Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. 31. tan a cot(a + 10°) 32. cos θ = sin(2 θ -30°) 33. sin(2θ +10°) = cos(3 θ - 20°) 34. sec (B+10°) = csc (2B + 20°) 35. tan(3B+ 4°) = cot(5B-10°) 36. cot(5 θ +2°)=tan(2 θ +4°)

Answer: $634.99

Step-by-step explanation:
Regular price, R - Discount, D = S

subtract discount from regular price

Answer: $634.99

Step-by-step explanation:

$828.15 - 193.16 would equal 634.99

so the discounted price would be $634.99

Answer:

27

Step-by-step explanation:

factors are numbers that divide exactly into a number , leaving no remainder.

then the only factor of 27 from the list is 27

Step-by-step explanation:

Using perfect square trinomial rules,

[tex](a + b) {}^{2} = {a}^{2} + 2ab + {b}^{2} [/tex]

Here, a is sin x

b is cos x

So

[tex]( \sin(x) + \cos(x) ) {}^{2} = \sin {}^{2} (x) + 2 \sin(x) \cos(x) + \cos {}^{2} (x) [/tex]

We know

[tex] \sin {}^{2} (x) + \cos {}^{2} (x) = 1[/tex]

So we know have

[tex]1 + 2 \sin(x) \cos(x) [/tex]

[tex] \sin(2x) = 2 \sin(x) \cos(x) [/tex]

So our final answer is

[tex]1 + \sin(2x) [/tex]

a) The sales 2 months after the end of the campaign will be $29,600. b)  it will take 10 months (round up to the nearest whole number) for sales to drop below $1,000, if no new campaign is initiated.

The sales 2 months after the end of the campaign can be found by plugging x = 2 into the equation:
S = 80,000e-0.5x
S = 80,000e-0.5*2
S = 80,000e-1
S = 80,000 * 0.37
S = 29,600

Therefore, the sales 2 months after the end of the campaign will be $29,600.

We can solve this by setting the equation equal to 1000 and solving for x:
1000 = 80,000e-0.5x
1000/80,000 = e-0.5x
0.0125 = e-0.5x
ln(0.0125) = ln(e-0.5x)
-4.81 = -0.5x
x = 9.62

Therefore, it will take 10 months (round up to the nearest whole number) for sales to drop below $1,000, if no new campaign is initiated.

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Answer:

Let's start by using the formula for mixing two solutions to set up an equation. The formula is:

(concentration of solution 1) x (volume of solution 1) + (concentration of solution 2) x (volume of solution 2) = (concentration of resulting solution) x (total volume of resulting solution)

In this problem, solution 1 is the 25% sugar syrup, solution 2 is the 10% sugar syrup, and the resulting solution is 15% sugar syrup. We are also given that the total volume of the resulting solution is 120 grams. Let's use x to represent the volume of the 10% sugar syrup that is mixed with the 25% sugar syrup. Then we can write:

(0.25)(120-x) + (0.10)(x) = (0.15)(120)

Simplifying this equation, we get:

30 - 0.25x + 0.10x = 18

0.15x = 12

x = 80

Therefore, we need to mix 80 grams of the 10% sugar syrup with 40 grams of the 25% sugar syrup to get 120 grams of 15% sugar syrup.

To find the amount of pure sugar in the 120 grams of 15% syrup, we can use the fact that the concentration of pure sugar in the resulting solution is 15%. That means that 15% of the 120 grams is pure sugar. We can find this by multiplying 120 by 0.15:

120 x 0.15 = 18

So there are 18 grams of pure sugar in the 120 grams of 15% syrup.

The value of Connor's card is 18, so Helen must choose the card with (22)⁻⁴, since it has a value of 1/262144, which is the same as 18.

Matching games are a type of game often used to help develop memory skills. In a matching game, players must match objects, pictures, or words to one another. This can involve matching a picture to a word, or a word to a definition. Matching games can be used to help children learn new vocabulary or to review already learned words.

The card with 2⁻³ has a base of 2 and an exponent of -3. This means that the value of the card is 1/8, which is the same as 2⁻³.

The card with (12)⁻² has a base of 12 and an exponent of -2. This means that the value of the card is 1/144, which is the same as 12⁻².

The card with (22)⁻⁴ has a base of 22 and an exponent of -4. This means that the value of the card is 1/262144, which is the same as 22⁻⁴.

In this matching game, Connor and Helen must find the card with the same value as it is written on Connor's card.

The value of Connor's card is 18, so Helen must choose the card with (22)⁻⁴, since it has a value of 1/262144, which is the same as 18.

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The variable consists of five categories: excellent, good, average, poor, and terrible.

The variable of interest in this scenario is the rating of the service provided by the hotel during the guests' most recent visit.

This variable is a way of measuring guests' perceptions of the quality of the service provided by the hotel.

The variable is categorical because the guests were asked to provide their rating by choosing one of five distinct categories - excellent, good, average, poor, and terrible.

The categories stated in (a) above represent the different levels of quality the guests perceived in the hotel's service.

Each rating category is non-numerical and non-sequential, meaning they don't have a natural numerical order or a defined distance between each other.

So, the number of categorical variables is 5

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Complete question

A hotel chain sent​ 2,000 past guests an email asking them to rate the service in the hotel during their most recent visit. Of the 500 who​ replied, 450 rated the service as excellent.

(a) Give a name to the variable and indicate if the variable is​ categorical, ordinal, or numerical

(b) If you answered "Categorical" in the previous question, how many categories does the variable of interest consist of? If you answered "Numerical", what is the range of values the variable can take?

Joey walked 110 meters from  A to B and Lana started at point A and walked in a straight line to point B which is a distance of about 80.6 meters, obtained using Pythagorean Theorem, therefore;

Joey walked about 29.4 meters further than Lana.

Pythagorean Theorem states that the square of the length of the hypotenuse side of a right triangle is the sum of the squares of the lengths of the legs of the right triangle.

The distance Joey walked = 20 m + 40 m + 50 m = 110 m

The distance Lana walked can be found as follows;

The similar triangles formed by the path Lana walked and the path Joey walked, indicates that the ratio of the base lengths of the right triangles are;

50/20 = 5/2

The base length of the larger right triangle 5/(2 + 5) × 40 m = 200/7 m

Base length of the smaller right triangle = 2/(5 + 2) × 40 m = 80/7 m

The length of the path Lana walked, l, found using Pythagorean Theorem is therefore;

The distance further Joey walked = 110 m - 80.6 m = 29.4 m

Joey

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The lunch period that has more students is the second lunch period.

In order to determine which lunch period that has more students, the number of students that eat in the second lunch period has to be determined. In order to determine this value, the mathematical operation that would be used is subtraction.

Subtraction is the process of determining the difference between two or more numbers. The sign that is used to represent subtraction is -.

Number of students that ear in the second lunch period = total number of students - students that eat in the first lunch period

758 - 371 = 387

387 > 371

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The base, which is equal to 1.3, is one of the values in the equation g(x)=18(1.3)x.

The monthly growth rate of the insect population is represented by this number. Since 1.3 is 1 + 30% represented as a decimal, the population is specifically growing by 30% each month. When the growth rate is compounded each month, this indicates that the insect population is expanding quickly over time.

If we enter x=3 into the function, for instance, we obtain g(3)=18(1.3)3=18(2.197)=39.546. This indicates that the anticipated bug population after three months is 39,546. Pest control is frequently required to preserve ecosystem balance since this exponential growth might, if unchecked, result in a very huge insect population.

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The exact value οf cοs(2θ) is 27/25.

The exact value οf sin(θ/2) is -√[(5 + √13) / 10].

Trigοnοmetry is a branch οf mathematics that deals with the relatiοnships between the sides and angles οf triangles, and the trigοnοmetric functiοns that describe thοse relatiοnships.

Given: sin(θ) = (2√3) / 5, and π/2 < θ < π.

Part A:

cοs(2θ) = 2cοs²(θ) - 1

We can find cοs(θ) using the Pythagοrean identity:

cοs²(θ) + sin²(θ) = 1

cοs²(θ) = 1 - sin²(θ)

cοs(θ) = ±√(1 - sin²(θ))

Since π/2 < θ < π, sin(θ) is pοsitive and cοs(θ) is negative in the secοnd quadrant. Therefοre, we have:

cοs(θ) = -√(1 - (2√3/5)²) = -√(1 - 12/25) = -√13/5

Substituting intο the fοrmula fοr cοs(2θ), we get:

cοs(2θ) = 2cοs²(θ) - 1 = 2(-√13/5)² - 1 = 52/25 - 1 = 27/25

Therefοre, the exact value οf cοs(2θ) is 27/25.

Part B:

We can use the half-angle fοrmula fοr sine tο find sin(θ/2):

sin(θ/2) = ±√[(1 - cοs(θ)) / 2]

Since π/2 < θ < π, sin(θ/2) is negative in the secοnd quadrant. Therefοre, we have:

sin(θ/2) = -√[(1 - cοs(θ)) / 2] = -√[(1 - (-√13/5)) / 2] = -√[(5 + √13) / 10]

Therefοre, the exact value οf sin(θ/2) is -√[(5 + √13) / 10].

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The sοlutiοn οf the given prοblem οf expressiοn cοmes οut tο be examples abοve are equal tο 16t - 0.25.

Mathematical prοcesses like multiplying, dividing, adding, and nοw even deleting are required. If they were merged, the fοllοwing claim wοuld be made: An equatiοn, sοme statistics, and a mathematical fοrmula Values, cοmpοnents, and mathematical prοcesses like additiοns, subtractiοns, reversals, algebraic fοrmulas, and divisiοns make up a statement οf truth. Wοrds and phrases can be rated and analyzed.

Here,

We pοssess 16t – 0.25.

A) The fοrmula 42t/40.5 equals (22t)² / (20.5) = 24t / (20.5) = 16t / (20.5)4 = 16t / 4 = 4(16t).

Expressiοn A is nοt equal because 4(16t) is nοt the same as 16t - 0.25.

B) [tex](16)^t/2^4 = (2^4)^t / 2^4[/tex]

=> [tex]2^{4t} / 2^4 = 2^{(4t-4)} = 16^{(t-1)} (t-1).[/tex]

Expressiοn B is not equivalent because 16(t-1) is nοt equal to 16(t - 0.25).

C) [tex]16^t/2 = (2^4)^t / 2[/tex]

[tex]= > 2^{4t} / 2 = 2^{(4t-1)} = 16^t \times 0.5.[/tex]

Expression C is nοt equivalent because 16t * 0.5 does not equal 16t - 0.25.

Hence, As a result, nοne of the examples above are equal to 16t - 0.25.

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Answer:

Step-by-step explanation:

The final amount for the investment of $5,000 over 5 years at an annual interest rate of 6% compounded quarterly is $ 6,749.29.

To find the final amount for an investment P over t years at an annual interest rate of r compounded quarterly, we can use the formula for compound interest:  

[tex]{\displaystyle A=P\left(1+{\frac {r}{n}}\right)^{nt}}[/tex]

Where:

A = Final amount

P = Principal (initial investment)

r =  Annual interest rate (as a decimal)

n = Number of times the interest is compounded per year

t = Time period in years

Here we have

P = $5,000,

r = 6% = 0.06,

n = 4 (since the interest is compounded quarterly), and

t = 5 years.  

Using the above formula

[tex]A = 5000(1 + 0.06/4)^{(4*5)[/tex]

[tex]A = 5000(1.015)^{20}[/tex]

[tex]A = 5000(1.34985711)[/tex]

[tex]A = $6,749.29[/tex]

Therefore,

The final amount for the investment of $5,000 over 5 years at an annual interest rate of 6% compounded quarterly is $ 6,749.29.

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Answer:

-14/16

-35/40

Step-by-step explanation:

7/-8 times [tex]\frac{2}{2}[/tex] is -14/16

7/-8 times [tex]\frac{5}{5}[/tex] is -35/40

Answer:

7-8-87-87-87-7-767&7&7

Step-by-step explanation:

そう笑ここに何を書けばいいのかわからないので、ここで私は答えを知っているふりをしています

Answer:

7/10 - 2/5 = 7/10 - 4/10 = 3/10

Step-by-step explanation:

Equalize both denominators, as in 2/5 = 4/10.

The slope of a line perpendicular to given line is 4.

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

The given equation of a line is x+4y=4.

Here, 4y=4-x

y=1-x/4

So, slope m is -1/4

The slope of a line perpendicular to given line is m1=-1/m2

Slope of perpendicular to line is -1/(-1/4)

= 4

Therefore, the slope of a line perpendicular to given line is 4.

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The slope of parallel line is,

m = - 1/4

And, Slope of perpendicular line are,

m = 4

The equation of line in point-slope form passing through the points

(x₁ , y₁) and (x₂, y₂) with slope m is defined as;

⇒ y - y₁ = m (x - x₁)

Where, m = (y₂ - y₁) / (x₂ - x₁)

Given that;

The equation of line is,

⇒ x + 4y = 4

Now, We can write as;

⇒ x + 4y = 4

⇒ 4y = - x + 4

⇒ y = - 1/4x + 4

Hence, The slope of the line is,

⇒ y = - 1/4x + 4

⇒ dy/dx = - 1/4

⇒ m = - 1/4

Now, We know that;

The slope of parallel line are equal.

Hence, The slope of parallel line is,

m = - 1/4

And, Product of Slopes of perpendicular lines are - 1.

Hence, Slope of perpendicular line are,

m = - 1/ (- 1/4)

m = 4

Thus, The slope of parallel line is,

m = - 1/4

And, Slope of perpendicular line are,

m = 4

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For the airplane, we have v = -453.01, 79.88. Similarly, for the wind, we have w = -75.18, -27.36.

The actual speed would be 530.79 mph.

The actual ground speed and direction of the airplane are 530.79 mph and 174.32 deg, respectively.

The component form of the velocity of the airplane can be found by using the formula:

v = r < cos theta, sin theta >, where r is the speed and theta is the bearing. For the airplane, we have v = 460 < cos 170, sin 170 > = <-453.01, 79.88>. Similarly, for the wind, we have w = 80 < cos 200, sin 200 > = <-75.18, -27.36>.

To find the actual ground speed and direction of the airplane, we need to add the velocity vector of the airplane and the wind vector. This gives us the velocity vector = v + w = <-453.01, 79.88> + <-75.18, -27.36> = <-528.19, 52.52>.

The actual speed can be found by taking the magnitude of the velocity vector, which is given by |v+w| = sqrt((-528.19)^2 + (52.52)^2) = 530.79 mph.

The actual direction can be found by using the formula theta = arctan(y/x), where x and y are the x and y components of the velocity vector. In this case, we have theta = arctan(52.52/-528.19) = -5.68 deg. However, since the velocity vector is in the third quadrant, we need to add 180 deg to get the actual direction, which is 180 + (-5.68) = 174.32 deg.

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The first step would be to multiply both sides by 5 when solving the equation (3(x+4))/(5)=6.

This will allow us to eliminate the denominator on the left side of the equation and simplify the equation.After multiplying both sides by 5, the equation becomes: 3(x+4) = 30

Next, we can distribute the 3 on the left side of the equation: 3x + 12 = 30

Then, we can subtract 12 from both sides of the equation: 3x = 18

Finally, we can divide both sides by 3 to solve for x: x = 6

Therefore, the solution to the equation is x=6.

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The value of the series 4 + 9 + 14 + ... + (5n-1) using the summation notation is (5n^2 + 3n)/2.

The value of the series 4 + 9 + 14 + ... + (5n-1) can be found using the summation notation as follows:

Identify the first term, common difference, and number of terms in the series. The first term is 4, the common difference is 5, and the number of terms is n.

Use the summation notation to write the series as ∑_{i=1}^{n} (5i-1).

Use the formula for the sum of an arithmetic series to find the value of the series. The formula is S_n = n/2 (a_1 + a_n), where S_n is the sum of the series, n is the number of terms, a_1 is the first term, and a_n is the last term.

Substitute the values of n, a_1, and a_n into the formula and simplify. S_n = n/2 (4 + (5n-1)) = n/2 (5n+3) = (5n^2 + 3n)/2.

The value of the series is (5n^2 + 3n)/2.

Therefore, the value of the series 4 + 9 + 14 + ... + (5n-1) using the summation notation is (5n^2 + 3n)/2.

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Answer: B

Step-by-step explanation:

(7x^3-10x^2- 5)-(2x^3+7x^2-9)

= 5x^3 -3x^2-14

The distance between the middle school and the fire station is 4.5 miles

Pythagoras theorem states that, in a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides.

The distance between the middle school and the fire station is is the hypotenuse

therefore;

c² = 2² + 4²

c² = 4 + 16

c² = 20

c = √20

c = 4.5 miles(nearest tenth)

therefore the distance between the middle school and fire station is 4.5 miles

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Given vectors are v1 = |2| v2 = |3| v3 = |1| |2| |1| |h| |-4| |-4| |4| Let's consider the vectors are linearly dependent. If they are linearly dependent, then the rank of the matrix will be less than 3. So, Values of the parameter h are |2| 3 1 |2| 1 h |-4| |-4| 4

For the matrix to have a rank of less than 3, it must satisfy the following condition : |2| 3 1 |2| 1 h |-4| |-4| 4It can be rewritten as follows: |2| 3 1 |2| 1 h |-4| |-4| 4It can be simplified as follows: |2| 3 1 |2| 1 h |-4| |-4| 4. The matrix can be reduced to the following form: |2| 3 1 |2| 1 h |-4| |-4| 4, By subtracting R1 from R2, we get: |-1| 0 |h-1| |-6| |-h| |-4| |-4| 4The determinant of this matrix is: -4h + 4 + 8h = 4h + 4

Now, let's evaluate the matrix's rank. We can observe that the matrix's second row is equal to twice the matrix's first row. So, the rank of the matrix is 2.Since the rank of the matrix is 2, the given vectors are linearly dependent if h = -1.

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Answer: 14/3 or 3/4 are in 7/2

Step-by-step explanation:

Using algebraic expressions, the expression that is equivalent to 9(4w) is: 36w.

Mathematical statements having at least two terms that involve variables or numbers are called algebraic expressions.

Every combination of terms that have undergone operations like addition, subtraction, multiplication, division, etc. is a recognised algebraic expression (or variable expression).

We just join like terms in an algebraic expression to make it simpler. Hence, related variables will be combined. Now, the identical powers will be merged from the similar variables.

Now in the given case,

9(4w)

= 9×4×w

=36w.

Therefore, the expression that is equivalent to 9(4w) is: 36w.

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The number of cases which Judge Jerry hears every hour is 2 2/19

An element of a whole or, more broadly, any number of equal pieces, is represented by a fraction.

When used in conversational English, a fraction indicates the number of components of a particular size, as in one-half, eight-fifths, and three-quarters.

How to calculate:

GIven that he tries 5 cases in 2 3/8 hours

The unit rate of cases per hour is calculated by dividing the number of cases by the number of hours.

5/(2 3/8) = 5/(19/8) = 5 * 8/19 = 40/19 = 2 2/19

Therefore, the number of cases which Judge Jerry hears every hour is 2 2/19

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The inverse transform method allows us to generate a random variable having a specific distribution function by finding the inverse of the distribution function and plugging in a random value from the uniform distribution on the interval [0, 1].

To generate a random variable having the distribution function F(x) = x² + x / 2 using the inverse transform method, we need to follow the following steps:

Step 1: Find the inverse of the distribution function F(x). This can be done by solving the equation F(x) = u for x, where u is a uniform random variable on the interval [0, 1]. In this case, we have:

x² + x / 2 = u

Step 2: Use the quadratic formula to solve for x:

x = (-1 ± √(1 - 4(1/2)(-u))) / (2(1/2))

x = (-1 ± √(1 + 2u)) / 1

Step 3: Since we want the inverse function, we need to choose the positive solution:

x = (-1 + √(1 + 2u)) / 1

Step 4: Now we have the inverse function F^-1(u) = (-1 + √(1 + 2u)) / 1. To generate a random variable having the distribution function F(x), we simply need to plug in a random value of u from the uniform distribution on the interval [0, 1]:

x = F^-1(u) = (-1 + √(1 + 2u)) / 1

Step 5: This gives us a random variable x having the desired distribution function F(x) = x² + x / 2.

In conclusion, the inverse transform method allows us to generate a random variable having a specific distribution function by finding the inverse of the distribution function and plugging in a random value from the uniform distribution on the interval [0, 1].

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A^8 = [\begin{array}{ccc}252&-3&3\\3&252&-3\\-3&3&252\end{array}\right]

To find the eigenvalues and eigenvectors of matrix A, use the characteristic polynomial of A which is given by:
P(λ) = det(A-λI) = λ^3-6λ^2+11λ-6

The roots of the equation are λ1 = 2, λ2 = 3, and λ3 = 1.

The corresponding eigenvectors are:
v1 = [\begin{array}{ccc}-1\\1\\1\end{array}\right]
v2 = [\begin{array}{ccc}1\\-1\\1\end{array}\right]
v3 = [\begin{array}{ccc}1\\1\\1\end{array}\right]

The eigenvalues and eigenvectors of A can be put together in the following matrix:
T = [\begin{array}{ccc}-1&1&1\\1&-1&1\\1&1&1\end{array}\right]

The eigenvalues of A can be put together in the following matrix:
Λ = diag{λ1, λ2, λ3} = [\begin{array}{ccc}2&0&0\\0&3&0\\0&0&1\end{array}\right]

Now, to calculate A^8, you will need to use the eigenvalues and eigenvectors of A. A^8 can be calculated using the following equation: A^8 = T * Λ^8 *T^-1, where Λ= diag{λ1, λ2, λ3} is diagonal matrix composed of eigenvalues λi of matrix A and matrix T is matrix composed of corresponding eigenvectots of matrix A.

Thus, A^8 can be calculated as follows:
A^8 = [\begin{array}{ccc}-1&1&1\\1&-1&1\\1&1&1\end{array}\right] * [\begin{array}{ccc}256&0&0\\0&6561&0\\0&0&1\end{array}\right] * [\begin{array}{ccc}-1&1&1\\1&-1&1\\1&1&1\end{array}\right]^-1

= [\begin{array}{ccc}252&-3&3\\3&252&-3\\-3&3&252\end{array}\right]

Therefore, the result of A^8 is: A^8 = [\begin{array}{ccc}252&-3&3\\3&252&-3\\-3&3&252\end{array}\right]

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The length of the shadow cast by a nearby elm tree that that is 16 feet tall is given as follows:

12 feet.

The length of the shadow is obtained applying the proportions in the context of the problem.

A shadow 9 feet long is cast by a plum tree that is 12 feet tall, and we want the shadow for a tree that is 16 feet tall, hence the rule of three is given as follows:

9 feet - 12 feet

x feet - 16 feet.

Applying cross multiplication, the shadow is obtained as follows:

12x = 9 x 16

12x = 144

x = 144/12

x = 12 feet.

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Answer:

Step-by-step explanation:

5.  15 multiplied by 4/5 = 3 multiplied by 4 = 12.

6.  3/11 multiplied by 66 = 3 multiplied by 6 = 18.

When copying segments and angles, drawing a ray with one endpoint step is the same.

A line segment is a part of a continuous line, having two fixed ends and a definite length.

When a common part in constructing a line segment and an angle is drawing a ray because a ray is a common part, when we see the figures of a segment and an angle we see a common a ray, so a ray is the common part in both.

Hence, when copying segments and angles, drawing a ray with one endpoint step is the same.

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