find derivative
f'(x) =​

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

An absolute maximum point is where the function reaches its highest value.

A function may have more than one location (x values) or point (ordered pairs) where these values occur, but there can be only one absolute minimum value and one absolute maximum value (in a given closed interval).

A point where the function achieves its highest value is known as an absolute maximum point. Similar to this, an absolute minimum point is the location at which the function's maximum value is obtained.

Minimum refers to the bare minimum that can be done. For instance, if anything requires seven dollars as the minimum payment, you cannot make a payment of six or less (you must pay at least seven). More than the bare minimum is acceptable, but not less. Maximum refers to the greatest quantity of anything.

Therefore, the correct answer is option C) (3, h(3)) is neither an absolute maximum nor an absolute minimum.

The complete question is:

Suppose that h(x) is a continuous function and is defined for all x greater than or equal to 1. You are given that h(x) has critical points at x = 1, x = 3, and x = 5. If h'(x) is negative on the interval 1 < x < 3, positive on the interval 3 < x < 5, and positive on the interval x > 5, what can be said about the point (3, h(3))?

A) (3, h(3)) is an absolute maximum.

B) (3, h(3)) is an absolute minimum.

C) (3, h(3)) is neither an absolute maximum nor an absolute minimum.

d) Nothing can be determined about the point (3, h(3)).

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The value of the missing length is 44.

Two or more triangles will be similar if on comparing their corresponding sides or measure of angles, some common properties can be observed. But when two or more shapes are similar, it do not imply that they are equal.

To find the value of the missing length, let the length LV be represented by s.

So that;

VN/ LN = NU/ MN

11/ (11 + s) = 8/ (24 + 8)

11/ (11 + s) = 8/ 32

cross multiply to have;

8(11 + s) = 11 *32

88 + 8s = 352

8s = 352 - 88

    = 264

s = 264/8

  = 33

Therefore, the missing length = 11 + 33

                        = 44

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∆AEC ≅ ∆AFC by reason of SAS. Thus, using CPCTC, we can prove that <AEC ≅ <AFC.

With the given information and the image attached below, we can state that the two triangles are congruent by SAS, and then go ahead to use the CPCTC to show that any of their corresponding parts are congruent as well.

The proof is given as:

Statement Reason

1. AB ≅ BC, AE ≅ FC 1. Given

2. AC ≅ AC 2. Reflexive property of congruence

3. <BAC ≅ <BCA 3. Base angles of isosceles ∆BAC

4. ∆AEC ≅ ∆AFC. 4. SAS

5. <AEC ≅ <AFC. 5. CPCTC

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Answer:

Jelly bean is 8 pound

Brach's variety is 4 pound

Step-by-step explanation:

jelly belly = x

brach's variety = y

5.99x + 3.99y = 63.88 _______(1)

x = 2y ______(2)

5.99x + 3.99y = 63.88

5.99(2y) + 3.99y = 63.88

11.98y + 3.99y = 63.88

15.97y = 63.88

15.97y/15.97 = 63.88/15.97

y = 4

Substitute y = 4 into equ 2

x = 2y

x = 2(4)

x = 8

Therefore, Steven bought 8 lb of jellybean and 4 lb of brach variety

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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The depth of snow in the yard one hour after the snow depth was 9 inches, obtained using the linear function used for finding the snow depth, D = 1.4·t + 3 is; 10.4

A linear function is a function that produces a straight line graph when plotted.

The function that can be used to calculate the depth of the snow D in inches is; D = 1.4·t + 3

Where;

D = The snow depth in inches

t = The time duration (hours)

The present depth of the snow = 9 inches

The depth of the snow one hour from now can be found as follows;

The function for the depth of the snow; D = 1.4·t + 3 is a linear function

Therefore, the slope, which is the coefficient of the input variable, t, when the coefficient of the output variable, D, is 1

Therefore, the slope = 1.4

The slope is the rate of change of the output variable D when the input variable increases by one unit

The slope therefore, indicates that the depth of the snow in the yard, increases by 1,4 inches per hour

If the depth of the snow is presently 9 inches, the depth of the snow 1 hour later is therefore; D = 9 inches + 1.4 inches = 10.4 inches

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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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The volumes of the hemisphere are (π/12)R³, 2π/3(R³) and π/3(R³)

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

x' + y' = R²

Using the disk method

Start by calculating the area of a single disk is given by the quarter circle of radius R in the first quadrant.

The area of the single disk is:

Area = π(R/2)²

The volume of the hemisphere using the disk method is given by:

V = ∫(π(R/2)² dx from 0 to R

This gives

V = (π/4) * ∫R^2 dx from 0 to R

Differentiate

V = (π/12)R³

Here, we first need to find the area of a single shell.

The area of the single shell is:

Area = 2πx * dx

The volume of the hemisphere using the single shell method is given by:

V = ∫(2π x * dx)dy from 0 to R

So, we have

V = 2π * ∫xdy*dx from 0 to R

Differentiate

V = 2π/3(R³)

Here, we can use any slicing method, such as slicing the hemisphere parallel to the y-axis.

The volume of the hemisphere using the general slicing method is given by:

V = ∫(π * (x^2) * dy) from 0 to R

So, we have

V = π * ∫(y^2)dy from 0 to R

Differentiate

V = π/3(R³)

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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:

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]

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 three integers are 89,406, 89408 and 89410.

What are integers?

The Latin term "Integer," which implies entire or intact, is where the word "integer" first appeared. Zero, positive numbers, and negative numbers make up the particular set of numbers known as integers.

Let 'm' be a positive integer.

Then, 2m is an even integer ( as multiplying by 2 makes the number even)

So, 2m+2 is the next integer and 2m+4 is the next.

Now, we have three consecutive integers as 2m, 2m+2 and 2m+4.

We are given that twice the first integer added to the second is 2,68,220, this means

⇒2(2m) + (2m+2) = 2,68,220

⇒4m + 2m + 2 = 2,68,220

⇒6m + 2 = 2,68,220

⇒6m = 2,68,218

⇒m = 44,703

So, 2m = 89,406

2m+2 = 89408

2m+4 = 89410

Hence, the three integers are 89,406, 89408 and 89410.

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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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Answer:

[tex]c > -1[/tex]

Step-by-step explanation:

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

The mathematical sentence can be written as the equation below,

k + t = 11

Here we want to write the the mathematical sentence below:

"The sum of a number k and t is 11"

As an algebraic equation, so let's analyze each part of the mathematical sentence.

"sum" refers to an addition, in this case between two numbers represented by k and t, then we can write:

k + t

And that is equal to 11, then the equation will just be:

k + t = 11

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The sentence into a mathematical equation is R = 160x

The sentence "The difference between a number and ten is four" can be translated into a mathematical equation as:

x - 10 = 4

To solve for x, we can add 10 to both sides of the equation. This will give us:

x - 10 + 10 = 4 + 10

x = 14

Therefore, the number is 14.

More generally speaking, we can say that the equation for this sentence is x - a = b, where a is the number we are subtracting (in this case 10) and b is the result of the subtraction (in this case 4). To solve for x, we add a to both sides of the equation, giving us x = a + b.

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If each code contains non-repeating digits there would be 720 possible gate codes.

What is a combination in mathematics?

Combination in mathematics is a way of selecting items from a set, without regard to the order in which they were chosen.

In your example, there are 10 possible digits (0-9) that can be used for each of the three numerical digits, but since the digits cannot repeat, the total number of possible gate codes is 1098 = 720. Since the "#" sign is not included in this calculation, the final answer is 720 possible gate codes.

Hence,  If each code contains non-repeating digits there would be 720 possible gate codes.

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[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{15\% of 19.37}}{\left( \cfrac{15}{100} \right)19.37} ~~ \approx ~~ 2.91[/tex]

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 quantity of chlorine that has a 60% concentration is 310 ml.

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.

Given that,  60% chlorine solution must be mixed with an 18% chlorine solution to produce 542.5 mL of 42% solution

Let "x" be the quantity of chlorine that has a 60% concentration

So, the quantity of chlorine that has 18% concentration = 542.5 - x

thus, As per the given data expression formed as given below:

0.6 * x + 0.18 (542.5 - x) = 542.5 * 0.42

0.6x - 0.18x + 0.18 *542.5 = 542.5 *0.42

0.42x = 130.20

x = 310

therefore, the quantity of chlorine that has a 60% concentration is 310 ml.

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10 number of hours khalil worked tutoring last week and 3 the number of hours khalil worked landscaping last week

given that

An inequality in mathematics is a relation that compares two numbers or other mathematical expressions in an unequal way. [1] The majority of the time, size comparisons between two numbers on the number line are made. Different types of inequalities are represented by a variety of notations, including:

The symbol a b indicates that an is smaller than b.

When a > b is used, it indicates that an is bigger than b.

A is not equivalent to b in any scenario. Since an is strictly less than or strictly greater than b, these relationships are known as stringent inequalities. Equivalence is not considered.

khalil is working two summer jobs, making $18 per hour tutoring

$9 per hour landscaping

last week khalil worked a total of 14 hours and earned a total of $180.

let x be number of hours khalil worked tutoring last week

y be the number of hours khalil worked landscaping last week

we can form equation by given information

x + y [tex]\leq[/tex] 13 - (1)

18x + 9y > 180 - (2)

the numbers that satisfy the above equations is (10,3)

10 number of hours khalil worked tutoring last week and 3 the number of hours khalil worked landscaping last week

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The potential function for f(x,y) is given by f(x,y) = x^2 + y^2 + 2xy.

The potential function can be used to find the maximum value of f by taking the partial derivatives with respect to x and y, setting them equal to zero, and solving for the x and y values that correspond to the maximum. This will give the x and y values that will produce the maximum value of f.

To prove this, we can start by taking the partial derivatives of f(x,y), which are given by:

f_x = 2x + 2y

f_y = 2y + 2x

Therefore, the potential function for f(x,y) is equal to the integral of the partial derivatives with respect to x and y:

∫f_xdx + ∫f_ydy = x^2 + y^2 + 2xy

This shows that the potential function for f(x,y) is given by f(x,y) = x^2 + y^2 + 2xy.

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Complete question: What is the potential function for the function f(x, y) = x + y + xy?

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.)

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.

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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Answer:

The distance that the baseball is from the pitching rubber, D, can be modeled using the equation:

D = V*t + 60.5

Where V is the horizontal velocity of the baseball in feet per second and t is the time in seconds that the baseball is in the air.

In this scenario, the baseball is thrown by the pitcher with a horizontal velocity of 132 feet/second, and then hit by the batter with a horizontal velocity of 141 feet/second. The baseball passes over the pitching rubber on its way out of the stadium and eventually passes over the center field fence, 450 feet away. So, the equation will be

D = (132+141) * t + 60.5

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:

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]

The 5th value is 15.5, and the 6th value is 15.7. The median is then (15.5 + 15.7)/2 = 15.6.

A Stem-Leaf plot is a graphical way to organize and summarize numerical data. It is composed of two columns, the stem and the leaf. The stem column contains the leading digits of the data values, while the leaf column contains the remaining digits.

For the given data set, the stem-leaf plot looks like this:

Stem  Leaf

14  8

15  2 8 5 7 9

16  0 1

To create this stem-leaf plot, the data values were first sorted in ascending order, and then the leading digits were placed in the stem column. The remaining digits were then placed in the leaf column, in increasing order. The key, 15|2=15.2, is an example of a data value from the set.

To calculate the median of the data set, we first need to find the number of values in the set. Since there are 10 values, the median will be the average of the 5th and 6th values.

The 5th value is 15.5, and the 6th value is 15.7. The median is then (15.5 + 15.7)/2 = 15.6.

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