athletes are running a race. A gold medal is to be given to the winner, a silver medal is to be given to the second-place finisher, and a bronze medal is to be given to the third-place finisher. Assume that there are no ties. In how many possible ways can the medals be distributed?

The number which represents the difference between the numbers given -3 and -2 as in the task content is; 1.

It follows from the task content that the difference between the given numbers -3 and -2 is to be determined by means of the number line.

On this note, since, it follows that the difference between two numbers in the number line is given by the absolute value of their arithmetic difference.

It follows that the number which is required in this scenario is; |-3-(-2)| = |-1| = 1.

This follows from the fact that the absolute value big any number is that number with a positive polarity.

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Using a trigonometric identity, and considering that the angle is in the fourth quadrant, the tangent of the angle is given as follows:

tan(theta) = -1/4

The following identity is used to relate the measures, considering an angle [tex]\theta[/tex]:

[tex]\sin^2{\theta} + \cos^2{\theta} = 1[/tex]

For this problem, the sine is given as follows:

[tex]\sin{\theta} = -\frac{1}{\sqrt{17}}[/tex]

Then the cosine, which we need to find the tangent, is found as follows:

[tex]\sin^2{\theta} + \cos^2{\theta} = 1[/tex]

[tex]\left(-\frac{1}{\sqrt{17}}\right)^2 + \cos^2{\theta} = 1[/tex]

[tex]\frac{1}{17} + \cos^2{\theta} = 1[/tex]

[tex]\cos^2{\theta} = \frac{16}{17}[/tex]

[tex]\cos{\theta} = \pm \sqrt{\frac{16}{17}}[/tex]

Since the angle is in the fourth quadrant, the cosine is positive, hence:

[tex]\cos{\theta} = \frac{4}{\sqrt{17}}[/tex]

It is the sine of the angle divided by the cosine of the angle, hence:

[tex]\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} = \frac{-\frac{1}{\sqrt{17}}}{\frac{4}{\sqrt{17}}} = -\frac{1}{4}[/tex]

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The denominator cannot be zero, so

[tex]e^x - 9 = 0 \implies e^x = 9 \implies x = \ln(9)[/tex]

is not in the domain of [tex]f(x)[/tex].

[tex]\ln(x)[/tex] is defined only for [tex]x>0[/tex], and we have

[tex]\ln(x+1) > 0 \implies e^{\ln(x+1)} > e^0 \implies x+1 > 1 \implies x>0[/tex]

so there is no issue here.

By the same token, we need to have

[tex]x+1 > 0 \implies x > -1[/tex]

Taking all the exclusions together, we find the domain of [tex]f(x)[/tex] is the set

[tex]\left\{ x \in \Bbb R \mid x > 0 \text{ and } x \neq \ln(9)\right\}[/tex]

or equivalently, the interval [tex](0,\ln(9))\cup(\ln(9),\infty)[/tex].

Check the picture below.

now hmmm let's observe, hmmmm 11 has two digits, and low and behold, the 2 is right in the middle, then 1 and 1 on flanking it.

let's check 111 hmmmm has three digits and has a 3 right in the middle as well, and low and behold, is flanked by a regressing count, namely an "outwards" count.

let's check 1111 and 11111, same happens, one has a digit of 4 in the middle and the other 5, and yes, the "outwards" count on each, outwards count towards 1 each time btw.

so, if we were to take that pattern, we can say that eight 1's, namely 11111111 x 11111111, will have an 8 in the middle, and will look like 1234567   8   7654321.

The sequence of angles from smallest to largest is m∠T,m∠S,m∠R.

Given three angles be m∠T=4x-52°,m∠S=x+38°,m∠R=2x+47°.

We are required to arrange the angles in ascending order.

Ascending order means values that are written in a way that smallest values come first and larger values comes last.

First we have to find the angles at one value of x.

The values will differ as the values of x differ so we have to choose point or value of x.

Suppose x=20.

Then the angles are:

m∠T=4x-52°

=4*20-52

=80-52

=28°

m∠S=x+38°

=20+38

=58°

m∠R=2x+47°

=2*20+47

=40+47

=87°

Hence the sequence of angles from smallest to largest is m∠T,m∠S,m∠R.

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Answer:

(1,2)

Step-by-step explanation:

Substitution:

x + 2y = 5  Solve for x

x = -2y + 5  Substitute -2y + 5 in for x in the second equation

4x - 12y = -20

4(-2y + 5) - 12y = -20  Distribute the 4

-8y + 20 - 12 y = -20  Combine the y term

-20y + 20 = -20  Subtract 20 from both sides

-20y = -40  Divide both sides by -20

y = 2

Plug y into either of the 2 original equations to get x.

x + 2y = 5

x + 2(2) = 5

x + 4 = 5

x = 1

The answer is (2,1).

Elimination:

x + 2y = 5      4x - 12y = -20.  We want to eliminate with the x or the y.  I am going to eliminate the x's that means that I have to multiply the first equation all the way through by -4

(-4)(x + 2y) = (5) (-4)  That makes the equivalent expression

-4x - 8y = -20  I will add that to 4x - 12y = -20

4x - 12y = -20

0x -20y = -40

-20y = -40

y = 2.  Plug 2 into either the 2 original equation to find x.  This time I will select the second original equation to find x.

4x -12y = -20

4x - 12(2) = -20

4x - 24 = -20

4x = 4

x = 1

The marked price of the ring given the cost as 8740 inclusive of 15% tax and discount of 5% is $7,283.3.

8740 = x + (15% of x) + (5% of x)

8740 = x + (0.15x) + (0.05x)

8740 = x + 0.15x + 0.05x

8740 = 1.20x

x = 8740 / 1.20

x = 7283.33333333333

Approximately,

x = $7,283.3

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Answer: No

Step-by-step explanation:

The sum of exterior angles in a polygon is always equal to 360 degrees.

By the definition and properties of absolute value, there are two possible resulting functions: (i) g(x) = 5 · |x|, (ii) g(x) = |5 · x|. (Correct choices: B, D)

Herein we must modify a parent function to get a resulting function by the use of rigid transformations, defined as those transformations applied on geometric loci, including functions, such that Euclidean distance is conserved.

Based on the statement seen in the picture, the enlargement of the parent function can be done by a rigid transformation known as dilation, whose definition is shown below:

g(x) = k · f(x), where k is the stretch factor and is greater than 1.     (1)

If we know that f(x) = |x| and k = 5, then the resulting function is:

g(x) = 5 · |x|     (1)

This expression is also equivalent to the expression g(x) = |5 · x| by definition of absolute value.

By the definition and properties of absolute value, there are two possible resulting functions: (i) g(x) = 5 · |x|, (ii) g(x) = |5 · x|. (Correct choices: B, D)

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The number line with the points 3/6, 3/4, and 3/12 graphed on it is attached.

When fractions are represented on a number line, we can plot them on a number line just like we do with whole numbers and integers. Parts of a whole are represented by fractions. As a result, fractions on the number line are represented by splitting a whole into equal parts, ranging from 0 to 1, and the number of those equal parts would match the denominator of the fraction.

To graph the given fractions on the number line, we first convert them to like fractions.

3/6, 3/4, and 3/12

= 6/12, 9/12, and 3/12, which are like fractions.

Now we divide the number line with the minimum mark as 1/12, the least fraction, with 12 as denominator.

Now, we can mark the points 6/12, 9/12, and 3/12, as shown.

Thus, the number line with the points 3/6, 3/4, and 3/12 graphed on it is attached.

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Answer: 2/8, 3/8, 4/8, 5/8, 7/8

Step-by-step explanation:

     Since they all have a common denominator, we can list them based on their numerator without having to worry about the denominator.

2/8 < 3/8 < 4/8 < 5/8 < 7/8

1. By de Moivre's theorem,

[tex]\left(4\left(\cos\left(\dfrac{7\pi}9\right) + i \sin\left(\dfrac{7\pi}9\right)\right)\right)^3 = 4^3 \left(\cos\left(\dfrac{21\pi}9\right) + i \sin\left(\dfrac{21\pi}9\right)\right) \\\\ ~~~~~~~~ = 64 \left(\cos\left(\dfrac{7\pi}3\right) + i \sin\left(\dfrac{7\pi}3\right)\right) \\\\ ~~~~~~~~ = 64 \left(\cos\left(\dfrac\pi3\right) + i \sin\left(\dfrac\pi3\right)\right) \\\\ ~~~~~~~~ = 64 \left(\dfrac12 + i\dfrac{\sqrt3}2\right) \\\\ ~~~~~~~~ = \boxed{32 + 32\sqrt3\,i}[/tex]

2. First write the given number in exponential/trigonometric form.

[tex]z = 5-5\sqrt3\,i[/tex]

has modulus

[tex]|z| = \sqrt{5^2 + \left(-5\sqrt3\right)^2} = \sqrt{100} = 10[/tex]

and since it lies in the second quadrant of the complex plane, its argument is

[tex]\arg(z) = \pi + \tan^{-1}\left(-\dfrac{5\sqrt3}5\right) = \pi + \tan^{-1}\left(-\sqrt3\right) = \pi - \dfrac\pi3 = \dfrac{2\pi}3[/tex]

So, we have

[tex]z = 5 - 5\sqrt3\,i = 10 e^{i2\pi/3} = 10 \left(\cos\left(\dfrac{2\pi}3\right) + i \sin\left(\dfrac{2\pi}3\right)\right)[/tex]

Now we apply de Moivre's theorem again, and make sure to account for the multivalued-ness of the exponential function. For [tex]k\in\{0,1,2,3,4\}[/tex], the fifth roots of [tex]z[/tex] are

[tex]z^{1/5} = 10^{1/5} e^{i(2\pi/3 + 2\pi k)/5}[/tex]

[tex]k=0 \implies z^{1/5} = 10^{1/5} e^{i2\pi/15} = \boxed{10^{1/5} \left(\cos\left(\dfrac{2\pi}{15}\right) + i \sin\left(\dfrac{2\pi}{15}\right)\right)}[/tex]

[tex]k=1 \implies z^{1/5} = 10^{1/5} e^{i8\pi/15} = \boxed{10^{1/5} \left(\cos\left(\dfrac{8\pi}{15}\right) + i \sin\left(\dfrac{8\pi}{15}\right)\right)}[/tex]

[tex]k=2 \implies z^{1/5} = 10^{1/5} e^{i14\pi/15} = \boxed{10^{1/5} \left(\cos\left(\dfrac{14\pi}{15}\right) + i \sin\left(\dfrac{14\pi}{15}\right)\right)}[/tex]

[tex]k=3 \implies z^{1/5} = 10^{1/5} e^{i20\pi/15} = \boxed{10^{1/5} \left(\cos\left(\dfrac{4\pi}3\right) + i \sin\left(\dfrac{4\pi}3\right)\right)}[/tex] [tex]k=4 \implies z^{1/5} = 10^{1/5} e^{i26\pi/15} = \boxed{10^{1/5} \left(\cos\left(\dfrac{26\pi}{15}\right) + i \sin\left(\dfrac{26\pi}{15}\right)\right)}[/tex]

The y factors of the given quadratic equation is -1 and 40/22.

According to the statement

we have given that the equation and we have to find the factors of those equation.

So, The given equation is:

[tex]Y =x^2-23x^2 + 18x+40[/tex]

where y is the factors of that equation.

So, we have to find the factors of the given quadratic equation.

[tex]x^2-23x^2 + 18x+40 = 0[/tex]

Then after solving it become

[tex]-22x^2 + 18x+40 = 0[/tex]

Then take negative sign common from it then

[tex]22x^2 - 18x - 40 = 0[/tex]

Make the factors of this quadratic equation

[tex]22x^2 + 22x -40x - 40 = 0[/tex]

Now take common values 22x and 40 from the equation then

[tex]22x (x + 1) -40 (x + 1) = 0[/tex]

Then it become

[tex](22x - 40) (x + 1) = 0[/tex]

Then the values of x become from the equation is

[tex]x = -1, x= 40/22[/tex]

The value of the factors y is -1 and 40/22.

So, The y factors of the given quadratic equation is -1 and 40/22.

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According to the data, it can be inferred that Lin is faster than Andre because she is going at 0.3 km/min speed while he is going at 0.25 km/min speed.

To calculate who goes faster we must divide the time into the distance traveled by each character as shown below:

According to the above, the point K of each would be equal to the distance that each of them travels in one minute. As shown in the graph, after 5min (Lin) and 8min (Andre), each one reaches their respective destination located 1.5km and 2km away, respectively, taking the point (0,0) as the starting point.

The graph is attached.

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The domain of the function is [-4, 4) and the range of the function is [-5, 2)

The domain

As a general rule, it should be noted that the domain of a function is the set of input values or independent values the function can take.

This means that the domain is the set of x values

From the graph, we have the following intervals on the x-axis

x = -4 (closed circle)

x =4 (open circle)

This means that the domain of the function is [-4, 4)

The range

As a general rule, it should be noted that the range of a function is the set of output values or dependent values the function can produce.

This means that the range is the set of y values

From the graph, we have the following intervals on the y-axis

y = -5 (closed circle)

y = 2 (open circle)

This means that the range of the function is [-5, 2)

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The value of the given expression is approximate equal to 1.021 × 10⁴ OR 10207.8

From the question, we are to determine the value of the given expression

The given expression is

10 to the 4th times the square root of 1.042

That is,

10⁴ × √1.042

The expression can be evaluated as shown below

10⁴ × √1.042

= 10⁴ × 1.02078

= 1.02078 × 10⁴

≈ 1.021 × 10⁴

OR

= 1.02078 × 10⁴ = 10207.8

Hence, the value of the given expression is approximate equal to 1.021 × 10⁴ OR 10207.8

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By solving the equations x+2y=5, 4x-12y=-20 through elimination or substitution method we will get the value of x=1 and y=2.

Given two equations x+2y=5 and 4x-12y=-20.

We are required to solve the equations through elimination and substitution method.

Firstly we will solve the equations through elimination method.

x+2y=5-------------1

4x-12y=-20--------2

Multiply the equation 1 by 4 and then subtract the equation 2 from equation 1.

4x+8y-4x+12y=20+20

20y=40

y=2

Use the value of y in 1

x=5-2*2

x=5-4

x=1

From substitution method.

First find the value of x from first equation in terms of y.

x=5-2y-----4

Now put this value in second equation to get the value of y.

4(5-2y)-12y=-20

20-8y-12y=-20

20-20y=-20

-20y=-40

y=2

Put the value of y in equation 4.

x=5-2y

x=5-2*2

x=5-4

x=1

Hence by solving the equations x+2y=5, 4x-12y=-20 through elimination or substitution method we will get the value of x=1 and y=2.

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The answers to the given addition operations are

7) 6 Hundredths add to 4 tenths add to 6 ones is equal to 6.46

8) 82 Hundredths add to 9 tenths add to 4 tens is equal to 41.72

From the question, we are to add the given numbers

7. 6 Hundredths add to 4 tenths add to 6 ones is equal to

6 Hundredths = 0.06

4 tenths = 0.4

6 ones = 6

Thus, we get

0.06 + 0.4 + 6 = 6.46

8. 82 Hundredths add to 9 tenths add to 4 tens is equal to

82 Hundredths = 0.82

9 tenths = 0.9

4 tens = 40

Thus, we get

0.82 + 0.9 + 40 = 41.72

Hence, the answers to the given addition operations are

7) 6 Hundredths add to 4 tenths add to 6 ones is equal to 6.46

8) 82 Hundredths add to 9 tenths add to 4 tens is equal to 41.72

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Answer:

B

Step-by-step explanation:

Comment

√15 = 3.87

√12 = 3.46              Subtract these 2

Difference = 0.41

The piecewise defined function is

f(t) = 800 * 1.5^t for 0 - 2

f(t) = 2700 * 1.05^t-3 for 3 to 9

On the domain of t = 0 to 2, we have:

f(0) = 800

f(1) = 1200

f(2) = 1800

The above function is a piecewise function with the following definition

f(0) = 800 --- initial value

r = 1200/800 = 1.5 --- rate

The function is then represented as:

f(t) = f(0) * r^t

This gives

f(t) = 800 * 1.5^t

On the domain of t = 3 to 9, we have:

f(3) = 2700

f(4) = 2835

The above function is a piecewise function with the following definition

f(3) = 800 --- initial value

r = 2835/2700 = 1.05 --- rate

The function is then represented as:

f(t) = f(3) * r^t-3

This gives

f(t) = 2700 * 1.05^t-3

Hence, the piecewise defined function is

f(t) = 800 * 1.5^t for 0 - 2

f(t) = 2700 * 1.05^t-3 for 3 to 9

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The answer is no.

Always remember when converting from percent to decimal, divide by 100%.

Hence, the student's thinking is not correct.

Answer:

no

Step-by-step explanation:

0.3 = 30% not 3%

to change a percentage to a decimal fraction, divide by 100

3% = [tex]\frac{3}{100}[/tex] = 0.03

Answer:

1 (3.66)

2(3.5)

3(5.25)

Answer:

9

Step-by-step explanation:

To find the rate of change we use the formula

f(x2) - f(x1)

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

x2 -x1

f(x) = 9x

x2 = 9  and x1 = 0

f(x2) = 9( 8) = 72

f(x1) = 9(0) =0

The rate of change is

72 - 0

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

8-0

72

----

8

9

The rate of change is 9

Answer:

9

Step-by-step explanation:

The average rate of change of function f(x) over the interval a ≤ x ≤ b is given by:

[tex]\dfrac{f(b)-f(a)}{b-a}[/tex]

Given:

Therefore:

Substitute the given values into the average rate of change formula:

[tex]\begin{aligned}\implies \dfrac{f(8)-f(0)}{8-0} & = \dfrac{9(8)-9(0)}{8-0}\\\\& = \dfrac{72-0}{8-0}\\\\& = \dfrac{72}{8}\\\\& = 9\end{aligned}[/tex]

Therefore, the average rate of change of the function f(x) over the interval 0 ≤ x ≤ 8 is 9.

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Answer:

8v-28/v^2-2v-24

Step-by-step explanation:

1. Find common denominator

2. Combine fractions with common denominator

3. Re-order terms so constants are on the left

Assuming you had a bill of $120 when you went to the restaurant with your three friends, the 20% tip would come out to $24.

First assume an amount that the meal you ate cost. Let's assume that amount if $120.

If the tip rate if 20%, it means that you need to find out what 20% of the bill of $120 is.

If you don't want to use a calculator, there are other ways you can calculate the tip.

An easy method would be to find 10% of the bill. To find 10%, all you have to do is to move the decimal point from right to left.

The decimal point in $120 is at the far right so the 10% amount would be:
= $12.0

Now that you have the 10% amount, remember that 20% is simply twice the amount of 10% so:

= 12 x 2

= $24

The total amount then becomes:

= 120 + 24

= $144

In conclusion, your tip amount would be $24.

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The two different ways of finding the area are,
Case 1 = assume horizontal rectangles,
Case 2 = assume vertical rectangles.

the rectangle is a four-sided polygon whose opposites sides are equal and has an angle of 90° between its sides.

Here,

case 1,
As shown in the image
Area = sum of horizontal rectangles
Area = 10 * 3 + 2 * 5 + 2 * 1
Area  = 30 + 10 + 2
Area = 42    

Case II,
As shown in right figure,
Area of the vertical rectangles
Area = 3 * 5 + 5 * 3 + 2 * 6
Area = 15 + 15 + 12
Area = 42
Here, the area in case 1 is equal to case 2.

Thus, the two different ways of finding the area have shown above.

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The function that has a minimum and is transformed to the right and down from the parent function is; g(x) = 8(x² - 3²) - 5

By inspection, only Functions B) and D) have a minimum.

For the first function in option B, we have:

g(x) = 4(x²- 6x + 9) + 1

This can be simplified to;

g(x) = 4( x - 3 )² + 1

Thus, we can say that this first function is transformed to the right and up from the parent function.

For the second function in option D, we have;

g(x) = 8(x² - 6x + 9 ) - 5

The function can be simplified to get;

g(x) = 8( x - 3 )² - 5

Thus, we can say that it is transformed to the right and down.

The function that has a minimum and is transformed to the right and down from the parent function is D

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Answer: option D

Step-by-step explanation:

Solution:

Let x be the number

Triple the number=3x

According to the question,

3x+5=50

⇒3x+5−5=50−5          [Subtract 5 from both sides]

⇒3x=45

⇒ x = 45/3

​           [Divide 3 from both sides]

⇒x=15

∴15 is the required number. When it is tripled and increased by 5, we get 50.

If the patio is as wide as the fire pit, and 5 feet long then area of backyard can be expressed as [tex]2x^{2}+16x+30[/tex].

Given that the patio is as wide as the fire pit, and 5 feet long.

We are required to form an equation which can describe the area of the backyard.

Equation is relationship between two or more variables that are expressed in equal to form. Equations of two variables look like ax+by=c. It may be linear equation, quadratic equation, cubic equation or many more depending on the powers of the variable.

If we see the figure carefully then we can find that the length of the backyard is 2x+6 and the breadth of the backyard is 5+x and backyard is in shape of rectangle.

Area of rectangle =Length *breadth

=(2x+6)*(5+x)

=30+10x+6x+2[tex]x^{2}[/tex].

Hence if the patio is as wide as the fire pit, and 5 feet long then area of backyard can be expressed as [tex]2x^{2}+16x+30[/tex].

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