Dustin and his friends each have a step counter, and they competed today to see who could
walk the farthest. At the end of the day, Dustin's StepPro step counter says he walked 2;
3
miles. His friends, Bobby and Marco, both have StepperPlus step counters. Bobby's step
counter says he walked 2.3 miles, while Marco's says he walked 2.5 miles.
Who walked the farthest?
Dustin
Submit
Bobby
Marco

The equation of the tangent line to the curve at the point (x1,y1) is

  y - y1 = - 3.54(x - x1)

What is a tangent?

The straight line that just touches the curve at a given location is referred to as the tangent line to a plane curve in geometry.

Given parametric equations,

x(t)=sin(8t)

y(t)=−8cos(−5t)

We now differentiate the above equations with respect to t.

dx/dt = 8cos(8t)

dy/dt = -8 * -5 * -sin(-5t) = -40sin(-5t)

the slope of tangent =  dy/dx of the curve.

dy/dx = (dy/dt)/(dx/dt) = -40sin(-5t) / 8cos(8t) = -5[ sin(-5t)/cos(8t) ]

Equation of a tangent to the curve

y - y1 = dy/dx (x-x1)

When t = (3π/4)

dy/dx = -5[ sin(-5*3π/4) / cos(8* 3π/4) ] = -5[ 0.707/1] = - 3.54

Therefore the equation of the tangent line at point (x1,y1) is

  y - y1 = - 3.54(x - x1)

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The value of x and y are 4.24 units and 4.24 units respectively in the right angled triangle.

An equation is an expression that shows the relationship between two or more numbers and variables.

Trigonometric ratio shows the relationship between the sides and angles of a right angled triangle.

For the triangle shown, using trigonometric ratio:

sin(45) = x / 6

x = 4.24 units

Also:

cos(45) = y / 6

y = 4.24 units

The value of x and y are 4.24 units and 4.24 units respectively

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The questions are:

Is the prize in the first five boxes?

Is the prize in the first two and a half boxes?

Is the prize in the first or second box?

To find the prize with the fewest number of questions, we can use the binary search algorithm.

The algorithm works as follows:

Ask if the prize is in the first five boxes or the last five boxes.

Based on the answer, eliminate half of the boxes from consideration.

Repeat this process until only one box remains.

The fewest number of questions you can ask is log base 2 of the number of boxes, which in this case is log base 2 of 10, or approximately 3 questions.

The questions are:

Is the prize in the first five boxes?

Is the prize in the first two and a half boxes?

Is the prize in the first or second box?

Note that the exact wording of the questions may differ depending on the specific rules of the game show, but the principle remains the same.

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Yes, the vectors v1 = (0,0,-3), v2 = (0,-3, 9), and v3 = (4,-2,-6) do span ℝ3.  This is because any 3 vectors in ℝ3 will span the entire space, meaning that any vector in ℝ3 can be written as a linear combination of these 3 vectors.

To determine if a set of vectors spans a space, we can create a matrix with the vectors as columns and row reduce it to see if there are any free variables. If there are no free variables, the vectors span the space.

The matrix with v1, v2, and v3 as columns is:

| 0 0 4 |

| 0 -3 -2 |

| -3 9 -6 |

Row reducing this matrix gives us:

| 1 0 0 |

| 0 1 0 |

| 0 0 1 |

Since, there are no free variables, the vectors v1, v2, and v3 do span ℝ3.

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the area of the given figure will be 13.8 cm².

What is are of triangle?

The territory included by a triangle's sides is referred to as its area. Depending on the length of the sides and the internal angles, a triangle's area changes from one triangle to another. Square units like m2, cm2, and in2 are used to express the area of a triangle.

The area of triangle = 1/2 (hb)

The sum of all angle of triangle = 180

A triangle with side 6.9 cm extends to form a second right triangle with a height of 4cm.

So the area will be A = 1/2 *6.9 * 4

A = 13.8 cm²

Hence the area of the given figure will be 13.8 cm².

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For the required value of the given complex numbers z₁z₂ = 32(cos 255° + isin 255°).

Complex numbers are those that expand on the idea of real numbers by incorporating an illogical element.

The formula for a complex number is "a + bi," where "a" and "b" are real numbers and "i" is the imaginary unit, which is equal to the square root of -1.

"A" stands for the complex number's real component, and "b" stands for the imaginary component.

We multiply the magnitudes of two complex numbers and add their arguments to determine their product.

Let's compute z₁z₂ with the supplied values.

Given:

z₁ = 8(cos 45° + isin 45°)

z₂ = 4(cos 210° + isin 210°)

We add the arguments and multiply the magnitudes to find z₁z₂:

Magnitude of z₁ = 8

Magnitude of z₂ = 4

Argument of z₁ = 45°

Argument of z₂ = 210°

Now, let's calculate z₁z₂:

Magnitude of z₁z₂ = (Magnitude of z₁) × (Magnitude of z₂) = 8 × 4 = 32

Argument of z₁z₂ = (Argument of z₁) + (Argument of z₂) = 45° + 210° = 255°

Therefore, z₁z₂ = 32(cos 255° + isin 255°).

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The correct question =

Given z₁ = 8(cos 45° + isin 45°) and z₂ = 4(cos 210° + isin 210°), what is z₁z₂?

Answer:

First option: ∠I = ∠K

Step-by-step explanation:

The translation dilates ΔGHI about point G using a positive scale factor to generate ΔGJK.

We can use process of elimination(POE) to get rid of obviously wrong choices

∠I = ∠K is a correct choice since dilation preserves the original shape so the angles two triangles are equal and therefore the first choice appears to be the correct answer

Let's look at the other choices to make sure:

GH is smaller than GJ so GH = 2GJ is obviously false

∠G cannot be equal to ∠K unless it is an isoscles triangle and such information is not provided

HI is less than JK so the last option is also not true

Correct answer:
First option: ∠I = ∠K

The range of the new score distribution will be greater than the range of the original score distribution.

The range of a set of scores is the difference between the highest and the lowest scores in the set. Adding 5 bonus points to the 10 highest scores in the original set will increase the highest scores and therefore increase the range of the new score distribution.

The range of the original score distribution represents the spread of scores between the highest and lowest scores in the original set, and increasing the highest scores will increase the spread of scores between the highest and lowest scores in the new set, resulting in a greater range.

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The simplest form of the function is f(x) = x2. The graph is a parabola often called the basic parabola.

A parabola is an approximately U-shaped, mirror-symmetrical plane curve in mathematics. It corresponds to a number of seemingly unrelated mathematical descriptions, all of which can be shown to define the same curves. A parabola can be described using a point and a line.

Three different parabolas exist. Vertex form, standard form, and intercept form are the three forms. You can access a unique key feature for the graph on each form.

Any point on a parabola is at an equal distance from both the focus, a fixed point, and the directrix, a fixed straight line. A parabola is a U-shaped plane curve.

Solve f{x}=x-2?

f(x)=x-2    x∈R

x≤0

x-2≤2

Range of f(x)=[∞,2)

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Costs that do not change with output are called fixed costs.

Hence the correct option is D.

Fixed costs are the costs that does not change when output gets changed. Fixed cost is also known as overhead cost. For example, insurance, rent, profit, depreciation, and normal cost.

Fixed costs are indirect costs and must be paid irrespective of the level of production, that is, these costs can be incurred even when the level of production is zero. The best example for this is rent of the factory and payment on the lease for the equipment of the factory must be paid even if the level of production is zero.

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The coordinates of the point on the unit circle corresponding to a central angle of 21 degrees are (0.93, 0.36), which is answer choice c.

The point on the unit circle corresponding to a central angle of 21 degrees can be found using the trigonometric functions sine and cosine.

In this case, the x-coordinate is equal to the cosine of the angle and the y-coordinate is equal to the sine of the angle.

Converting degrees to radians, we have:

x = cos(21) = 0.93

y = sin(21) = 0.36

A unit circle in mathematics is a circle with a radius of one, or one unit. The unit circle is frequently the circle with radius 1 that is located at the origin (0, 0) in the Cartesian coordinate system on the Euclidean plane, notably in trigonometry.

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None of the points listed are solutions to the equation y = 12x^3.

The equation y = 12x^3 is a polynomial equation, which describes a curve in a graph. The points listed are just random points and do not satisfy the equation. To find the solutions to the equation, we need to find the x-values that make y equal to the given value.

The equation y = 12x^3 is a cubic equation, which is a polynomial equation with degree 3. It describes a curve in a graph, where the x-values are inputs and the y-values are outputs. To find the solutions to the equation, we need to find the x-values that make y equal to the given value.

For example, if we plug in x = 1 into the equation, we get y = 12 * 1^3 = 12. This means that the point (1, 12) is a solution to the equation y = 12x^3. We can repeat this process for different x-values to find all the solutions.

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The required probability that the hand is a full house is; 0.00144 or 0.144%.

Probability refers to a possibility that deals with the occurrence of random events.

The probability of all the events occurring need to be 1.

Consider the provided information we have Five cards are drawn from an ordinary deck of 52 playing cards.

Thus full house is a hand that consists of two of one kind and three of another kind.

The total number of ways to draw 5 cards are 78 was.

Now we want two of one kind and three of another.

Therefore, the hand has the pattern AAABB, where A and B are from distinct kinds. The number of such hands are:

3744/25989611

= 0.00144

Hence, the required probability is 0.00144 or 0.144%.

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Answer:

Step-by-step explanation:true

Answer:

False

Step-by-step explanation:

To find the interquartile range, compute the positive difference between the first and third quartiles.

8639 is the value of 75 U.S. dollars in Japanese yen.

The term "yen," which refers to Chinese round coins, dates back in time (yuan). The yen was first coined in 1869, following the Meiji Restoration, and was formally recognised as the fundamental unit in the 1871 monetary reform.

"Yen" is Japanese for "circle" or "round thing." With the "New Currency Act" of 1871, the Meiji government formally adopted this currency in an effort to stabilise the nation's monetary position at the time.

The difference between US and Japanese interest rates has been a major factor in the decline of the yen. In an effort to combat the growing cost of living, the US Federal Reserve has aggressively increased its key interest rate from 0.25% to 3.25% since March.

As we see from the table that,

1 U.S dollar = 115.19 Japanese Yen

$1 = 115.19 ¥

so, $75 = 115.19 × 75¥

$75 = 8639.25¥  ≈ 8639¥

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An equation of the question is 250 × 0.94^n = 150 and in approx 8 years it will have about 150 students.

The population of its high school decreases by 6% each year.

The senior class has 250 students now.

Each year decrease in population = 6% = 0.06

So it can be = 1 - 0.06 = 0.94

Total student now = 250

Let n be the number of years before the class will have  students.

So the equation should be = 250 × 0.94^n

After years left students = 150.

Now the equation should be

250 × 0.94^n = 150

Now solving the equation.

Divide by 250 on both side, we get

0.94^n = 150/250

0.94^n = 3/5

Taking log on both side, we get

log(0.94^n) = log(3/5)

Using the formula log(m^n) = n logm and log(m/n) = log m - log n

n log(0.94) = log3 - log5

Now finding the value of log using calculator

n (-0.062) = 1.099 - 1.61

(-0.062)n = -0.511

Divide by (-0.062) on both side, we get

n = 8.242

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The complete question is:

As a town gets smaller, the population of its high school decreases by 6% each year. The senior class has 250 students now. In how many years will it have about 150 students? Write an equation. Then solve the equation without graphing. Write an equation to represent this situation. Let n be the number of years before the class will have  students.

Answer:

see explanation

Step-by-step explanation:

the equation of proportion is of the form

y = kx ← k is the constant of proportionality

(a)

- [tex]\frac{5}{2}[/tex] x = y , or

y = - [tex]\frac{5}{2}[/tex] x ← is in standard form

thus x and y are directly proportional

with constant of proportionality k = - [tex]\frac{5}{2}[/tex]

(b)

y = x ← is in standard form

thus x and y are directly proportional

with constant of proportionality k = 1

The combination of two linear equation (x-1) and (x-3) is equal to zero according to the extended equation (x2 - 4x + 3) = 0.

1. Expand (x-1)²: x² - 2x + 1

2. Multiply the result with (x-3): x² - 2x + 1 * x - 3 = x³ - 5x² + 3x - 3

3. Simplify the equation: x³ - 5x² + 3x - 3 = 0

It is possible to solve the equation (x-1)2(x-3) = 0 by multiplying the two linear equations (x-1) and (x-3). (x-1) can be expanded as x2 - 2x + 1 since it is a square. This value will be obtained by multiplying it by (x-3): x3 - 5x2 + 3x - 3. This indicates that the two linear expressions (x-1) and (x-3) have the same product, which is equal to zero. We can make the extended form equal to zero and then solve for x to solve the equation. As a result, we will have two solutions: x = 1 and x = 3. This indicates that the equation will be true when either x = 1 or x = 3. The product of two linear expressions (x-1) and (x-3) is equal to zero, as shown by the expansion of the equation (x-1)2(x-3) = 0 to (x2 - 4x + 3) = 0. Set the expanded form equal to zero and search for x to solve the equation. This will yield two answers: x = 1 and x = 3.

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The first part of the process of investigating a question using data is C. Collecting the data.

The first ideas regarding the questions to ask guide the data exploration process. It continues with planning, data gathering and exploration, and feature reporting. There are many different ways to explain and summarize the procedure.

Data is information that has been transformed into a cycle that is useful for transfer or processing in computing. Data is information that has been transformed into binary digital form for use with modern computers and communication mediums. The topic of data may be used in either the

Therefore, option C is correct.

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missing options:

A. Describing the data

B. Describing the data

C. Collecting the data

D. Rephrasing the question

Answer:

the answer is 2a and 4b

Step-by-step explanation:

(1) By 1716 different ways can the offices be filled if each person can hold at most one office.

(2) By 132 number of ways Caden can be the chairperson.

(3) By 1320 of the 1716 total ways the offices can be filled,

(1) If each person can hold at most one office, there are 13 people to choose from for the first office, 12 people remaining to choose from for the second office, and 11 people remaining to choose from for the third office.

Therefore, the number of different ways the offices can be filled is:

13 * 12 * 11 = 1716 ways

(2) If Caden is the chairperson, there are 12 people to choose from for the second office and 11 people remaining to choose from for the third office.

Therefore, the number of ways Caden can be the chairperson is:

12 * 11 = 132 ways

(3) If Caden is an officer, there are 12 people to choose from for the first office, 11 people remaining to choose from for the second office, and 10 people remaining to choose from for the third office.

Therefore, the number of ways Caden can be an officer is:

12 * 11 * 10 = 1320 ways

So, in 1320 of the 1716 total ways the offices can be filled, Caden is one of the elected officers.

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The conditional probability that the first and fourth balls drawn will be white given that the sample drawn contains exactly 3 white balls will be  [C(5, 3) * C(2, 2) / C(12, 5)] / (C(12, 3) / C(12, 5))

The conditional probability of the first and fourth balls drawn being white given that the sample drawn contains exactly 3 white balls can be calculated as follows:

Let A be the event that the first and fourth balls drawn are white and B be the event that the sample drawn contains exactly 3 white balls. We want to find P(A|B), the probability of A given B.

Using the formula for conditional probability:

P(A|B) = P(A and B) / P(B)

To calculate P(A and B), we can use the formula for the number of combinations:

C(n, k) = n! / (k! (n - k)!)

Where n is the total number of balls, k is the number of white balls, and ! denotes the factorial symbol.

In this case, n = 5 (the sample size), k = 3 (the number of white balls), and m = 2 (the number of white balls in the first and fourth positions).

P(A and B) = C(5, 3) * C(2, 2) / C(12, 5)

P(B) = C(12, 3) / C(12, 5)

Putting it all together:

P(A|B) = P(A and B) / P(B)

= [C(5, 3) * C(2, 2) / C(12, 5)] / (C(12, 3) / C(12, 5))
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The possible approximate lengths of b are: 2.3 units and 7.8 units

We know that the law of sines for triangle is:

The ratios of the length of all sides of a triangle to the sine of the respective opposite angles are in proportion.

This means, for triangle ABC,

[tex]\frac{sin~ A}{a} =\frac{sin~B}{b} =\frac{sin~ C}{c}[/tex]

where a is the length of side BC,

b is the length of side AC,

c is the length of side AB.

For triangle ABC consider an equation from sine law,

[tex]\frac{sin~ A}{a} =\frac{sin~ C}{c}[/tex]

here, c = 5.4, a = 3.3, and m∠A = 20°

[tex]\frac{sin~ 20}{3.3} =\frac{sin~ C}{5.4}\\\\\frac{0.3420}{3.3} =\frac{sin~ C}{5.4}[/tex]

0.3420 × 5.4 = 3.3 × sin(C)

sin(C) = 0.5596

∠C = arcsin(0.5596)

∠C = 34.03°  OR 145.9°

∠C ≈ 34° OR 146°

We know that the sum of all angles of triangle is 180 degrees.

so, ∠A + ∠B + ∠C = 180°

when m∠C = 34°,

20° + ∠B + 34.03° = 180°

∠B = 125.97°

m∠B = 126°

when m∠C = 146°,

20° + ∠B + 146° = 180°

m∠B = 14°

Now consider equation,

[tex]\frac{sin~ A}{a} =\frac{sin~ B}{b}\\\\\frac{sin~20^{\circ}}{3.3} =\frac{sin~ 126^{\circ}}{b}[/tex]

b × 0.3420 = 0.8090 × 3.3

b = 7.8 units

when m∠B = 14°,

[tex]\frac{sin~20^{\circ}}{3.3} =\frac{sin~ 14^{\circ}}{b}[/tex]

b × 0.3420 = 0.2419 × 3.3

b = 2.3 units

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The complete question is:

Law of sines: StartFraction sine (uppercase A) Over a EndFraction = StartFraction sine (uppercase B) Over b EndFraction = StartFraction sine (uppercase C) Over c EndFraction

In ΔABC, c = 5.4, a = 3.3, and measure of angle A = 20 degrees. What are the possible approximate lengths of b? Use the law of sines to find the answer.

2.0 units and 4.6 units

2.1 units and 8.7 units

2.3 units and 7.8 units

2.6 units and 6.6 units

Calculating the correlation coefficient, which runs from -1 to 1, will reveal how strong a correlation is.

Positive Correlation, A relationship between two variables such that as one variable increases, the other variable also increases.

Negative Correlation, A relationship between two variables such that as one variable increases, the other variable decreases.

No Correlation, A relationship between two variables such that there is no relationship or pattern between the two variables.

The strength of a correlation can be determined by calculating the correlation coefficient, which ranges from -1 to 1. A value of -1 indicates a perfect negative correlation, a value of 1 indicates a perfect positive correlation, and a value of 0 indicates no correlation. The closer the correlation coefficient is to either -1 or 1, the stronger the correlation is.

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Answer:

(a) 6

(b) 49

Step-by-step explanation:

(a) The largest integer which is a factor of 96 and 294 should be their HCF.

Therefore, 96 = 2 × 2 × 2 × 2 × 2 × 3

294 = 2 × 3 × 7 × 7

Therefore, HCF = 2 × 3 = 6

(b) 96n should be the LCM of 96 and 294.

We know, product of two numbers = HCF × LCM

or, LCM = (96 × 294)/6 = 4704

So, 96n = 4704

or n = 49

Hope it helps.

If you have any query, feel free to ask.

Answer: 21.5%

Step-by-step explanation:

this is for number 25.

Answer:

Step-by-step explanation:

Figure 1

Total area = Area of square = 12 * 12 =144

Area of Shaded region =

Area of Square – Area of circle =

144 - pi r^2 = 144 – π 6*6=144-36π = 30.96

(Here the value \pi  of is 3.14)

Probability = Area of Shaded region/Total area = 30.96/144=0.215

Rounded to the nearest 10th the answer is 21.5%

Area of circle = pi{10}^2=100 \pi=314

Area of arc AB = Area of quarter circle – Area of triangle OAB

=( pi{10}^2\ *1/4)-1/2 * 10*10 = 25pi – 50 = 28.5

 Probability = Area of arc/Area of circle = 28.5/314 = 0.090

   Rounded to nearest 10th the answer is 9%

The width of the rectangle is 2 units when length of the rectangle is 10 units.

As per the given data:

The length of a rectangle varies inversely with its width.

The length of the rectangle is 5 units when its width is 4 units

The length of the rectangle when its width is 10 units

Here we have to determine the width of a rectangle.

Let l be the length and w be the width of the rectangle.

Now, from the given direction, we have

[tex]$l=\frac{k}{w}[/tex]

Substitute l = 5 units, w = 4 units, to find the constant k

5 = [tex]$\frac{k}{4}[/tex]

k = 20 units

Now, we have to find the width of the rectangle when l = 10 units

10 = [tex]$\frac{20}{w}[/tex]

w = [tex]$\frac{20}{10}[/tex]

w = 2 units.

Therefore the width of the rectangle is 2 units.

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The length of a rectangle varies inversely with its width. if the length of the rectangle is 5 units when its width is 4 units , find the width of the rectangle when its length is 10 units?

26x - 52y  is the equivalent expression.

2(13a - 5) is the equivalent expression.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

-13 (-2x + 4y)

= -13 x -2x+ -13 x 4y

= 26x - 52y

Now,

26a - 10

Taking out a common factor.

= 2 (13a - 5)

Thus,

26x - 52y and 2(13a - 5) are the equivalent expression.

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The probability of Margo being paired with Irma will be = 1/29

As per the data given,

Total number of students participated in the quiz = 30

Since each student must be paired with exactly one other student, there are 29 possible students that Margo could be paired with.

Since Irma is Margo's best friend, there is only one possibility for Margo's pairing that would result in her being paired with Irma.

The probability of Margo being paired with Irma is 1 out of 29 possible pairings, so the answer is = 1/29

This is the exact probability, expressed as a common fraction.

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