based on your answer(s) to part (a), complete the following statement to represent the length of candle remaining in terms of the burned length b .

We want to reduce x by 12%. Thus, the reduced price is:

x - 0.12x = 0.88x

More informally, if you take away 12%, you're left with 88%.

Answer:

We want to reduce x by 29%. Thus, the reduced price is:

x - 0.29x = 0.71x

More informally, if you take away 29%, you're left with 71%.

The solution of the below trignometric equation are θ= mπ , θ= nπ/3

Given tan θ + tan 2θ = tan 3θ

=> tan θ + tan 2θ - tan 3θ = 0

(tanθ + tan2θ)−(tanθ + tan2θ)/(1 - tanθ tan2θ) =0

or (tanθ + tan2θ)(1 − tanθ tan2θ−1)=0

or tanθ tan2θ(tanθ + tan2θ) = 0

tanθ = 0

∴θ=nπ, tan2θ=0,

∴2θ=nπ or θ=nπ/2

tanθ + tan2θ=0

or sin(θ+2θ)=0

∴θ=nπ/3

​

However, the values of given by =n/2 for odd values of n do not fulfill the preceding equation as it will result in ∞ = ∞

Hence the required solution is:

θ= 2mπ/2=mπ

θ= nπ/3

​

where m and n is an integer

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Complete question:

Find the general solution of the equation tan θ + tan 2θ = tan 3θ

A quadratic graph is a type of parabolic graph that can be represented by the equation y = ax^2 + bx + c, where a, b, and c are constants and x is the variable. The graph of this equation is a parabola that opens either up or down, depending on the value of a.

The equation y = x^2 + 2x - 3 is an example of a quadratic graph. It can be written in the standard form of a quadratic equation as y = x^2 + 2x - 3, where a = 1, b = 2 and c = -3.

The x^2 term in the equation is known as the quadratic term, and it is responsible for the parabolic shape of the graph. The x term and the constant term, 2x and -3, respectively, determine the location of the vertex and the direction of the parabola.

The vertex of the parabola is the point where the parabola changes direction. It can be found by using the formula x = -b/2a, which in this case is x = -2/2 = -1 and y = f(-1) = -1.

The graph of y = x^2 + 2x - 3 is a parabola that opens upwards and the vertex of the parabola is (-1,-1)

In summary, a quadratic graph is a parabolic graph represented by the equation y = ax^2 + bx + c, where a, b, and c are constants. The x^2 term gives the parabolic shape to the graph, the x term and the constant term determine the vertex and the direction of the parabola respectively.

Iron, which is a magnetic substance, makes up most of the core of the Earth, where the magnetic field of the planet first appears. The fluid outer core's movement around the solid inner core causes electric currents, which in turn creates and maintains the Earth's magnetic field.

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The statement '9 less than 10 and 6 times the letter n' in the mathematical form will be 9 < 10 + 6n.

Inequality is defined as an equation that does not contain an equal sign. Inequality is a term that describes a statement's relative size and can be used to compare these two claims.

The act of converting a specified statement into an expression or equation is known as a sentence-to-equation transformation.

The statement is '9 less than 10 and 6 times the letter n'.

Convert the statement into a mathematical form. Then we have

9 < 10 + 6n

Simplify the inequality, then we have

9 < 10 + 6n

6n > - 1

n > - 1/6

The statement '9 less than 10 and 6 times the letter n' in the mathematical form will be 9 < 10 + 6n.

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The correct option is (A). The correct answer is Ursa Minor, the Little Bear, containing the bright star Polaris.

Whereas stars spin or revolve from the northern hemisphere is the center of the sky, also known as the North Celestial Pole. Additionally, it marks the precise location of the Polaris or Northern Star constellation in full view. In addition, this served as a point of reference for the construction of sundials in earlier times.

When people go outside, especially at night, and try to find the Polaris constellation, it is evident that the stars close to this star formation circle in an east-to-west direction, in contrast to the northern star, which makes no movement whatsoever as the evening wears on.

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There are a total of 720 possible gate codes for your apartment complex.

Gate codes are security codes used to control access to a restricted area. Gate codes are used in both residential and commercial settings to ensure the safety of the occupants and to protect the property. Gate codes can be used to unlock and lock a gate, door, or other access point, and typically consist of a combination of letters, numbers, and/or symbols.

This is because each code consists of three non-repeating digits (0-9).

For the first digit, there are 10 possible choices (0-9).

For the second digit, there are 9 possible choices (excluding the number chosen for the first digit).

For the third digit, there are 8 possible choices (excluding the numbers chosen for the first and second digits).

This means that the total number of possible gate codes is 10 x 9 x 8 = 720.

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Answer:

Ex1

[tex] ln( \frac{ {y}^{4} { {e}^{2} } }{ \sqrt[3]{ {x}^{2} } } ) \\ = ln( {y}^{4} {e}^{2} ) - ln( \sqrt[3]{ {x}^{2} } ) \: \: \: \: (by \: quotient \: rue) \\ = ln( {y}^{4} ) + ln( {e}^{2} ) - ln( \sqrt[3]{ {x}^{2} } ) (by \: product \: rule) \\ = 2 + ln( {y}^{4} ) - ln( \sqrt[3]{ {x}^{2} } ) \\ = 2 + 4 ln(y) - \frac{3}{2} ln(x) \: \: \: \: (power \: rule)[/tex]

Ex2

[tex] log_{4}(x + 2) + log_{4}(x - 1) = 1 \\ log_{4}((x + 2)(x - 1)) = 1 \: (by \: product \: rue) \\ log_{4}( {x}^{2} + x - 2 ) = 1 \\ {x}^{2} + x - 2 = {4}^{1} \\ {x }^{2} + x - 6 = 0 \\ x = 3 \: \: or \: x = - 2[/tex]

Answer:

2 + x = 5, x + (negative 4) = 7, Negative 5 + x = negative 2 are equivalent equations.

In the first equation 2+ x = 5, x = 3

In the second equation x + (negative 4) = 7, x = 11

In the third equation Negative 5 + x = negative 2, x = negative 3

Answer:

9 ft, 3 ft 7, ft  

Step-by-step explanation:

Perimeter = 19 ft

Let side 2 = x

Side 1 = 3x

Side 3 = x + 4

perimeter = 3x + x + x + 4 = 19

5x + 4 = 19

5x = 15

x = 3

Side 1: 3x = 3(9) = 9

Side 2: x = 3

Side 3: x + 4 = 3 + 4 = 7

Answer:

9 ft, 3 ft 7, ft  

The amount of money that the worker will earn in a week if he works 4 hours overtime is; $3180

We are given;

Number of work hours per week = 40 hours

Regular rate of work = $15.90 per hour

Overtime rate of work = 1.5(15.90) per hour

Since the regular rate is $15.90 per hour, then for a week, the money earned is; (15.90 * 40)

Now, he also works 4 hours overtime in a week and as such amount earned for overtime in the week = 4 * 1.5(15.90)

Thus;

Total amount earned in the entire week = (15.90 * 40) + (4 * 1.5(15.90))

= $3180

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Answer:

3 times.

Step-by-step explanation:

30, 31, 32, 33, 34, 35, 36, 37, 38, 39,40,41,42,43,44,45,46,47,48,49,50,51,

52,53,54,55,56,57,58,59,60,61,62,63,64,65.

37,47,57.

3 times

Original Unit Price:

    $10.29 / 4.2 kg = $2.45/kg

New Unit Price:

    $9.18 / 3.6 kg = $2.55/kg

The price went up by $0.10/kg.  If we compare that (divide that) by the starting unit price of $2.45, we get:

       $0.10 / $2.45 ≈ 0.040816326530612

                              ≈ 4.0816326530612 %

(You can round as accurately as your teacher needs.)

Line A is parallel to line B

We know, the slope of the line joining two points [tex](x_{1} ,y_{1} )[/tex] , [tex](x_{2} ,y_{2} )[/tex] is:

[tex]m = \frac{y_{2} - y_{1} }{x_{2}- x_{1}}[/tex]

Line A is passing through (2, 10) and (4,13).

The slope of line A

[tex]m_{1} = \frac{y_{2} - y_{1} }{x_{2}- x_{1}} = \frac{13 - 10}{4-2} =\frac{3}{2}[/tex]

Line B is passing through (4,9) and (6, 12)

The slope of line B

[tex]m_{2} = \frac{y_{2} - y_{1} }{x_{2}- x_{1}} = \frac{12 - 9}{6-4} =\frac{3}{2}[/tex]

Line C is passing through (2, 10), and (4,9).

The slope of line C

[tex]m_{3} = \frac{y_{2} - y_{1} }{x_{2}- x_{1}} = \frac{9- 10}{4-2} =\frac{-1}{2}[/tex]

We also know that if the slope of two lines is equal that is [tex]m_{1} = m_{2}[/tex] then the lines are parallel and

if the slope of the two lines has a relation [tex]m_{1} * m_{2} = -1[/tex] , the lines are perpendicular to each other.

Here, since the slope of lines A and lines B are equal.

Hence, line A is parallel to line B.

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The linear model for the cost, [tex]C[/tex], of waste collection as a function of the number of kilograms, [tex]w[/tex] is  [tex]C = 4w +70[/tex] .

Given,

The monthly charge for a waste collection service for 100 kg of waste = $470

so, [tex](x_{1} , y_{1} ) = (100, 470)[/tex]

The monthly charge for a waste collection service for 175 kg of waste = $770

so, [tex](x_{2} , y_{2} ) = (175, 770)[/tex]

In this question, We are supposed to find a linear model for the cost, C, of waste collection as a function of the number of kilograms, w.

So, we will use two point slope form :

[tex]y - y_{1} = \frac{y_{2} - y_{1} }{x_{2}- x_{1} } (x-x_{1} )[/tex]

Substitute the values:

[tex]y - 470 = \frac{770- 470 }{175- 100 } (x-100)[/tex]

[tex]y - 470 = \frac{300 }{75} (x-100)[/tex]

[tex]y - 470 = 4 (x-100)[/tex]

[tex]y - 470 = 4x-400[/tex]

[tex]y = 4x +70[/tex]

Now, replace [tex]y[/tex] with C and [tex]x[/tex] with w

[tex]C[/tex] [tex]= 4w +70[/tex]

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Answer:

Robert's salary = $ 9426.65

Step-by-step explanation:

First we need to solve for n ( noel's salary)

And from there we can just sub n value into Robert's salary equation which is (n - 512) [ he earns $512 less than noel]

Answer:

[tex]c > -1[/tex]

Step-by-step explanation:

[tex]2c-10 < 3c-9\\-10 < c-9\\-1 < c\\c > -1[/tex]

When two lines are perpendicular to one another, they are said to intersect at a right angle or a 90-degree angle.

An angle is split in half equally by an angle bisector. Thus, divide the angle's degree count by 2 to determine the location of the angle bisector. At 80 degrees of the angle, the angle bisector is located.

An equal split of a line is created by a bisector. As a result, when we refer to a line segment's perpendicular bisector as AB, we mean that it bisects or divides AB into two equal halves.

According to the angle bisector theorem, if a point is on an angle bisector, it is equidistant from the angles' sides. The opposite is also true; if a point is equally far from each of the triangle's sides, it is on the angle's bisector.

Two lines are said to intersect at a right angle or a 90-degree angle when they are perpendicular to one another. A bisector, on the other hand, is a line that splits a line into two equally sized half.

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The solutions are:

a) y = (lnx + 42 - ln7)/x

b) y = (lnx + 42 - ln6)/x

What are differential equations?
A differential equation is an equation that relates a function to its derivatives.

To verify that every member of the family of functions y=(lnx+C)/x is a solution of the differential equation x^2y' + xy = 1, we can substitute the function into the equation and see if it holds true for all values of x.

y=(lnx+C)/x

y'= (1/x - lnx - C)/x^2

x^2(1/x - lnx - C)/x^2 + x(lnx+C)/x = 1

(1 - xlnx - xC) + (lnx+C) = 1

lnx + C = 1/x

The equation holds true for all x, so it is a solution of the differential equation.

a) To find a solution of the differential equation that satisfies the initial condition y(7)=6, we can use the fact that y=(lnx+C)/x is a solution and plug in the initial condition.

y(7) = (ln7+C)/7 = 6

ln7 + C = 42

C = 42 - ln7

y = (lnx + 42 - ln7)/x

b) To find a solution of the differential equation that satisfies the initial condition y(6)=7, we can use the fact that y=(lnx+C)/x is a solution and plug in the initial condition.

y(6) = (ln6+C)/6 = 7

ln6 + C = 42

C = 42 - ln6

y = (lnx + 42 - ln6)/x

Hence, the solutions are:

a) y = (lnx + 42 - ln7)/x

b) y = (lnx + 42 - ln6)/x

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The number of ways that the people can be arranged is given as follows:

120,960.

The number of possible permutations of x elements from a set of n elements is given by:

[tex]P_{(n,x)} = \frac{n!}{(n-x)!}[/tex]

To choose four people from a set of 10, as the order matters, the permutation formula is used, hence:

10!/6! = 5040.

Then we have to consider the four strains, which can be arranged in 4! = 24 ways, hence the number of ways for which the statistics can be arranged is of:

24 x 5040 = 120,960.

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Answer:

f(g(8)) = 26

Step-by-step explanation:

to calculate f(g(8)) , evaluate g(8) , then substitute the value obtained into f(x)

g(8) = 8 - 6 = 2 , then

f(2) = 3(2)² + 2 + 12 = 3(4) + 14 = 12 + 14 = 26

The probability that Jack will score more than 45 sixes in 240 spins of the spinner is about 0.13%.

What is the probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 indicates that an event is impossible, and 1 indicates that an event is certain.

The probability of getting a six on one spin of a fair six-sided spinner is 1/6. If Jack spins the spinner 240 times, the expected number of sixes he will get is 240 * 1/6 = 40.

We can use a normal approximation to the binomial distribution to find the probability that Jack will get more than 45 sixes. The mean of the distribution is 40 and the standard deviation is the square root of the variance, which is the product of the number of spins and the probability of getting a six on each spin, divided by the number of trials:

Standard deviation = sqrt(240 * 1/6 * 5/6 / 240) = sqrt(40/240) = sqrt(1/6)

Using the normal approximation, the probability that Jack will get more than 45 sixes is equal to the probability that a standard normal random variable is greater than (45 - 40) / sqrt(1/6) = 5 / sqrt(1/6) = 5 * sqrt(6).

Using a standard normal table or a calculator, this probability can be found to be approximately 0.0013, or 0.13%.

Hence, the probability that Jack will score more than 45 sixes in 240 spins of the spinner is about 0.13%.

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The midpoint method is a numerical integration technique for solving ordinary differential equations. To use it to solve the given equation, we start by discretizing the interval [0, L] into smaller steps of size h = 2, and use the midpoint of each interval to estimate the deflection at that point. Here's the algorithm:

Initialize z = 0, y = 0, and dydz = 0

Repeat the following steps for z = z + h until z = 30:

a. Calculate f1 = dydz

b. Calculate ymid = y + h/2 * f1

c. Calculate f2 = d2y/dz2 = f/2EI * (L-z-h/2)^2

d. Update y = y + h * f2

e. Update dydz = dydz + h * f2

The final value of y is the estimated deflection at z = 30.

The midpoint method is a numerical integration technique for solving ordinary differential equations. To use it to solve the given equation, we start by discretizing the interval [0, L] into smaller steps of size h = 2, and use the midpoint of each interval to estimate the deflection at that point. Here's the algorithm:

Initialize z = 0, y = 0, and dydz = 0

Repeat the following steps for z = z + h until z = 30:

a. Calculate f1 = dydz

b. Calculate ymid = y + h/2 * f1

c. Calculate f2 = d2y/dz2 = f/2EI * (L-z-h/2)^2

d. Update y = y + h * f2

e. Update dydz = dydz + h * f2

The final value of y is the estimated deflection at z = 30.

Plugging in the given parameter values, we have:

f = 60, E = 1.25 * 10^8, I = 0.05, L = 30

h = 2

z = 0, y = 0, dydz = 0

while z < 30:

  f1 = dydz

  ymid = y + h/2 * f1

  f2 = f/2EI * (L-z-h/2)^2

  y = y + h * f2

  dydz = dydz + h * f2

  z = z + h

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The equation for line q is y = mx + b where m is the slope and b is the y-intercept. Since the line passes through the point (4,6) and (0,6), we can find the slope by using the formula m = (y2 - y1) / (x2 - x1) = (6 - 6) / (4 - 0) = 0.

Since the line is perpendicular to line q, the slope of the new line is the negative reciprocal of 0, which is undefined. Therefore, the new line is a vertical line, and the equation is in the form x = a constant. Since the line passes through the point (-2,3), the equation is x = -2. Therefore, the answer is A: x = -2.

Answer:

The difference in interest received from an investment of $750 from an account paying 4% simple interest per annum and an account paying 0.3% monthly compound interest, over a one-year period is $19.75. The account paying 0.3% monthly compound interest will pay the most interest, yielding $102.50 in total.

The average velocity of the top of the ladder on this time interval is 8.75.

What is an average velocity?

The directional speed of an item in motion, as measured by a specific unit of time and observed from a certain point of reference, is what is referred to as velocity.

Here, we have

Given: time interval [a,9] so that the average velocity of the top of the ladder on this time interval is -20 ft/sec.

v = Δy/Δt = 20 = (0 - √625 - (7+2a)²)/(9-a)

-20 = (-√4(144 - 7a - a²)/(9-a)

10 = √(16 + a)(9 - a)/(9-a)

10 = √(16 + a)/(9 - a)

a = √884/101 = 8.75

Hence, the average velocity of the top of the ladder on this time interval is 8.75.

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For logx = 2, the value of x would be 100.

Option (C) is correct.

What is a logarithm?

A logarithm is a mathematical function that calculates the power to which a given number (called the base) must be raised to produce a given value. The most common base used in logarithms is base 10, but logarithms can be defined with any base. If the base of a logarithm is not specified, it is assumed to be base 10. The logarithm of a number is represented by the symbol log.

The value of x in the equation logx = 2 can be found by solving for x. The logarithm of a number is the exponent that the base must be raised to in order to produce that number. So, in the equation logx = 2, x is equal to the base raised to the power of 2.

So,

logx = 2

x = 10^2

x = 100

if the base of the logarithm is not specified it is assumed to be base 10.

So, logx = 2 means x = 10^2 = 100

Hence, for logx = 2, the value of x would be 100.

Option (C) is correct.

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The baggage handling services of On-Time Airlines are conducting a study to determine the correlation between the number of passengers arriving at a given time and the number of baggage handlers needed.

The study is performed by an airport executive who samples different times during the day and different days of the week including weekends. The main objective is to ensure that passengers do not wait an unreasonable amount of time for their baggage.

The study records the number of passengers arriving within any 1-hour time block. The data collected is then analyzed using a statistical technique called regression equation analysis, which is a method to estimate the strength and direction of the relationship between two or more variables.

The outcome or the result of the analysis is not provided in the given information.

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A general solution is one which involves the integer 'n' and gives all solutions of a trigonometric equation.

Given data :

dy / dx = 9 + [tex]\frac{9 + (5x)^{1/2} }{8 + (13y)^{1/2} }[/tex] ; x > 0

The general solution of a differential equation in implicit form is obtained as follows

dy / dx = h(x) / g(y)

g(y)dy = {8 + (13y)^{1/2}

h(x) dx = {9 + (5x)^{1/2}

F(x,y) = C

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ΔRED ≅ ΔBUL, by rule SSA criterion.

In geometry, two figures or objects are said to be congruent if their shapes and sizes match, or if one is the mirror image of the other.

We have the triangles,

ΔRED and ΔBUL.

And given:

RE ≅ BU

RD ≅ BL

And ∠RDE ≅ ∠BLU.

Hence,

ΔRED ≅ ΔBUL, by SSA criterion.

Therefore, ΔRED ≅ ΔBUL, by SSA criterion.

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