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Statistics Chapter 11: Simulation Practice

Both Roger and Jeff will have run a distance of 1/3 mile when Jeff catches up to Roger after 2 minutes.

To solve this problem, we can determine the relative speeds of Roger and Jeff.

Since Roger runs one mile in 9 minutes, his speed is 1/9 miles per minute.

Similarly, Jeff runs one mile in 6 minutes, so his speed is 1/6 miles per minutes.

Let's assume that Jeff catches up to Roger after t minutes.

In that time, Roger would have run (1/9) [tex]\times[/tex] t miles, and Jeff would have run (1/6) [tex]\times[/tex] t miles.

Since Jeff gives Roger a 1-minute head start, we can express their distances covered as:

Distance covered by Roger = (1/9) [tex]\times[/tex] (t+1) miles

Distance covered by Jeff = (1/6) [tex]\times[/tex] t miles

For Jeff to catch up to Roger, their distances covered must be equal. So we can set up the equation:

(1/9) [tex]\times[/tex] (t+1) = (1/6) [tex]\times[/tex] t

To solve for t, we can cross-multiply and simplify:

6(t+1) = 9t

6t + 6 = 9t

6 = 9t - 6t

6 = 3t

t = 2

Therefore, it will take 2 minutes for Jeff to catch up to Roger.

Substituting t = 2 back into the equations, we can find the distances covered by each:

Distance covered by Roger = (1/9) [tex]\times[/tex] (2+1) = 1/3 mile

Distance covered by Jeff = (1/6) [tex]\times[/tex] 2 = 1/3 mile

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The solution to the system is (a) (4/5, 5, -4, -4)

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

The augmented matrix

Where, we have

[tex]\left[\begin{array}{cccc|c}1&0&0&0&4/5\\0&1&0&0&5\\0&0&1&0&-4\\0&0&0&1&-4\end{array}\right][/tex]

From the above, we have the diagonals to be 1

And other elements to be 0

This means that the equation has been solved

So, we have

a = 4/5, b = 5, c = -4 and d =-4

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The values of x and y are 9√3 and 9. Thus, option C is the answer.

From the figure, we know that angle A = 60°, angle C = 30° and side AC =18 units.

We have to use trigonometric ratios to find the values of x and y.

sin30° = Opposite side/Hypotenuse  

But, we also know that sin30° = 1/2

Substituting the value in the above equation, we get

1/2 = y/18

Thus, the value of y = 18/2 = 9.

Now, sin60° = Opposite Side/ Hypotenuse

sin60° = √3/2

Substituting the value, we get

√3/2 = x/18

Thus, x = 9√3.

Therefore, the values of x and y in the given triangle are 9√3 and 9.

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Step-by-step explanation:

b. To determine the coordinates of the points on the graph, we need to consider the information given.

At the beginning of the second lap, one minute and three seconds had passed. Since each lap is 4.7 kilometers long, we can calculate the distance traveled at the beginning of the lap using the formula:

Distance = (Number of laps - 1) × Length of each lap

For the beginning of the second lap:

Distance = (2 - 1) × 4.7 km = 4.7 km

At the beginning of the third lap, one minute and three seconds had passed. Therefore, the distance traveled at the beginning of the third lap is:

Distance = (3 - 1) × 4.7 km = 9.4 km

The coordinates of the points on the graph would be:

Point A: (4.7 km, 4.7 km)

Point B: (9.4 km, 9.4 km)

On the graph, the x-axis would represent the distance traveled at the beginning of the lap, and the y-axis would represent the distance traveled at the end of the lap.

c. To find the equation of the line between the two points, we can use the slope-intercept form of a linear equation:

y = mx + b

where m is the slope and b is the y-intercept.

Using the coordinates of the two points (4.7 km, 4.7 km) and (9.4 km, 9.4 km), we can calculate the slope:

m = (y2 - y1) / (x2 - x1)

= (9.4 km - 4.7 km) / (9.4 km - 4.7 km)

= 4.7 km / 4.7 km

= 1

The slope of the line is 1.

Since the line passes through the origin (0, 0), the y-intercept (b) is 0.

Therefore, the equation of the line is:

y = 1x + 0

y = x

This line represents a one-to-one relationship between the distance traveled at the beginning of the lap and the distance traveled at the end of the lap.

d. To estimate how long it would take Fernando to finish the entire race if he continues to average the same speed, we need to find the time it takes for one lap.

In the given scenario, Brandon timed Fernando's lap for one minute and three seconds. Since there are 3 laps in total, we can divide the total time by the number of laps to find the average time per lap:

Average time per lap = Total time / Number of laps

= 1 minute and 3 seconds / 3

= 21 seconds

Since each lap is 4.7 kilometers long, we can calculate the speed:

Speed = Distance / Time

= 4.7 km / (1 minute + 21 seconds)

To estimate the time it would take Fernando to finish the entire race, we divide the total race distance by the speed:

Total race time = Total race distance / Speed

= 500 km / (4.7 km / (1 minute + 21 seconds))

To calculate the time, we need to convert 1 minute and 21 seconds to a decimal form:

1 minute + 21 seconds = 1 minute + (21 seconds / 60 seconds)

= 1 minute + 0.35 minutes

= 1.35 minutes

Substituting the values:

Total race time = 500 km / (4.7 km / 1.35 minutes)

= 500 km × (

1.35 minutes / 4.7 km)

= 500 km × 0.28723...

≈ 143.615 minutes

Therefore, it would take approximately 143.615 minutes for Fernando to finish the entire race if he continues to average the same speed.

i) the original distance when the lighthouse is due West of the ship is 7 km

ii) The time in minutes when the lighthouse is due West of the ship is 21 minutes

iii) The distance in km of the ship from the lighthouse when the lighthouse is due West of the ship is 29.97 km

To solve this problem, we'll use the concepts of relative velocity and trigonometry. Let's break down the problem into three parts:

i) Finding the original distance when the lighthouse is due West of the ship:

The ship's velocity is given as 21 km/h at a bearing of 052°. Since the ship observed the lighthouse due North, we know that the angle between the ship's initial heading and the lighthouse is 90°.

To find the distance, we'll consider the ship's velocity in the North direction only. Using trigonometry, we can determine the distance as follows:

Distance = Velocity * Time = 21 km/h * (20 min / 60 min/h) = 7 km (to three significant figures).

ii) Finding the time in minutes when the lighthouse is due West of the ship:

To find the time, we need to consider the change in angle from 052° to 312°. The difference is 260° (312° - 052°), but we need to convert it to radians for calculations. 260° is equal to 260 * π / 180 radians. The ship's velocity in the West direction can be calculated as:

Velocity in West direction = Velocity * cos(angle) = 21 km/h * cos(260 * π / 180) ≈ -19.98 km/h (negative because it's in the opposite direction).

To find the time, we can use the formula:

Time = Distance / Velocity = 7 km / (19.98 km/h) = 0.35 h = 0.35 * 60 min = 21 minutes (to three significant figures).

iii) Finding the distance in km of the ship from the lighthouse when the lighthouse is due West of the ship:

We can use the formula for relative velocity to find the distance:

Relative Velocity = sqrt((Velocity in North direction)² + (Velocity in West direction)²)

Using the values we calculated earlier, we have:

Relative Velocity = sqrt((21 km/h)² + (-19.98 km/h)²) ≈ 29.97 km/h (to three significant figures).

Therefore, the ship is approximately 29.97 km away from the lighthouse when the lighthouse is due West of the ship.

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Answer:

We start with 17 marbles, 4 of which are red. So P(first marble is red) = 4/17. Since the red marble is not replaced, there are now 16 marbles, 3 of which are red. So P(second marble is red) = 3/16.

The correct calculation is

P(red, then red) = 4/17 × 3/16

The equation in terms of x that represents the given relationship is 114 = (1 + 3x) * (Width)

Let's break down the information given:

Height of the rectangular box = 7 ft

Length of the rectangular box = 1 ft longer than thrice the width (x)

Volume of the rectangular box = 798 ft³

To write an equation that represents the given relationship, we need to relate the length, width, and height to the volume.

The volume of a rectangular box is given by the formula: Volume = Length * Width * Height.

Given that the height is 7 ft, we can substitute this value into the equation.

Volume = (Length) * (Width) * (7)

Now, let's focus on the length. It is described as 1 ft longer than thrice the width.

Length = 1 + (3x)

Substituting this value into the equation, we have:

Volume = (1 + (3x)) * (Width) * (7)

Since the volume is given as 798 ft³, we can set up the equation as follows:

798 = (1 + (3x)) * (Width) * 7

Simplifying further, we get:

798 = 7 * (1 + 3x) * (Width)

Dividing both sides of the equation by 7, we have:

114 = (1 + 3x) * (Width)

Therefore, the equation in terms of x that represents the given relationship is:

114 = (1 + 3x) * (Width)

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Answer:

∠1 = 100

∠2 = 110

Step-by-step explanation:

∠1  + 80 = 180(angles on a straight line add to 180)

∠1 = 180 - 80

∠1 = 100

∠2 = 110 (vertically opposite angles are equal)

Answer:

20 dimes

Step-by-step explanation:

1 dime = $0.1

1 quarter = $0.25

Let there be x dimes

Then the number of quarters is 32-x

Also, $5 = 0.1x + 0.25(32-x)

⇒ 5 = 0.1x + 8 - 0.25x

⇒ 0.25x - 0.1x = 8 - 5

⇒ 0.15x = 3

⇒ x = 3/0.15

⇒ x = 20

Answer:

c

Step-by-step explanation:

assume base 10

-logy = x

[tex] \frac{1}{ log(y) } = x[/tex]

log base x y = 5 turns into the format x^ 5 = y

implement that to get c

The most appropriate name for the function that takes the length of a race and returns the time needed to complete it would be "time(length)".

When choosing a name for a function, it is important to consider clarity and readability. The name should accurately describe the purpose of the function and provide a clear indication of what it does.

In this case, the function is expected to take the length of a race as input and return the time needed to complete it as output. Among the given options, "time(length)" is the most suitable choice.

a. length(time): This name suggests that the function takes time as input and returns the length. However, in this scenario, we are interested in finding the time needed to complete the race based on its length, so this option is not the best fit.

b. Time(race): This name implies that the function takes a race as input and returns the time. While it conveys the idea of finding the time, it doesn't explicitly mention that the input is the length of the race, making it less clear.

c. time(length): This option accurately describes the purpose of the function, indicating that it takes the length of the race as input and returns the corresponding time. It is concise, clear, and aligns with the conventional naming conventions for functions.

d. cost(time): This name suggests that the function calculates the cost based on time, which is not relevant to the scenario of finding the time needed to complete a race.

Therefore, "time(length)" is the most suitable and appropriate name for the function.

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A "⊂-maximal element" in set theory refers to an element in a set that cannot be strictly contained within any other element of the set, indicating a maximum extent or boundary within that set.

In the context of set theory and Tarski's notion of finiteness, a "⊂-maximal element" refers to an element within a set that cannot be strictly contained within any other element of the set. Let's break down the meaning of this term.

Consider a set S and a partial order relation ⊆ (subset relation) defined on the power set P(S) of S. A "⊂-maximal element" u of a set A ⊆ S is an element that is not strictly contained within any other element of A with respect to the subset relation. In other words, there is no element v in A such that u is a proper subset of v.

Formally, for any u ∈ A, if there is no v ∈ A such that u ⊂ v, then u is a ⊂-maximal element of A. This means that u is as large as possible within A and cannot be extended by including additional elements.

In the context of T-finite sets, the existence of a ⊂-maximal element in every nonempty subset of the set guarantees that the set has a well-defined structure and does not continue indefinitely without boundaries.

It ensures that there is a definitive maximum element within each subset, which is a key characteristic of finiteness.

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The measure of angle LB is 40°.

To find the measure of angle LB, we can use the fact that the measure of an inscribed angle is half the measure of its intercepted arc.

In this case, we are given that the measure of arc AC is 80°, so we can conclude that the measure of angle LB is half of that.

Since angle LB is an inscribed angle that intercepts arc AC, we can write the equation:

mLB = 1/2 [tex]\times[/tex] mAC

Substituting the given value, we have:

mLB = 1/2 [tex]\times[/tex] 80°

mLB = 40°

Therefore, the measure of angle LB is 40°.

In the context of the corporate logo, angle LB represents a portion of the circular shape of the logo.

By knowing the measure of arc AC, we can determine the measure of angle LB, which helps in accurately representing the logo in terms of angles and proportions.

This information is crucial for design and branding purposes, as it ensures consistency and precision in the presentation of the logo across various media and materials.

Overall, understanding the measures of angles and arcs in the logo design allows for effective communication and replication of the logo's visual elements, ensuring brand recognition and consistency in its representation.

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Answer:

Step-by-step explanation:

The given augmented matrix is already in reduced row echelon form. We can interpret it as a system of equations as follows:

1x + 0y + 0z + (4/5)w = 0

0x + 1y + 0z + 5w = 0

0x + 0y + 1z - 4w = 0

0x + 0y + 0z + w = -4

From the last row, we can see that w = -4. Substituting this value back into the previous rows, we get:

1x + 0y + 0z + (4/5)(-4) = 0

0x + 1y + 0z + 5(-4) = 0

0x + 0y + 1z - 4(-4) = 0

Simplifying these equations, we have:

x - (16/5) = 0

y - 20 = 0

z + 16 = 0

From the second equation, y = 20. From the third equation, z = -16. Substituting these values into the first equation, we get x - (16/5) = 0, which implies x = 16/5.

Therefore, the solution of the system is (16/5, 20, -16, -4).

So, the correct choice is:

A. The system has exactly one solution. The solution is

(16/5, 20, -16, -4)

Using the concept of scale factor, the value of t in polygon s is:  7.2

The amount by which the shape is expanded or contracted is referred to as the scale factor. This is usually utilized when 2D shapes such as circles, triangles, squares and rectangles need to be enlarged. If y = Kx is an equation, K is the scaling factor for x.

From the given image, since Polygon s is a scaled copy of polygon R, then we can say that using the concept of scale factor, we have:

9/t = 12/9.6

t = (9 * 9.6)/12

t = 7.2

Thus, we can be sure that is the value of t of the polygon S using the concept of scale factor

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Answer:

D. 1/2

Step-by-step explanation:

9^x - 1 = 2

9^x = 3

x = log{sub}9 (3)

x = 1/2 (9^(1/2)=3)

The answer is:

3x + 12

Work/explanation:

To simplify this expression, we will use the distributive property:

[tex]\sf{3(x+4)}[/tex]

Distribute the 3:

[tex]\sf{3\cdot x + 3 \cdot 4}[/tex]

[tex]\sf{3x+12}[/tex]

The calculated value o x in the circle is 7

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

The circle

The value of x can be calculated using teh equation of the intersection of chords

So, we have

122 = 1/2(6x + 8 + 194)

Evaluate

244 = 6x + 202

So, we have

6x = 42

Divide by 6

x = 7

Hence, the solution for x is 7

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To solve these problems, we can use Coulomb's Law, which states that the magnitude of the electric field produced by a charged particle is directly proportional to the charge and inversely proportional to the square of the distance from the particle.

a) Without finding the charge of the particle, we can use the inverse square relationship. If the electric field magnitude is 2.0 N/C at a distance of 50 cm, then at half the distance (25 cm), the electric field would be four times stronger. Therefore, the electric field at 25 cm would be 4 * 2.0 N/C = 8.0 N/C.

b) Doubling the charge would result in doubling the electric field magnitude. So, if the electric field at a distance of 50 cm was initially 2.0 N/C, it would become 4.0 N/C after doubling the charge.

c) To determine the magnitude of the particle's charge, we need to use the equation for the electric field:

E = k * (|q| / r^2)

where E is the electric field magnitude, k is the electrostatic constant, |q| is the magnitude of the charge, and r is the distance from the particle.

Using the known values of E = 2.0 N/C and r = 50 cm (or 0.5 m), we can rearrange the equation to solve for |q|:

|q| = E * r^2 / k

Substituting the values and the known value of k, we can calculate the magnitude of the charge.

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The exact values of csc(theta) and cot(theta) are: csc(theta) = 4√7/7

cot(theta) = -3√7/7.

To find the exact values of csc(theta) and cot(theta), given that cos(theta) = -3/4 and theta is an angle in quadrant two, we can use the trigonometric identities and the Pythagorean identity.

We know that cos(theta) = adjacent/hypotenuse, and in quadrant two, the adjacent side is negative. Let's assume the adjacent side is -3 and the hypotenuse is 4. Using the Pythagorean identity, we can find the opposite side:

[tex]opposite^2 = hypotenuse^2 - adjacent^2opposite^2 = 4^2 - (-3)^2opposite^2 = 16 - 9opposite^2 = 7[/tex]

opposite = √7

Now we have the values for the adjacent side, opposite side, and hypotenuse. We can use these values to find the values of the other trigonometric functions:

csc(theta) = hypotenuse/opposite

csc(theta) = 4/√7

To rationalize the denominator, we multiply the numerator and denominator by √7:

csc(theta) = (4/√7) * (√7/√7)

csc(theta) = 4√7/7

cot(theta) = adjacent/opposite

cot(theta) = -3/√7

To rationalize the denominator, we multiply the numerator and denominator by √7:

cot(theta) = (-3/√7) * (√7/√7)

cot(theta) = -3√7/7

Therefore, the exact values of csc(theta) and cot(theta) are:

csc(theta) = 4√7/7

cot(theta) = -3√7/7

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The perimeter of shape C is 30 cm shorter than the total perimeter of A and B.

The perimeter of a polygon is given by the sum of all the lengths of the outer edges of the figure, that is, we must find the length of all the edges of the polygon, and then add these lengths to obtain the perimeter.

Hence the perimeter of each shape is given as follows:

Then the difference is given as follows:

34 + 26 - 30 = 30 cm.

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Answer:

? = 220°

Step-by-step explanation:

the measure of the inscribed angle QRS is half the measure of its intercepted arc QS , then

QS = ? = 2 × QRS = 2 × 110° = 220°

The  system of equation: -x + 7 = -2x³ + 5x² + x - 2 has three solutions

The correct answer is option D.

To solve the system of equations:

-x + 7 = -2x³ + 5x² + x - 2

We need to simplify and rearrange the equation to find its solutions. Let's start by combining like terms:

-x + 7 = -2x³ + 5x² + x - 2

Simplifying the left side:

7 = -2x³ + 5x² - x - 2

Next, let's arrange the equation in descending order of the variable's exponent:

-2x³ + 5x² - x - 2 = 7

Now, let's move all terms to one side of the equation to set it equal to zero:

-2x³ + 5x² - x - 2 - 7 = 0

Simplifying further:

-2x³ + 5x² - x - 9 = 0

To determine the number of solutions, we can analyze the degree of the equation. Since it is a cubic equation, it can have a maximum of three real solutions.

The given system of equations is:

-x + 7 = -2x³ + 5x² + x - 2

To find the solutions, we need to set the equation equal to zero. Let's rearrange the terms:

2. -2x³ + 5x² - x - 9 = 0

Now, we can try to factor or use numerical methods to solve the equation. However, factoring a cubic equation can be complex and time-consuming. In this case, we'll use numerical methods to approximate the solutions.

One common numerical method is the Newton-Raphson method, which involves making an initial guess for the solutions and iterating to converge on a more accurate solution.

Using numerical software or calculators, we can find the approximate solutions of the equation as follows:

x ≈ -1.607, x ≈ 1.279, and x ≈ 3.328

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m∠BCD = 31.57° (approx). Hence, the answer of the angle is 31.57 degrees.

In the given diagram, BD is extended through point D to point E, m∠CDE = (9x - 12)°, m∠BCD = (2x + 3)°, and m∠DBC = (3x + 5)°. We need to find m∠BCD.

       We know that m∠BCD = (2x + 3)°

       Substituting x = 92/7, we get:

       m∠BCD = (2 × 92/7 + 3)° = (184/7 + 3)° = 221/7°

       Therefore, m∠BCD = 31.57° (approx). Hence, the answer is 31.57.

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The area of the figure is 114 square units.

To determine the area of the figure, we need to identify its shape.

From the given dimensions, it appears that we have three rectangular sections.

The first section has a length of 3 units and a width of 8 units, giving us an area of 3 [tex]\times[/tex] 8 = 24 square units.

The second section has a length of 3 units and a width of 9 units, resulting in an area of 3 [tex]\times[/tex] 9 = 27 square units

The third section has a length of 3 units and a width of 21 units, yielding an area of 3 [tex]\times[/tex] 21 = 63 square units.

To find the total area of the figure, we need to sum up the areas of the individual sections:

Total area = 24 + 27 + 63 = 114 square units.

Therefore, the area of the figure is 114 square units.

It's important to note that without a clear description or diagram of the figure, it's challenging to provide an accurate interpretation.

The given dimensions could represent various arrangements, and the resulting area would vary accordingly.

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Answer:

if two parallel lines are cut by a transversal,then, exterior angles on the same side of the transversal are supplementary!

Step-by-step explanation:

I don't really know how to explain it but I hope this helped!