FLUENCY
A function is a rule that for every input it assigns

(1) exactly one output
(2) at least one output
(3) two or more outputs
(4) an infinite number of outputs

We conclude that the angle at the top measures 65.8°.

A right triangle or right-angled triangle, or more formally an orthogonal triangle, formerly called a rectangled triangle, is a triangle in which one angle is a right angle, i.e., in which two sides are perpendicular. The relation between the sides and other angles of the right triangle is the basis for trigonometry.

here, we have,

What angle does the top part?

The whole tree measures 117 ft, and it is broken 34 ft above the ground, forming this way a right triangle.

Then one cathetus of the right triangle measures 34ft, and the hypotenuse measures 117ft - 34ft = 83ft

The angle at the top is θ, the adjacent cathetus to it is the one that measures 34 ft, then to get the angle we can use the relation:

cos(θ) = (adjacent cathetus)/(hypotenuse)

Replacing what we know, we get:

cos(θ) = (34 ft)/(83ft) = 34/83

Solving for θ, we get:

θ = Acos( 34/83) = 65.8°

So we conclude that the angle at the top measures 65.8°.

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The equation solved for the variable h is h = [tex]\sqrt[]{\frac{2A -b^2}{b} }[/tex]

To solve for a variable in a given equation is to make it the subject of formula.

The variable is made to stand alone on one end of the equality sign.

We have the equation given as;

A=1/2bh(h+b)

First, cross multiply

2A = bh(h+b)

2A = bh² + b²

collect like terms

2A - b² = bh²

Now divide by the coefficient of h²

h² = 2A - b²/b

Find the square root of both sides

h = [tex]\sqrt[]{\frac{2A -b^2}{b} }[/tex]

Hence, the equation is h = [tex]\sqrt[]{\frac{2A -b^2}{b} }[/tex]

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Answer:

1. ~ ABC = 180 degrees

2. AB/AD = BC/AC

3. 3/9 = x/6

4. x = 2

5. The scale factor of ADE to ABC is 1:2.

Step-by-step explanation:

1. ~ABC = 180 degrees means that the sum of all the angles of a triangle add up to 180 degrees.

2. AB/AD = BC/AC means that if two sides of a triangle are divided by another side, the ratio will be the same for both sides of the triangle.

3. 3/9 = x/6 means that if you divide 3 by 9, you can find out what x is when it is divided by 6.

4. x = 2 means that when 3 is divided by 9 and the result is divided by 6, the answer is 2.

5. The scale factor of ADE to ABC is 1:2 means that ADE is twice the size of ABC and all related measurements between the two triangles are in a proportion of 1:2.

The simplified form of the expression (4 - a²b³)/5 = (9 - 3b³)/7 is

b³(15 - 7a²) = 17.

An equation is written in the form of variables and constants separated by the operation of multiplication and division,

An equation states that terms in different forms on both sides of the equality sign are equal.

Multiplication and division do not separate the terms of an equation.

Given, An equation (4 - a²b³)/5 = (9 - 3b³)/7,  in variables 'a' and 'b'.

7(4 - a²b³) = 5(9 - 3b³).

28 - 7a²b³ = 45 - 15b³.

15b³ - 7a²b³ = 45 - 28.

b³(15 - 7a²) = 17.

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Based on the information, it can be inferred that the volume of a cube with a side length of 30 cm is 27,000cm³ and the area would be 90cm².

To calculate the volume and area of the cube we must perform the following mathematical procedures:

Volume:

We must multiply side * base * depth = volume.

Area:

We must multiply the base by the side

According to the above, the area and volume of this cube would be 90cm² and 27,000cm³ respectively.

Note: This question is incomplete. Here is the complete information:

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The effect on the graph for the given function if the slope of the function is changed to -10 include the following: D. the line is less steep.

The effect on the graph for the given function if the y-intercept of the function is changed to a smaller positive number is: C. the line shifts down.

In Mathematics, a slope is sometimes referred to as rate of change and it is typically used to describe both the direction, ratio, and steepness of the function of a straight line based on its coordinates or points.

Generally speaking, a slope simply refers to a measure of the steepness of a straight line. This ultimately implies that, the higher the slope of a line, the more steep it is and vice-versa.

Based on the information provided about this function y = 8 - 2x, we can logically deduce that the line would be shifted downward when its y-intercept is changed to a smaller positive number.

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Mack should make 4 necklaces and 8 wristbands to maximize his profit while staying within his time and quantity constraints.

His profit would be:

Profit = 3(4) + 2(8) = $20.

One or many equations having the same number of unknowns that can be solved simultaneously are called simultaneous equations. And the simultaneous equation is the system of equations.

From the information given, we know:

a) Necklace requires 40 minutes to make.

Let x be the number of necklaces made, then the total time spent on necklaces is 40x.

b) Wristband requires 25 minutes to make.

Let y be the number of wristbands made, then the total time spent on wristbands is 25y.

c) Mack has a total of 360 minutes to make necklaces and wristbands. Therefore, the equation for time is:

40x + 25y = 360

d) Mack wants to make no more than 12 items, so we have the inequality:

x + y ≤ 12

e) Mack makes a profit of $3.00 per necklace and $2.00 per wristband. Therefore, the equation for profit is:

Profit = 3x + 2y

Now we have the following system of equations:

40x + 25y = 360

x + y ≤ 12

Profit = 3x + 2y

From equation (2), we have y = 12 - x.

Substitute y = 12 - x into equations (1) and (3):

40x + 25(12 - x) = 360

Profit = 3x + 2(12 - x)

Simplify and solve for x:

40x + 300 - 25x = 360

15x = 60

x = 4

Substitute x = 4 into equation (2) to find y:

y = 12 - x = 12 - 4 = 8

Therefore, His profit would be, Profit = 3(4) + 2(8) = $20.

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Answer:

= 5n³ + 3n² - 4

Step-by-step explanation:

(-3n³-3n²) + (8n³ + 6n² - 4)

= -3n³-3n²+8n³+6n²-4

= -3n³+8n³-3n²+6n²-4

= 5n³+3n²-4.

The two angels are similar because they have the same magnitude of corresponding angles. The length of x is 6 cm.

For two or more similar triangles, it has two conditions.

The two triangles as shown in the picture, ΔABC and ΔEDC.

Explain why the two triangles are similar! Find the length of x!

We have

The magnitude of ∠BCA and ∠DCE are equal. It is because they are at one point of intersection.

Since, ∠BCA and ∠DCE are equal and m∠A = m∠E, the other angles are also same.

Thus, the two triangle are similar. They have the same magnitude of corresponding angles.

Then, we find x. See the picture in the attachment!

3/10 = 1.8/x

x = (10 × 1.8)/3

x = 18/3

x = 6 cm

Hence, they are similar because they have the same corresponding angles. The x has the length of 6 cm.

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The system has no solution only when h = 3 and k  ≠ 20.

The system has unique solution when h ≠ 3 and k is any real number.

System has many solutions only when h = 3 and k = 20

As per the data given:

[tex]& x_1+h x_2=5 \\[/tex]

[tex]& h x_1+12 x_2=k[/tex]

Consider augmented matrix

[tex]\quad[A \mid b]= & {\left[\begin{array}{cc|c}1 & h & 5 \\h & 12 & k\end{array}\right] } \\[/tex]

[tex]R_2 \rightarrow R_2-4 R_1 & {\left[\begin{array}{cc|c}1 & h & 5 \\0 & 12-4 h & k-20 \\\end{array}\right] }\end{aligned}[/tex]

(a) System has no solution if rank(A) ≠ rank(A|b)

Therefore if 12 - hk = 0 and k - 20 ≠ 0 then

rank(A) ≠ rank(A|b) and system has no solution

12 = 4h and k  ≠ 20

h = 3 and k  ≠ 20

Answer: E. The system has no solution only when h = 3 and k  ≠ 20.

(b) The system has unique solution if rank(A) = rank(A|b)

Number of variables = 2

This is possible only when 12 ≠ 4h and h ≠ 3

The system has unique solution when h ≠ 3 and k is any real number.

(c) The system has many solution when rank(A) = rank(A|b) ≤ (Number of variables = 2)

Therefore this is true only when 12 - 4h = 0 and k - 20 = 0 i.e. h = 3 and  k = 20

System has many solutions only when h = 3 and k = 20

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Choose h and k such that the system has (a) no solution, (b) a unique solution, and (c) many solutions.

[tex]x_1+h x_2=5 \\[/tex]

[tex]4 x_1+12 x_2=k[/tex]

a. Select the correct answer below and fill in the answer box(es) to complete your choice. (Type an integer or simplified fraction.)

A. The system has no solutions only when [tex]$k \neq \square$[/tex] and h is any real number.

B. The system has no solutions only when h = [tex]\square$[/tex] and k=

C. The system has no solutions only when h [tex]\neq \square$[/tex] and [tex]$k \neq$[/tex]

D. The system has no solutions only when [tex]$\mathrm{h}=\square$[/tex] and k is any real number.

E. The system has no solutions only when [tex]$h=\square$[/tex] and [tex]$k \neq$[/tex]

F. The system has no solutions only when [tex]$k=\square$[/tex] and h is any real number.

G. The system has no solutions only when [tex]$h \neq \square$[/tex] and k is any real number.

H. The system has no solutions only when [tex]$h \neq \square$[/tex] and k= ___

Answer: 12

Step-by-step explanation:

The probability that the couple will have 3 normal girls is approximately 0.422

Since both parents are normal, they both have to be heterozygous carriers for the recessive allele that causes albinism. We can represent the normal allele with the letter N and the albino allele with the letter n. Then, we can write the genotypes of the parents as Nn x Nn.

The daughter is albino, so her genotype must be nn. The son is normal, so his genotype must be Nn.

We want to find the probability that the couple will have 3 normal girls. We can use the multiplication rule of probability to find this probability:

P(3 normal girls) = P(normal girl) x P(normal girl) x P(normal girl)

The probability of having a normal girl is 3/4 since there are three possible genotypes that result in a normal phenotype (NN, Nn, Nn) out of a total of four possible genotypes. The probability of having a normal boy is also 3/4, for the same reason.

Therefore, the probability of having 3 normal children (in this case, girls) is:

P(3 normal girls) = (3/4) x (3/4) x (3/4) = 27/64 ≈ 0.422

So, the probability that the couple will have 3 normal girls is approximately 0.422, or about 42.2%.

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The vectors that are parallel to z that has has initial point (5, 4, 1) and terminal point (-2, -4, 4) are -7i - 6j - 2k and -14i - 16j + 6k. So, the correct options are B and D.

To determine which vectors are parallel to z, we need to compare their direction to the direction of z. Two vectors are parallel if they have the same direction or are opposite in direction.

The direction of z can be found by subtracting the initial point from the terminal point:

z = (-2 - 5)i + (-4 - 4)j + (4 - 1)k

z = -7i - 8j + 3k

Now let's compare the direction of each given vector to the direction of z:

a. 7i + 6j + 2k: This vector is not parallel to z because it has a different direction.

b. -7i- 6j -2k: This vector is parallel to z because it has the opposite direction.

c. 14i +16j+ 6k: This vector is not parallel to z because it has a different direction.

d. -14i -16j + 6k: This vector is parallel to z because it has the opposite direction.

e. i + 4j - 2k: This vector is not parallel to z because it has a different direction.

f. i - 4j + 2k: This vector is not parallel to z because it has a different direction.

Therefore, the vectors that are parallel to z are b. -7i - 6j - 2k and d. -14i - 16j + 6k.

To confirm these results using a graphing utility, we can plot z and each of the given vectors on a 3D coordinate system and visually check their directions.

Here is an example using Wolfram Alpha:

vectorplot3d {<-7, -8, 3>, {x, -10, 10}, {y, -10, 10}, {z, -10, 10}}

vectorplot3d {<7, 6, 2>, {x, -10, 10}, {y, -10, 10}, {z, -10, 10}}

vectorplot3d {<-7, -6, -2>, {x, -10, 10}, {y, -10, 10}, {z, -10, 10}}

vectorplot3d {<14, 16, 6>, {x, -10, 10}, {y, -10, 10}, {z, -10, 10}}

vectorplot3d {<-14, -16, 6>, {x, -10, 10}, {y, -10, 10}, {z, -10, 10}}

vectorplot3d {<1, 4, -2>, {x, -10, 10}, {y, -10, 10}, {z, -10, 10}}

vectorplot3d {<1, -4, 2>, {x, -10, 10}, {y, -10, 10}, {z, -10, 10}}

This will generate a 3D plot with all the vectors plotted. We can see that only the vectors with opposite directions to z, namely -7i - 6j - 2k and -14i - 16j + 6k, are parallel to z.

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The area of the trapezoid is given as follows:

160 cm².

The area of a trapezoid is obtained adding the bases, multiplying by the height, and dividing by 2.

For the trapezoid in this problem, the parameters are given as follows:

Hence the area is given as follows:

A = 0.5 x (8 + 24) x 10

A = 160 cm².

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The time to travel 279 miles is 4.5 hours

We can use the formula:

time = distance ÷ speed

To find the time it will take to travel 279 miles, given that it takes 6 hours to travel 372 miles.

The speed is constant, so we can find it by dividing the distance by the time for the first trip:

speed = distance ÷ time = 372 miles ÷ 6 hours = 62 miles/hour

Now we can use the speed to find the time it will take to travel 279 miles:

time = distance ÷ speed = 279 miles ÷ 62 miles/hour ≈ 4.5 hours

Hence, it will take approximately 4.5 hours to travel 279 miles at this speed.

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The quotient is x² - 3 - 4 and remainder is 1.

So, The value of f (3) is, 1

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

The value of f(3) is told by the remainder theorem to be the remainder from division of f(x) by (x -3). That division is shown in the attachment. The remainder is 1, hence we have;

⇒ f(3) = 1

The number at upper left in the synthetic division table is the value of x for which we are evaluating f(x).

When the polynomial is written in Horner Form:

⇒ f(x) = ((x -6)x +13)x -11

Here, the coefficient of x at each point in the evaluation is the same as the number on the bottom row of the synthetic division. The result of the evaluation is the same as the remainder from the division.

Now, Add the obtained result to the next coefficient of the dividend and write down the sum as;

3 | 1  - 6  13         - 11

         3  - 9        3x4 = 12

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

   1  - 3    4    (- 11)  + 12 = 1

Thus, We have complete table and have obtained the resulting coefficient are,

1 , - 3, 4, 1

Thus, The quotient is x² - 3 - 4 and remainder is 1.

So, The value of f (3) is, 1

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The solution to the system of equation y = -2x + 3 and y = 5x - 4 is (1,1).

Given the system of equation in the question;

y = -2x + 3

y = 5x - 4

To solve the system of equation, plug equation one to into equation two and solve for x.

y = 5x - 4

Plug in y = -2x + 3

-2x + 3 = 5x - 4

Solve for x

-2x - 5x = - 4 - 3

-7x = -7

x = -7 / -7

x = 1

Now, plug x = 1 into equation one and solve for x.

y = -2x + 3

Plug in x = 1

y = -2( 1 ) + 3

y = -2 + 3

y = 1

Therefore, the value of x is 1 and y is also 1.

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37. A possible solution is y(x) = ax + b, where a and b are constants. Taking two derivatives gives y''(x) = 0, which satisfies the differential equation.

38. A possible solution is y(x) = Ce^x, where C is a constant. Taking the derivative of y(x) gives y'(x) = Ce^x, which satisfies the differential equation.

39. A possible solution is y(x) = 3 + 2x^2, which can be checked by taking the derivative and plugging it into the differential equation: xy'(x) + y(x) = x(4x) + (3 + 2x^2) = 3x^2 + 2x^2 + 3 = 5x^2 + 3, which is equal to 3x^2 when x=0.

40. A possible solution is y(x) = sin(x), which can be checked by taking the derivative and plugging it into the differential equation: (y'(x))^2 + (y(x))^2 = cos^2(x) + sin^2(x) = 1.

41. A possible solution is y(x) = ex - 1, which can be checked by taking the derivative and plugging it into the differential equation: y'(x) + y(x) = e^x + e^x - 1 = 2e^x - 1.

42. A possible solution is y(x) = Asin(x) + Bcos(x), where A and B are constants. Taking the second derivative gives y''(x) = -Asin(x) - Bcos(x), which satisfies the differential equation.

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Answer:

Charlotte's mix has a bluer shade of purple

Step-by-step explanation:

Since we are asked whose paint is a bluer shade it would be easier to express the ratios of the paint mix as blue:red

Charlotte: 1 cup red for every 2 cups blue ==> 2 cups blue for every 1 cup red

This can be expressed as the ratio
[tex]\dfrac{ 2 \;blue}{1\;red} = \dfrac{2}{1} = 2[/tex]

[tex]-------------------[/tex]

Pablo : 2 cups red for every 3 cups blue ==> 3 cups blue for every 2 cups red

The ratio of blue to red
= [tex]\dfrac{3 \;blue}{2\;red} = \dfrac{3}{2} = 1\dfrac{1}{2} \;\; (or\; 1.5)[/tex]

Since
[tex]2 > 1\dfrac{1}{2} \;\;\;(or\; 2 > 1.5)[/tex]  

Charlotte's mix has a bluer shade of purple

Answer:

3150

Step-by-step explanation:

...2(42x11+21x11+42x21)

BTW: This isn´t college-level math this is like 2nd-grade math.

The measure of XZ is equivalent to 24. Option C is correct

The given triangles in question are similar in nature. For every similar triangles, the ratio of the similar sides of the triangle is equal to a constant as shown:

k+3/36 = k+5/44

Cross multiply and expand

36(k + 5) = 44(k + 3)

36k + 180 = 44k + 132

36k - 44k = 132-180|

-8k = -48

k = 6

For the measure of XZ

k/XZ = k+5/44

6/XZ = 11/44
6/XZ = 1/4
XZ = 24

Hence the measure of XZ from the given diagram is equivalent to 24

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There is a probability of 98.1% or around 0.98 of making at least one shot.

We can solve this problem using the binomial distribution. Let X be the number of shots the basketball player makes out of 6 attempts. Then X follows a binomial distribution with parameters n = 6 (number of trials) and p = 0.553 (probability of success).

The probability of making at least one shot can be calculated as the complement of the probability of missing all six shots. That is:

P(X ≥ 1) = 1 - P(X = 0)

Using the binomial probability formula, we can calculate P(X = 0) as:

P(X = 0) = (6 choose 0) * (0.553)^0 * (1 - 0.553)^(6-0) = 0.019

where (6 choose 0) = 1 is the number of ways to choose 0 shots out of 6.

Therefore,

P(X ≥ 1) = 1 - P(X = 0) = 1 - 0.019 = 0.981

So the probability of making at least one shot is approximately 0.981 or 98.1%.

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The value of the function f(x) = 3x^2 - 2x + 1 at x = -1, will be 6

In mathematics, an expression is a phrase that has at least two numbers or variables and at least one arithmetic operation. Addition, subtraction, multiplication, or division are all examples of math operations. An expression's structure is as follows: (Number/variable, Math Operator, Number/variable) is an expression.

Algebraic expressions are expressions that are made up of variables and constants. Any value can be assigned to a variable. The value of an expression changes depending on the values assigned to the variables it includes. There are an unlimited number of points on a number line.

To evaluate the function f(x) = 3x^2 - 2x + 1 at x = -1,

we simply substitute -1 for x in the expression:

f(-1) = 3(-1)^2 - 2(-1) + 1

= 3(1) + 2 + 1

= 6

Therefore, f(-1) = 6.

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Answer:

[tex]\huge\boxed{\sf x\³ - 10x\² + 19x + 6}[/tex]

Step-by-step explanation:

= (x - 3)(x² - 7x - 2)

= x(x² - 7x - 2) - 3(x² - 7x - 2)

= x³ - 7x² - 2x - 3x² + 21x + 6

= x³ - 7x² - 3x² - 2x + 21x + 6

[tex]\rule[225]{225}{2}[/tex]

The two numbers of the form a+bi and c+di where a,b,c, and d are real numbers, and i is the imaginary unit, always resulting in the correct mathematical result in the same form.

This is because complex numbers are a mathematical construct and the operations defined on them, such as addition, subtraction, multiplication, and division, follow the rules of arithmetic. These operations preserve the form of the numbers, so that the result of any two complex numbers, when operated upon, will always be in the form of a complex number.

For example, when adding two complex numbers (a + bi) and (c + di), the real parts add up to form a new real part, and the imaginary parts add up to form a new imaginary part, resulting in a new complex number in the form (a + c) + (b + d)i.

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The solution is, the value of x=4.

Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a2 + b2 = c2.

here, we have,

We can modify the Pythagorean Theorem to fit our needs and solve x. We know our c and our a so to find x or b

we use c^2-a^2=b^2.

from the given figure , we get,

Lets input our known.

5^2-3^2=x^2.

Lets start solving by squaring.

25-9=x^2

Subtract

16=x^2

Take the square root

4=x

Hence, The solution is, the value of x=4.

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As a result of answering the question, we may state that equation In ten attempts, there is a 0.0407 percent chance of getting more than four kings, which is a very little chance.

A mathematical equation is a process that links two statements and indicates equality using the equals symbol (=). A theoretical statement that proves the equality of two formulas is known as an equation in algebra. For instance, the equal sign separates the numbers 3x + 5 and 14 in the equation 3x + 5 = 14.

A. With n=10 and p=4/52, the binomial distribution of the number of kings seen in the 10 trials, K, is followed (since there are 4 kings in a standard deck of 52 cards). K's average is:

mean(K) is equal to[tex]n * p (10 * 4/52 = 0.7692).[/tex]

Meaning: On average, we anticipate seeing 0.77 kings per 10 trials.

B. This is the variance of K:

The formula for var(K) is [tex]n * p * (1 - p) = 10 * 4/52 * (1 - 4/52) = 0.7255.[/tex]

K's standard deviation:

[tex]\sqrt(var(K)) = \sqrt(0.7255) = 0.8524[/tex] where sd(K) =

D. The binomial distribution can be used to calculate the likelihood of receiving more than four kings:

P(K > 4) = 1 - P(K 4) = 1 - 0.0407 for binom.cdf(4, 10, 4/52).

Interpretation: In ten attempts, there is a 0.0407 percent chance of getting more than four kings, which is a very little chance.

D. The binomial distribution can be used to get P(K 3):

Binom.cdf(3, 10, 4/52) = P(K 3) = 0.8789

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In geometry, ​8 ft. 3 ft. specifies the dimension of an object. That is the  length and the width.

8 ft. 3 ft. is a dimension, specifically a length and a width. It could refer to the size of a rectangular object, such as a room or a piece of furniture.

In geometry, such a dimension is often used to calculate the area of the rectangle.

The area of a rectangle can be found by multiplying its length by its width.

So, in this case, the area of the rectangle with dimensions 8 ft. * 3 ft. would be 8 ft * 3 ft = 24 ft².

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The relationship between X and Y in the equations y = 2x and y = x + 2 is that they represent two different linear equations.

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

y = 2x

y = x + 2

Using the above as a guide, we have the following:

Hence, the relationship is different linear equations

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