequalities to find the restrictions, hence:

[tex]4x^2 - 12x \neq 0[/tex]

[tex]4x(x - 3) \neq 0[/tex]

[tex]4x \neq 0 \rightarrow x \neq 0[/tex]

[tex]x - 3 \neq 0 \rightarrow x \neq 3[/tex].

[tex]-x^2 + 5x - 6 \neq 0[/tex].

[tex]x^2 - 5x + 6 \neq 0[/tex]

[tex](x - 3)(x - 2) \neq 0[/tex]

[tex]x - 2 \neq 0 \rightarrow x \neq 2[/tex].

[tex]x - 3 \neq 0 \rightarrow x \neq 3[/tex].

Hence the correct option is:

B. [tex]x \neq -3, x \neq -2, x \neq 0[/tex].

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Four CDs must be bought to receive a gift card from an online store.

The total costs are equal to the sum of the shipping cost and the total related to the number of acquired CDs (n). The number of gift cards (m) is equal to the total costs divided by minimum spent money, that is, $ 50. We need to solve the following inequation to find the minimum quantity of CDs:

(12 · n + 5)/50 > 1

12 · n + 5 > 50

12 · n > 45

n > 45/12

n > 3.75

Four CDs must be bought to receive a gift card from an online store.

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Answer:

No

Step-by-step explanation:

This table does not represent a function. The domain(x) can have only one range(y) in order for a set of data to represent a function. However, the domain value 2 has the range of both 1 and 4. Therefore, the table does not represent a function.

So to run 10km, you need to run 6.25 times the trail. (Or 7 if you only accept whole numbers as answers, we need to round up).

If you run the trail x times, then you will run:

y = x*1mi

Now remember that:

1 mi = 1.6 km

Replacing that, we get the linear equation:

y = x*1.6km

Now we want to find the value of x such that:

x*1.6km = 10km

Dividing both sides by 1.6km

x = 10km/1.6km = 6.25

So to run 10km, you need to run 6.25 times the trail. (Or 7 if you only accept whole numbers as answers, we need to round up).

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Answer:

Answer is..... D

Step-by-step explanation:

every single flipping may two results, head or tail.So last one has equal chance.

The figure is represented by the inequality y ≥ 5 · x² - 40 · x - 45. (Correct choice: C)

In accordance with the figure, we have an inequation of the form y ≥ f(x). Now we proceed to find the quadratic equation of the parabola:

f(x) = a · (x + 1) · (x - 9)

- 125 = a · (4 + 1) · (4 - 9)

- 125 = a · 5 · (- 5)

- 125 = - 25 · a

a = - 5

f(x) = 5 · (x + 1) · (x - 9)

f(x) = 5 · (x² - 8 · x - 9)

f(x) = 5 · x² - 40 · x - 45

The figure is represented by the inequality y ≥ 5 · x² - 40 · x - 45. (Correct choice: C)

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Step-by-step explanation:

The formula for the angle between vectors

[tex] \alpha = \cos {}^{ - 1} ( \frac{uv}{ |u| |v| } ) [/tex]

To multiply vectors, multiply the first component and multiply the second component and add them.

(-4*-1) + (-3*5)= 4-15=-11.

To find magnitude of vectors, use the Pythagorean theorem

[tex]u = \sqrt{ { - 4}^{2} + { - 3}^{2} } = 5[/tex]

[tex]v = \sqrt{ - 1 {}^{2} + 5 {}^{2} } = \sqrt{26} [/tex]

so

[tex] |u| |v| = 5 \sqrt{26} [/tex]

Know we have,

[tex] \alpha = \cos {}^{ - 1} ( \frac{7}{5 \sqrt{26} } ) [/tex]

[tex] \alpha = 105.94[/tex]

in degrees,

[tex] \alpha = 1.849[/tex]

in radians

Answer:

115.6° (1 d.p.)

Step-by-step explanation:

To find the angle between two vectors:

**Please see attached for the triangle diagram**

Given vectors:

[tex]\textbf{u}=-4\textbf{i}-3\textbf{j}[/tex]

[tex]\textbf{v}=-\textbf{i}+5\textbf{j}[/tex]

Use Pythagoras Theorem to find the magnitude of each vector:

[tex]\implies |\textbf{u}|=\sqrt{(-4)^2+(-3)^2}=5[/tex]

[tex]\implies |\textbf{v}|=\sqrt{(-1)^2+5^2}=\sqrt{26}[/tex]

[tex]\overrightarrow{\text{UV}}=\textbf{v}-\textbf{u}=(-\textbf{i}+5\textbf{j})-(-4\textbf{i}-3\textbf{j})=3\textbf{i}+8\textbf{j}[/tex]

[tex]|\overrightarrow{\text{UV}}|=\sqrt{3^2+8^2}=\sqrt{73}[/tex]

Cosine Rule (for finding angles)

[tex]\sf \cos(C)=\dfrac{a^2+b^2-c^2}{2ab}[/tex]

where:

Find angle θ using the cosine rule:

[tex]\implies \cos(\theta)=\dfrac{|\textbf{u}|^2+|\textbf{v}|^2-|\overrightarrow{\text{UV}}|^2}{2|\textbf{u}||\textbf{v}|}[/tex]

[tex]\implies \cos(\theta)=\dfrac{5^2+\left(\sqrt{26}\right)^2-\left(\sqrt{73}\right)^2}{2(5)\left(\sqrt{26}\right)}[/tex]

[tex]\implies \cos(\theta)=\dfrac{-22}{10\sqrt{26}}[/tex]

[tex]\implies \theta=\cos^{-1}\left(\dfrac{-22}{10\sqrt{26}}\right)[/tex]

[tex]\implies \theta=115.5599652...^{\circ}[/tex]

Therefore, the angle between the vectors is 115.6° (1 d.p.).

Answer:

75/4

Step-by-step explanation:

Assuming BA and ED are parallel, we know thay the two triangles are similar.

So,

[tex]\frac{25-x}{x}=\frac{36}{12} \\ \\ \frac{25-x}{x}=3 \\ \\ 25-x=3x \\ \\ 4x=25 \\ \\ x=\frac{25}{4} \\ \\ BD=25-\frac{25}{4}=\frac{75}{4}[/tex]

Answer:

There are many different answers to this question but here is one that I thought of.

g(x)=x^2

f(x)=x/(x+4)

f(g(x))=F(x): plug in x^2 into each of the x in f(x) and and answer should equal F(x)

Step-by-step explanation:

since the function F(x) had 2 x^2 in it, I thought that g(x) could easily be x^2 that way u can subsitute the x in f(x) later to x^2 and get the right answer.

Step-by-step explanation:

1. if the number of pages in the 1st day is 'x', then the 2d day - 'x+10', the 3d day - 'x+20', the 4th day - 'x+30', the 5th day - 'x+40' and the last day - 'x+50' pages;

2. if the sum of all the pages is 300, then it is possible to make up the equation:

x+x+10+x+20+x+30+x+40+x+50=300;

3. x=25, it means:

1st day - 25;

2d day - 35;

3d day - 45;

4th day - 55;

5th day - 65;

6th day - 75 pages.

Answer:

c

Step-by-step explanation:

Aight, let's hop to it:

So we got

[tex] \frac{5x}{45} = \frac{20}{36} = = > \\ \frac{x}{9} = \frac{5}{9} = = > \\ x = 5[/tex]

and boom

The amount of the radioactive material in the vault after 140 years is 210 pounds

We have that the function is given as a model;

f(x) = 300(0.5)x/100

Where

Let's substitute the value of 'x' in the model

f(x) = 300(0.5)x/100

[tex]f(x) = \frac{300(0.5) * 140}{100}[/tex]

[tex]f(x) =\frac{21000}{100}[/tex]

f(140) = 210 pounds

This mean that the function of 149 years would give an amount of 210 pounds rounded up to the nearest whole number.

Thus, the amount of the radioactive material in the vault after 140 years is 210 pounds

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Using the binomial distribution, we have that:

It is the probability of exactly x successes on n repeated trials, with p probability of a success on each trial.

The expected value of the binomial distribution is:

[tex]\mu = np[/tex]

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

For this problem, the parameters are:

n = 20, p = 0.3.

Hence:

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The speed of the sports car is 53.6 miles per hour

The key concept for solving such questions is the relationship between speed, distance and time.

The relationship between them is Distance = Speed x Time

Distance covered by Melissa = 0.5 miles + Distance covered by minivan

Let speed of Melissa be x miles per second

x. 50 = 0.5 + 50x50/3600

x .50 = 0.5 + 25/36

x = 1/100 + 1/72

x = 0.001 + 0.013

x = 0.014 miles per second

x = 3600 x 0.014 = 53.6 miles per hour

Thus the speed of the sports car is 53.6 miles per hour

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The distribution is skewed because it's on the right side of the polygon and this illustrates that's it's positively skewed.

It should be noted that the polygon can be gotten by plotting the data.

After the analysis of the distribution, the distribution is skewed to the right.

This illustrates that it is skewed.

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Finding the inverse function of [tex]f(x) = 5\sqrt{x + 3} - 2[/tex], it is best described by graph B.

Supposing we have a function y = f(x), to find the inverse, we exchange x and y, and isolate y.

In this problem, the function is:

[tex]f(x) = 5\sqrt{x + 3} - 2[/tex]

[tex]y = 5\sqrt{x + 3} - 2[/tex]

Exchanging x and y:

[tex]x = 5\sqrt{y + 3} - 2[/tex]

Working through the function to isolate y:

[tex]5\sqrt{y + 3} = x + 2[/tex]

[tex]\sqrt{y + 3} = \frac{x + 2}{5}[/tex]

[tex](\sqrt{y + 3})^2 = \left(\frac{x + 2}{5}\right)^2[/tex]

[tex]y + 3 = \frac{(x + 2)^2}{25}[/tex]

[tex]y = \frac{(x + 2)^2}{25} - 3[/tex]

[tex]f^{-1}(x) = \frac{(x + 2)^2}{25} - 3[/tex]

Which is best represented by graph B.

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The score Macro needs in his next game to have a mean of 130 is 150.

Mean is the average of a set of numbers. It is determined by adding the numbers together and dividing it by the total number

Mean = sum of the numbers / total number

130 = (128 + 127 + 123 + 122 + x) / 5

Where x represents the fifth score

130 = (500 + x) / 5

130 x 5 = 500 + x

650 = 500 + x

650 - 500 = x

x = 150

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The following expressions set up as an equation to solve for the unknown numbers in the cases given below:

let

5x + 6 = 46

5x = 46 - 6

5x = 40

x = 40/5

x = 8

2/3x - 5 = 11

2/3x = 11 + 5

2/3x = 16

x = 16 ÷ 2/3

x = 16 × 3/2

x = 48/2

x = 24

1/3x + 4 = 45

1/3x = 45 - 4

1/3x = 41

x = 41 ÷ 1/3

x = 41 × 3/1

x = 123

3x - 12 = 18

3x = 18 + 12

3x = 30

x = 30/3

x = 10

40 - 2x = 16

-2x = 16 - 40

-2x = -24

x = -24/-2

x = 12

(x + 18) / 6 = 7

(x + 18) = 7 × 6

x + 18 = 42

x = 42 - 18

x = 24

2/3x - 8 = 6

2/3x = 6 + 8

2/3x = 14

x = 14 ÷ 2/3

x = 14 × 3/2

= 42/2

x = 21

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Answer:

  A.  Function f has exactly two complex solutions.

Step-by-step explanation:

The function will have real solutions where the graph crosses the x-axis. It will always have a total number of solutions equal to its degree. The ones that are not real are complex.

The graph has its vertex above the x-axis and extends upward from there. It never crosses the x-axis, so there are no real solutions. That means both solutions are complex.

The measure of dispersion that measures how much the data differ from the mean is called the standard deviation.

Standard deviation is a statistical measure of dispersion as opposed to the measure of central tendency like mean, median and mode.

The standard deviation is a measure of how spread out data values are around the mean.

It is defined as the square root of the variance and represented with the Greek letter σ.

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The solution to the system of inequalities is given as Graph II.

A group of two or more inequalities in one or more variables is referred to as a system of inequalities.

When a problem demands a variety of solutions and those solutions must satisfy many constraints, systems of inequalities are utilized.

The system of inequality,

2x - y < 4

x + y < -1

First we draw the graph of both line. So make table of each line

For line 2x - y < 4

x  :    -1        0        1

y  :    -6     -4       -2

Test point (0,0)

0 - 0 < 4

0 < 4

True (Shade towards origin)

For line x + y < -1

x  :    -1        0        1

y  :    0        -1       -2

Test point (0,0)

0 + 0 < -1

    0 < -1

False (Shade away from origin)

Plot the points graph.

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Full Question:

The graphs associated with the question are attached.

Rational expressions are often used in combining rates of work. Therefore, it's true.

It should be noted that a rational expression is simply defined by a rational fraction.

They're are used in combining rates of work. Fir example, if Mr John performs 1/2 of his work and does 1/3 on another day. This can be expressed as:

= 1/2 + 1/3

= 5/6

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Answer:

Step-by-step explanation:

$274 for 40 hours of work as a unit rate

in practice you look for the hourly wage, you find it by dividing the wages divided by the working hours

274 : 40 = 6.85 units per hour