Simplify the following expression. 3 11/5 go in to 3/-9/5

Answer:

n = 2

Step-by-step explanation:

[tex]\frac{1}{2}[/tex] (n - 4) - 3 = 3 - (2n + 3) ← distribute parenthesis on both sides

[tex]\frac{1}{2}[/tex] n - 2 - 3 = 3 - 2n - 3 ( simplify both sides )

[tex]\frac{1}{2}[/tex] n - 5 = - 2n ( multiply through by 2 to clear the fraction )

n - 10 = - 4n ( add 4n to both sides )

5n - 10 = 0 ( add 10 to both sides )

5n = 10 ( divide both sides by 5 )

n = 2

The exponential function giving the number of bacteria after x hours is given as follows:

[tex]y = 250(2)^x[/tex]

An exponential function has the definition presented according to the equation as follows:

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

In which the parameters are given as follows:

The parameter values for this problem are given as follows:

a = 250, b = 2.

Hence the function is given as follows:

[tex]y = 250(2)^x[/tex]

The problem asks for the exponential function giving the number of bacteria after x hours is given as follows:

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Step-by-step explanation:

The inscribed angle MPN intercepts twice as many degrees of arc as its measure

 so  MN = 62 degrees

   the lower NP is 180 degrees

      the remainder of the 360 degree circle is MP

               360 - 180 - 62 = MP = 118 degrees

Answer:

[tex]m\overset\frown{MP} =118^{\circ}[/tex]

Step-by-step explanation:

The diagram shows a circle with an inscribed angle NPM and an intercepted arc NM.

To find the measure of arc MP, we first need to find the measure of the intercepted arc NM.

According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the intercepted arc. Therefore:

[tex]m \angle NPM = \dfrac{1}{2} \overset\frown{NM}[/tex]

         [tex]31^{\circ} = \dfrac{1}{2} \overset\frown{NM}[/tex]

       [tex]\overset\frown{NM}=62^{\circ}[/tex]

The minor arcs in a semicircle sum to 180°. Therefore:

[tex]\overset\frown{MP} + \overset\frown{NM} = 180^{\circ}[/tex]

Substitute the found measure of arc MN into the equation:

[tex]\overset\frown{MP} +62^{\circ} = 180^{\circ}[/tex]

[tex]\overset\frown{MP} +62^{\circ} -62^{\circ}= 180^{\circ}-62^{\circ}[/tex]

[tex]\overset\frown{MP} =118^{\circ}[/tex]

Therefore, the measure of arc MP is 118°.

[tex]\hrulefill[/tex]

Additional information

Answer:

The equation QS = 32 is false.

Step-by-step explanation:

In a right triangle, the Pythagorean theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Given:

QT = 16

RT = 16

QS = 32

We can apply the Pythagorean theorem to the right triangle:

RT^2 + QT^2 = QS^2

Substituting the given values:

16^2 + 16^2 = 32^2

Calculating:

256 + 256 = 1024

However, we find that 512 (the sum of the squares of the other two sides) does not equal 1024 (the square of the hypotenuse). This contradicts the Pythagorean theorem, which means the equation QS = 32 is false.

An example of a data set where the mean is equal to one of the numbers in the set is 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, with a mean of 11.

Here's an example of a data set where the mean is equal to one of the numbers in the set:

Data set: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

In this data set, the mean (average) value is calculated by summing up all the numbers in the set and dividing by the total number of values. In this case, the sum of the numbers is 110, and since there are 10 numbers in the set, the mean is 110/10 = 11.

As we can see, the number 11 is present in the data set itself and coincidentally, it is also the mean value of the set. This happens because the other numbers are symmetrically distributed around the mean, balancing out to yield the same value.

It's important to note that this is just one example, and there can be various data sets where the mean matches one of the numbers. The occurrence of such a scenario depends on the values within the data set and their distribution.

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Question: Data set [2, 4, 6, 8, 10, 12, 14, 16, 18, 20]

Consider the data set provided above. Is there any number in the data set that is equal to the mean of the data set?

For the following equations:

1) In general, the solutions can be expressed as θ = π/6 + 2πn or θ = 5π/6 + 2πn, where n is an integer.

2) The solutions within the interval (π/2) ≤ θ < 3π can be represented as θ = 7π/6 or θ = 11π/6, both in terms of π.

To solve the equation cscθ = 2, we need to find the values of θ that satisfy the equation.

1) General Form for All Solutions:

The reciprocal of sine is cosecant (csc), so we can rewrite the equation as 1/sinθ = 2. To solve for θ, we can take the reciprocal of both sides:

sinθ = 1/2

Now, we need to determine the values of θ where the sine function equals 1/2. The sine function is positive in the first and second quadrants, so we'll focus on those quadrants.

In the first quadrant (0 ≤ θ < π), the reference angle with a sine of 1/2 is π/6.

In the second quadrant (π < θ < 2π), the reference angle with a sine of 1/2 is also π/6.

To account for all solutions, we can add multiples of the period of sine (2π) to the reference angles. Therefore, the general form for all solutions is:

θ = π/6 + 2πn  or  θ = 5π/6 + 2πn

where n is an integer representing the number of periods of sine added.

2) Solutions on the Interval (π/2) ≤ θ < 3π in Terms of π:

For the given interval, we need to find the values of θ that satisfy the equation and lie within the interval (π/2) ≤ θ < 3π.

From the general form, we can see that the solutions that satisfy the interval are:

θ = π/6 + 2π  or  θ = 5π/6 + 2π

Simplifying these expressions gives us:

θ = 7π/6  or  θ = 11π/6

Therefore, the solutions on the interval (π/2) ≤ θ < 3π in terms of π are:

θ = 7π/6π or θ = 11π/6π.

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The trapezoid ABCD have the measure of angles m∠A and m∠B equal to 79° and 66° respectively.

The sum of the interior angles of a quadrilateral is equal to 360°, so the sum of angles, m∠A, m∠B, m∠C, and m∠D is equal to 360°.

2x - 19 + x + 17 + 3x + 7 + 2x - 37 = 360°

8x - 32 = 360°

8x = 360° + 32°

8x = 392

x = 392/8 {divide through by 8}

x = 49

m∠A = 2(49) - 19 = 79°

m∠A = 49 + 17 = 66°

Therefore, the trapezoid ABCD have the measure of angles m∠A and m∠B equal to 79° and 66° respectively.

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A lap around the inside edge of lane 2 is 2.0 meters farther than a lap around the inside edge of lane 1.

To find the difference in distance between a lap around the inside edge of lane 2 and lane 1, we can compare their circumferences. The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius.

1. First, let's calculate the radius of lane 1. Since the width of lane 1 is given as 1.0 m, we can deduce that the radius of lane 1 is half of this width, which is 0.5 m.

2. Using the radius of lane 1, we can calculate its circumference. Plugging the radius (0.5 m) into the circumference formula, we get C1 = 2π(0.5) = π meters.

3. Next, let's calculate the radius of lane 2. Since lane 2 is not given a specific width, we need additional information to determine its radius.

4. Assuming that lane 2 has the same center as lane 1, we can calculate its radius by adding the width of lane 1 to the radius of lane 1. This gives us a radius of 0.5 + 1.0 = 1.5 m for lane 2.

5. Now, using the radius of lane 2, we can calculate its circumference. Plugging the radius (1.5 m) into the circumference formula, we get C2 = 2π(1.5) = 3π meters.

6. Finally, to find the difference in distance between a lap around lane 2 and lane 1, we subtract the circumference of lane 1 from the circumference of lane 2: ΔC = C2 - C1 = 3π - π = 2π meters.

Therefore, a lap around the inside edge of lane 2 is 2π (approximately 6.28) meters farther than a lap around the inside edge of lane 1.

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g'(x) = 21(7x¹ + 6)² * (4x³ + 1)⁵ + (7x¹ + 6)³ * 60x²(4x³ + 1)⁴

Hence, g'(x) is the derivative of the given function g(x) and can be represented by the expression above.

To find the derivative of the function g(x) = (7x¹ + 6)³ (4x³ + 1)⁵, we can apply the product rule and the chain rule.

Let's start by applying the product rule. If we have two functions u(x) and v(x), the derivative of their product is given by:

(d/dx) [u(x) v(x)] = u'(x) v(x) + u(x) v'(x)

For our function g(x) = (7x¹ + 6)³ (4x³ + 1)⁵, we can consider u(x) = (7x¹ + 6)³ and v(x) = (4x³ + 1)⁵.

Now, let's find the derivatives of u(x) and v(x):

u'(x) = 3(7x¹ + 6)² * (7) = 21(7x¹ + 6)²

v'(x) = 5(4x³ + 1)⁴ * (12x²) = 60x²(4x³ + 1)

Now, we can apply the product rule to find g'(x):

g'(x) = u'(x) v(x) + u(x) v'(x)

= (21(7x¹ + 6)²) * (4x³ + 1)⁵ + (7x¹ + 6)³ * (60x²(4x³ + 1)⁴)

Simplifying further, we can expand the expressions:

g'(x) = 21(7x¹ + 6)² * (4x³ + 1)⁵ + (7x¹ + 6)³ * 60x²(4x³ + 1)⁴

Hence, g'(x) is the derivative of the given function g(x) and can be represented by the expression above.

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The expression "dom(s) = {i : i < n} for some n ∈ N" represents the domain of a set or function called 's'.

Let's break down the notation:

- "dom(s)" refers to the domain of the set or function 's'. The domain represents the set of all possible input values or elements for which 's' is defined or applicable.

- "{i : i < n}" is a set comprehension notation. It means that the set contains elements 'i' such that 'i' is less than 'n'. In other words, it represents a set of all natural numbers 'i' that are smaller than a specific value 'n'.

- "for some n ∈ N" denotes that there exists a natural number 'n' (n ∈ N) such that the set 'dom(s)' consists of elements 'i' that are less than 'n'.

In summary, the expression states that the domain of set or function 's' consists of all natural numbers 'i' that are smaller than a specific natural number 'n'. The actual value of 'n' is not specified and can vary depending on the specific context or problem.

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The volume of the oblique cone in this problem is given as follows:

D. 100.5 in³.

The volume of a cone of radius r and height h is given by the equation presented as follows:

V = πr²h/3.

Applying the Pythagorean Theorem, the diameter of the cone is obtained as follows:

d² + 6² = 10²

d² = 100 - 36

d² = 64

d = 8.

The radius is half the diameter, hence it's measure is given as follows:

r = 4 in.

The height of the cone is given as follows:

h = 6 in.

Hence the volume is given as follows:

V = π x 4² x 8/3

V = 100.5 in³ (rounded down).

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The relationship shown in the diagram can be described as follows: For each input value, there is a corresponding output value. The output value is obtained by performing certain operations on the input value according to the rules specified in the diagram.

2.2.1: The relationship shown in the diagram represents a function where each input value corresponds to a specific output value. The diagram may include various operations or rules to transform the input values into their respective output values.

2.2.2: Using the information provided in the flow diagram, we can complete the table as follows:

  - For input 0, the output is 1.

  - For input 1, the output is 2.

  - For input 2, the output is 4.

  - For input 4, the output is LO.

  - For input 5, the output is 182.

  - For input 182, the output is 2.

  - For input 2, the output is 4.

  - For input -1, the output is 12-10.

  - For input 12-10, the output is 2.

2.2.3: To calculate the input value for the given output value of -21, we follow these steps:

  - Start with the output value -21.

  - Reverse the operations or rules specified in the diagram to transform the output back into the input.

  - Apply the reverse operations in the opposite order to obtain the input value.

Please note that without a specific diagram or additional information, it is challenging to provide precise steps for reversing the operations or rules. The steps may vary depending on the complexity and specifics of the diagram.

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To prove that ABC = DEF by AAS, then (b) A = D

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

The triangles ABC and DEF

The AAS theorem states that "If one side in one triangle is proportional to one side in another triangle and two corresponding angles in both are congruent, then the two triangles are similar"

From the figure, we have

E = B = 50 degrees

AB = DE = 10

So, another pair of angles must be congruent

In this case, the pair are (b) A = D

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Answer:

[tex]BC=5.1[/tex]

[tex]B=23^{\circ}[/tex]

[tex]C=116^{\circ}[/tex]

Step-by-step explanation:

The diagram shows triangle ABC, with two side measures and the included angle.

To find the measure of the third side, we can use the Law of Cosines.

[tex]\boxed{\begin{minipage}{6 cm}\underline{Law of Cosines} \\\\$c^2=a^2+b^2-2ab \cos C$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}[/tex]

In this case, A is the angle, and BC is the side opposite angle A, so:

[tex]BC^2=AB^2+AC^2-2(AB)(AC) \cos A[/tex]

Substitute the given side lengths and angle in the formula, and solve for BC:

[tex]BC^2=7^2+3^2-2(7)(3) \cos 41^{\circ}[/tex]

[tex]BC^2=49+9-2(7)(3) \cos 41^{\circ}[/tex]

[tex]BC^2=49+9-42\cos 41^{\circ}[/tex]

[tex]BC^2=58-42\cos 41^{\circ}[/tex]

[tex]BC=\sqrt{58-42\cos 41^{\circ}}[/tex]

[tex]BC=5.12856682...[/tex]

[tex]BC=5.1\; \sf (nearest\;tenth)[/tex]

Now we have the length of all three sides of the triangle and one of the interior angles, we can use the Law of Sines to find the measures of angles B and C.

[tex]\boxed{\begin{minipage}{7.6 cm}\underline{Law of Sines} \\\\$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c} $\\\\\\where:\\ \phantom{ww}$\bullet$ $A, B$ and $C$ are the angles. \\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides opposite the angles.\\\end{minipage}}[/tex]

In this case, side BC is opposite angle A, side AC is opposite angle B, and side AB is opposite angle C. Therefore:

[tex]\dfrac{\sin A}{BC}=\dfrac{\sin B}{AC}=\dfrac{\sin C}{AB}[/tex]

Substitute the values of the sides and angle A into the formula and solve for the remaining angles.

[tex]\dfrac{\sin 41^{\circ}}{5.12856682...}=\dfrac{\sin B}{3}=\dfrac{\sin C}{7}[/tex]

Therefore:

[tex]\dfrac{\sin B}{3}=\dfrac{\sin 41^{\circ}}{5.12856682...}[/tex]

[tex]\sin B=\dfrac{3\sin 41^{\circ}}{5.12856682...}[/tex]

[tex]B=\sin^{-1}\left(\dfrac{3\sin 41^{\circ}}{5.12856682...}\right)[/tex]

[tex]B=22.5672442...^{\circ}[/tex]

[tex]B=23^{\circ}[/tex]

From the diagram, we can see that angle C is obtuse (it measures more than 90° but less than 180°). Therefore, we need to use sin(180° - C):

[tex]\dfrac{\sin (180^{\circ}-C)}{7}=\dfrac{\sin 41^{\circ}}{5.12856682...}[/tex]

[tex]\sin (180^{\circ}-C)=\dfrac{7\sin 41^{\circ}}{5.12856682...}[/tex]

[tex]180^{\circ}-C=\sin^{-1}\left(\dfrac{7\sin 41^{\circ}}{5.12856682...}\right)[/tex]

[tex]180^{\circ}-C=63.5672442...^{\circ}[/tex]

[tex]C=180^{\circ}-63.5672442...^{\circ}[/tex]

[tex]C=116.432755...^{\circ}[/tex]

[tex]C=116^{\circ}[/tex]

[tex]\hrulefill[/tex]

Additional notes:

I have used the exact measure of side BC in my calculations for angles B and C. However, the results will be the same (when rounded to the nearest degree), if you use the rounded measure of BC in your angle calculations.

To obtain the graph of h(x) = -2x - 4, start with the graph of y = x. Stretch vertically by multiplying each x-coordinate by 2. Then reflect it across the x-axis and shift it down by 4 units

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

h(x) = -2x - 4

The parent function is

f(x) = x

First, stretch vertically by a factor of 2

So, we have

f'(x) = 2x

Next, we reflect across the x-axis

So, we have

f''(x) = -2x

Lastly, we shift down by 4 units

So, we have

h(x) = -2x - 4

The graph is attached

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The solution to the quadratic equation is x = 8 only.

The polynomial equation whose highest degree is two is called a quadratic equation or sometimes just quadratics. It is expressed in the form of:

[tex]\bold{ax^2 + bx + c = 0}[/tex]

The above equation is a quadratic equation, and can be solve by either formula method or factorization method or completing the square method.

We will be solving using the factorization method:

[tex]\sf x^2 - 16x + 64 = 0[/tex]

We are going to find two numbers such that its sum is equal to -16 and its  product is 64

The two numbers are: -8 and -8

[tex]\sf -8 + (-8) = -16[/tex]

and [tex]\sf -8(-8)=64[/tex]

We will replace -16x by -8x and -8x

[tex]\sf x^2 - 16x + 64 = 0[/tex]

[tex]\sf x^2 - 8x - 8x + 64 = 0[/tex]

[tex]\sf (x^2 - 8x) (-8x + 64) = 0[/tex]

In the first parenthesis, x is common so we will factor out x while in the second parenthesis -8 is common and it will be factored out.

That is:

[tex]\sf x ( x- 8) -8(x - 8) = 0[/tex]

[tex]\sf (x-8)(x-8) = 0[/tex]

[tex]\sf x -8 =0[/tex]

Add 8 to both sides

[tex]\sf x -8 + 8 = 0 + 8[/tex]

[tex]\rightarrow \bold{x =8}[/tex]

Hence, the solution to the quadratic equation x² - 16x + 64 = 0 is x = 8 only.

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The complete question is

Find the solution(s) to x² - 16x + 64 = 0.

A. x = 8 only

B. x = 8 and x = -8

C. x = 4 and x = 16

D. x = -2 and x = 32​

The solution for f in terms of g is: f = -6g / 5. Out of the answer options provided, none of them exactly match this solution.

To solve for f in the equation 6f + 9g = 3g + f, we can simplify the equation and isolate the variable f.

Starting with the given equation as follows:

6f + 9g = 3g + f

We can combine like terms by subtracting f from both sides and subtracting 3g from both sides:

6f - f = 3g - 9g

Simplifying further we get:

5f = -6g

To solve for f, we divide both sides of the equation by 5:

f = -6g / 5

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Answer:

The diagram illustrates that the two sides with lengths 4 and 3 can never meet to form a vertex, therefore the correct answer is C.

The value of x can be found by rearranging the formula S = 6[tex]x^2[/tex] to x = √(S/6).

1. The formula for the surface area of a cube is given as S = 6[tex]x^2[/tex], where S represents the surface area and x represents the side length.

2. To solve for x, we need to isolate it on one side of the equation.

3. Divide both sides of the equation by 6: S/6 = [tex]x^2[/tex].

4. To eliminate the exponent of 2, take the square root of both sides: √(S/6) = x.

5. Therefore, the value of x is given by x = √(S/6).

6. If you have a specific value for S, you can substitute it into the equation to find the corresponding value of x.

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The Jack's speed was approximately 85.44 km/h, and Bob's speed was approximately 89.44 km/h.

Let's assume that Jack's speed during the race was "x" km/h. Since Bob was driving 4 km/h faster than Jack, his speed was "x + 4" km/h.

When Bob got a flat tire, he had to stop and repair it, which took 30 minutes. During this time, Jack continued to race. Since both of them finished the race in a tie, it means that Bob caught up to Jack after his tire was repaired.

In 30 minutes, Jack traveled a distance of (x/2) km because the time is half of an hour. During the same time, Bob was stationary due to the tire repair.

Once Bob's tire was fixed, he started racing again and caught up to Jack. At this point, both Bob and Jack had covered the same distance.

So, Bob's time to complete the race was the same as Jack's time plus the 30-minute tire repair time.

We can set up the equation:(1000 km) / (x + 4 km/h) = (1000 km - x/2 km) x km/h

Simplifying the equation, we get:1000x = (1000 - x/2)(x + 4)

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Answer: C

Step-by-step explanation:

Point D is at -14 on the number line.

In Mathematics, the midpoint of a line segment with two end points can be calculated by adding each end point on a line segment together and then divide by two (2).

Since E is the midpoint of line segment CD, we can logically deduce the following relationship:

Line segment CD = Line segment C + Line segment D

Midpoint E = (point C + point D)/2

By substituting the given points into the equation above, we have the following:

-3 = (8 + D)/2

-6 = 8 + D

D = -6 - 8

D = -14

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Among the surveyed students, 50% are boys and 50% are girls. Out of the boys, 30% plan to attend the school play. Therefore, the probability that a student surveyed plans to attend the school play given that the student is a boy is 60%. Option C.

To determine the probability that a student surveyed plans to attend the school play given that the student is a boy, we need to examine the data provided in the table.

From the table, we can see that the probability of a student attending the school play is 70% in total, and the probability of not attending is 30% in total.

Out of the total surveyed students, 50% are boys and 50% are girls. Among the boys, 30% plan to attend the school play, while 20% do not plan to attend.

To calculate the probability that a student plans to attend the school play given that the student is a boy, we divide the number of boys attending the school play by the total number of boys:

Probability = (Boys attending) / (Total boys)

Probability = 30% / 50%

Probability = 0.6 or 60%

Therefore, the probability that a student surveyed plans to attend the school play given that the student is a boy is 60%. Option C is correct.

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Step-by-step explanation:

Slope 1/2    point    5,1  

  in point slope form would be

    (y-1) = 1/2 (x-5)

Tom's percentage score is above 75%, which is the passing threshold, we can conclude that Tom has indeed passed the assessment.

To determine if Tom has passed the assessment, we need to calculate his percentage score out of the total marks.

Percentage Score = (Tom's Score / Total Marks) * 100

Given that Tom's score is 98 marks and the total marks are 128:

Percentage Score = (98 / 128) * 100 ≈ 76.5625%

Since Tom's percentage score is above 75%, which is the passing threshold, we can conclude that Tom has indeed passed the assessment.

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Hello!

Answer:

11/12

Step-by-step explanation:

To solve this problem, we would need to find the denominator of the 2 numbers. In this case, it is 12. We will now convert the fractions to twelfths.

2/3 = 8/12

1/4 = 3/12

After we add 8/12 and 3/12, we get our answer 11/12.

Answer:

1,4,9,16,25,36,49,64,81,100

Triangle ABC and ADE are similar triangles

Similar triangles have the same corresponding angle measures and proportional side lengths.

This means that for two triangles to be similar, the corresponding angles must be equal and the ratio of corresponding sides of similar triangles are equal.

It has been shown that angles in ABC and ADE are equal.

To show that the ratio of corresponding sides are equal

6/12 = 8/16 = 10/20

The ratios all give a value of 1/2

Therefore we can say that the triangles ABC and ADE are similar.

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The rate at which the distance between the cars is increasing 2 hours later is 0 mi/h, by using the Pythagorean theorem.

To find the rate at which the distance between the cars is increasing, we can use the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

Let's assume that after 2 hours, the distance traveled by the southbound car is d_south and the distance traveled by the westbound car is d_west.

Since the southbound car travels at a speed of 50 mi/h for 2 hours, we have d_south = 50 mi/h [tex]\times[/tex] 2 h = 100 mi.

Similarly, the westbound car travels at a speed of 20 mi/h for 2 hours, so d_west = 20 mi/h [tex]\times[/tex] 2 h = 40 mi.

Now, we can use the Pythagorean theorem to find the distance between the two cars:

[tex]distance^2 = d_south^2 + d_west^2distance^2 = 100^2 + 40^2distance^2 = 10000 + 1600distance^2 = 11600[/tex]

distance ≈ sqrt(11600) ≈ 107.68 mi

To find the rate at which the distance between the cars is increasing, we differentiate the equation with respect to time:

2 [tex]\times[/tex] distance [tex]\times[/tex] [tex]\(\frac{{d(\text{{distance}})}}{{dt}}\)[/tex] = 2 [tex]\times[/tex] d_south [tex]\times[/tex] [tex]\(\frac{{d(d_{\text{{south}}})}}{{dt}}\)[/tex] + 2 [tex]\times[/tex] d_west [tex]\times[/tex] (d(d_west)/dt)

Since d_south and d_west are constant (no mention of their rates of change), we can simplify the equation to:

2 [tex]\times[/tex] distance [tex]\times[/tex] [tex]\(\frac{{d(\text{{distance}})}}{{dt}}\)[/tex] = 0

Therefore, the rate at which the distance between the cars is increasing 2 hours later is 0 mi/h.

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The standard error represents the average deviation of the sample means from the true population mean. Rounding this value to four decimal places, the standard error of X¯¯¯ is approximtely 0.6500 mpg.

A smaller standard error indicates that the sample means are more likely to be close to the population mean.To calculate the standard error of X¯¯¯, the mean from a random sample of 25 fill-ups by one driver, we can use the formula:

Standard Error (SE) = σ / sqrt(n),

where σ is the standard deviation and n is the sample size.

In this case, the standard deviation (σ) is given as 3.25 mpg, and the sample size (n) is 25.

Plugging in these values into the formula, we have:

SE = 3.25 / sqrt(25).

Calculating the square root of 25, we get:

SE = 3.25 / 5.

Performing the division, we find:

SE ≈ 0.65.

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