M114) solve
cos (2x) = √2 - cos (2x)

By considering ages and equations we have,

Uncle Rob will be 62 years old when Dave is 32.

Dave is listed as being 15 years old, while his uncle Rob is listed as being three times Dave's age. Rob's age as of right now may be calculated using the formula below:

Age of Rob is Dave times three.

Using Dave's age as a plug-in, we obtain following equations,

Rob's age is equal to 15 + 3 = 45.

Rob is currently 45 years old, according to this.

We need to know how many years it will be before Dave turns 32 in order to calculate Rob's age at that time. By deducting Dave's present age from his anticipated age, we may determine this:

Years from now = Dave's age in the future minus Dave's age currently

By entering the specified values, we obtain:

The number of years from now is equal to 32 - 15 = 17 years.

Dave will turn 32 in 17 years, according to this.

We may multiply Rob's current age by the number of years from now to determine his age at that point:

When Dave reaches the age of 32, Rob will be that age plus the number of years.

When we enter the calculated values, we obtain:

When Dave is 32 years old, Rob will be 45 + 17 years old, or 62 years old.

  So, Dave uncle rob 's age will be 62 years old.

As a result, when Dave's age is 32, his uncle Rob's age will be 62.

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Therefore , the solution of the given problem of median comes out to be  the typical grade received by pupils is 80.

The median value of an arranged dataset is the precise value that makes up its mean. The ratio between the minimum and maximum 50% of the data is what is known as a likelihood measure, or an average. Based if there is an odd or perhaps an even number of points, multiple formulas are used to calculate the median and mode.

Here,

The range and scatter of statistical data are represented visually by this analysis (continuous data).

It establishes four quartile groups.

Min - 25th Percentile, Quartile Group 1 (Q1)

Group 2 of the Quartile: 25th Percentile (Q1) to 50th Percentile (Q2, Median)

50th percentile (Q2) and 75th percentile make up Quartile Group 3. (Q3)

75th percentile (Q3) for Quartile Group 4: Maximum

Q3-Q1 is the interquartile range in this (IQR)

the 40-point cutoff

a maximum score of 110

25% of the class received a score between 40 and 63.

50% of the pupils got scores under 81, and 50% got scores over 81.

75% of pupils had scores between 40 and 95. additionally, 25% of students achieved 95 or higher.

The typical grade received by pupils is 80.

Therefore , the solution of the given problem of median comes out to be  the typical grade received by pupils is 80.

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The complete question is " Professor A. has 40 students in her class. After giving a first quiz, she calculated 5 measures of relative standing: min =3, Q1 = 12, Median = 14, Q3 = 16, Max = 20. Draw a box plot for quiz scores of her class. What are the highest and the lowest quiz score? What is the score that separates the bottom 75% from the top 25%? How many students scored more than the median? How many students scored more than 14 points? What percent of her students received less than 12 points? Would the score of 4 be an outlier? (use the IQR method to explain)"

The volume of the spherical glass bulb is approximately 14.1 cubic cm.

To find the volume of a spherical glass bulb, we can use the formula for the volume of a sphere:

[tex]V = 4/3 * pi * r^3[/tex]

Where V is the volume, pi is a mathematical constant (approximately equal to 3.14), and r is the radius of the sphere.

We know the diameter of the bulb is 31.5mm and to find the radius of the sphere, we divide the diameter by 2.

So, radius (r) = 31.5mm / 2 = 15.75mm

Now we can substitute this value into the formula for the volume of a sphere:

[tex]V = 4/3 * pi * (15.75mm)^3[/tex]

We can convert the value of radius from mm to cm by multiplying it by 10^-3

[tex]V = 4/3 * pi * (15.75 * 10^{-3})^3[/tex]

Now we can calculate the value of V

V ≈ 14.1 cubic cm

Therefore, the volume of the spherical glass bulb is approximately 14.1 cubic cm.

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the answer is  4/7x                                                      

Answer:

25 and 9

Step-by-step explanation:

to find ( u ￮ w )(8)

evaluate w(8) then substitute the value obtained into u(x)

w(8) = [tex]\sqrt{8+8}[/tex] = [tex]\sqrt{16}[/tex] = 4 , then

u(4) = 4² + 9 = 16 + 9 = 25

------------------------------------------------------------

to find (w ￮ u )(8)

evaluate u(8) then substitute the value obtained into w(x)

u(8) = 8² + 9 = 64 + 9 = 73 , then

w(73) = [tex]\sqrt{73+8}[/tex] = [tex]\sqrt{81}[/tex] = 9

Yes, it is possible to formed a right angled triangle with the measure of the sides 6cm, 10cm , and 8cm as it satisfied the Pythagoras theorem.

( 10 )² = ( 6 )² + ( 8 )².

As given in the question,

Measure of the sides of the given triangle is equal to :

6cm , 8cm , and 10cm.

In the given triangle longest side of the triangle is with measure 10cm.

Condition for a triangle to be right angled triangle is satisfied Pythagoras theorem given by :

Hypotenuse represent the longest side of the triangle.

(Hypotenuse) ² = ( base )² + ( Altitude )²

Right hand side

= ( 6 )² + ( 8 )²

= 36 + 64

= 100

= 10²

= (Longest side )²

It satisfied the Pythagoras theorem condition.

Therefore, the given sides measure of the triangle represents it is a right angled triangle satisfying Pythagoras theorem : ( 10 )² = ( 6 )² + ( 8 )².

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Answer:

The solution to the system of equations g(x) = x - 1 and f(x) = x^2 - 5 is the point where the two functions intersect, which is the point (1,0).

To confirm this, we can substitute x = 1 into each equation and see that they both equal 0:

g(1) = 1 - 1 = 0

f(1) = 1^2 - 5 = -4 + 1 = -3 = 0

So (1,0) is the solution to the system of equations.

Answer:

r=-7

See explanation below

Step-by-step explanation:

slope= y2-y1/x2-x1

1-r/5-1=2

1-r/4=2. (simplify)

1-r=8 (cross multiply)

-r=7

r=-7

The rate of change is -1/2

Initial point is (-2,-1)

What is slope ?

A line's steepness is determined by its slope. In mathematics, slope is determined by "rise over run" (change in y divided by change in x).

Consider any two pairs of points:

(-2,-1)  and (0,-2)

The slope is the same as the rate of change

Rate of change = change in y values/change in x values

                         = (-2 - (-1) ) / (0-(-2))

                         = ( -2 + 1) / (+2)  

Rate of change  = (-1/2)

The rate of change is -1/2

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The slope m of the tangent to the curve will be m = -3/a^2.

Calculating the slope of the tangent line to the curve y = 3/x at a point where x = a > 0 is an important mathematical concept. To do this, we will first need to find the derivative of the curve y = 3/x.

Let's start by taking a look at the equation for the slope of a line: m = (y2-y1) / (x2-x1). We'll use this equation to find the slope of the tangent line at the point (a, 3/a).

To do this, we'll need to calculate the derivative of the given equation, which is y' = -3/x^2.

At the point (a, 3/a), the derivative is equal to -3/a^2. Therefore, the slope of the tangent line at the point (a, 3/a) is m = -3/a^2.

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The probability that a randomly chosen person 25 years of age or older is in the labor force 67.035%.

The chance of an event happening given that another event has already occurred (via assumption, supposition, statement, or evidence) is known as conditional probability in the field of probability theory.  This particular approach depends on event B happening in some way connected to another event A. In this case, a conditional probability analysis with respect to event B is possible.

The probability of A under the condition B is typically expressed as P(A|B)[2] or PB if the event of interest is A and the event B is known to have occurred or is assumed to have done so (A). This can alternatively be thought of as the percentage of possibility that B and A will collide.

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Answer:

= 625x^4 + 500x^3 y + 150x^2•y^2 + 20xy^3 + y^4

Step-by-step explanation:

1. Use the Binomial Theorem.
( 5 x ) 4 + 4 ( 5 x ) 3 y + 6 ( 5 x ) 2 y 2 + 4 ( 5 x ) y 3 + y 4

STEPS

Apply the product rule to 5x.
5^4 x^4 + 4(5x)^3y + 6( 5x )^2y^2 + 4(5x)^y3 + y^4

Raise 5 to the power of 4.

625x^4 + 4(5x)^3y + 6(5x)^2y^2 + 4(5x)y^3 + y^4

Apply the product rule to 5x.

625^x^4 + 4( 5^3•x^3) y + 6(5x)^2y^2 + 4(5x)^y^3 + y^4

Raise 5 to the power of 3.

625x^4 + 4(125x^3) y + 6(5x)^2y^2 + 4(5x)y^3 + y^4

Multiply 125 by 4.

625x^4 + 500x^3•y + 6 ( 5 x )^2y^2 + 4(5x) y^3 + y^4

Apply the product rule to 5x.

625x^4 + 500x^3y + 6(5^2• x^2 )y^2 + 4(5x)y^3 + y^4

Raise 5 to the power of 2.
625 x 4 + 500 x 3 y + 6 ( 25 x 2 ) y 2 + 4 ( 5 x ) y 3 + y 4

Multiply 25 by 6.

625 x 4 + 500 x 3 y + 150 x 2 y 2 + 4 ( 5 x ) y 3 + y 4

Multiply 5 by4 .

625 x 4 + 500 x 3 y + 150 x 2 y 2 + 20 x y 3 + y 4

2. Simplify each term.
625x^4 + 500x^3 y + 150x^2•y^2 + 20xy^3 + y^4

The expression that is equivalent to the given question is option D. (7^5)/ (3^4)

Indices is a topic in mathematics which has its applications in performing  arithmetic operations on numbers with powers. This process requires some laws; which are referred to as the law of indices. These laws have their various applications as required.

By applying the appropriate law of indices to the given question, the equivalent expression can be determined as;

From the numerator;

(7^2)(7^6)(3^2) = (7^8)(3^2)

So that;

(7^8)(3^2)/ (7^3)(3^6) = (7^(8-3)) (3^(2-6))

                                  = (7^5)(3^-4)

The equivalent expression is;

(7^5)(3^-4) = (7^5)/ (3^4)

The equivalent expression is (7^5)/ (3^4). Thus option D.

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On solving the the provided question we can say that -  here in the slope the equation will be y - 3 = 1(x+6) = y - x - 9 = 0

A line's slope determines how steep it is. A mathematical equation for the gradient is called "gradient overflow" (the change in y divided by the change in x). The slope is the ratio of the vertical change (rise) between two points to the horizontal change (run) between those same two spots. When a straight line's equation is written as y = mx + b, the slope-intercept form of an equation is employed to express it. The y-intercept is located at a location where the line's slope is m, b is b, and (0, b). For instance, the slope of the equation y = 3x - 7 and the y-intercept (0, 7).The line's slope is m. b is b at the point where the y-intercept is, and (0, b). As an illustration, the equation y = 3x - 7 has a slope of 3, and its y-intercept is (0, 7).

points are (-6,3)

y - y1 = m(x-x1)

y - 3 = 1(x+6)

y - x - 9 = 0

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The following intervals of the quadratic equation are listed below:

Herein we find a representation of a quadratic equation with a vertical axis and on a Cartesian plane, of which we must look for every positive and negative interval. Quadratic equation has the feature of having an absolute extreme, either a minimum or a maximum.

The strategy to determine to determine whether the interval is positive and negative is summarized below:

Then, the quadratic equation has two positive intervals on x ∈ (- ∞, - 5) and x ∈ (1, + ∞) and a negative interval: x ∈ (- 5, 1).

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The following question can be solved using concepts of mathematical reasoning.

Mathematical reasoning is the process of using logical and systematic thinking to arrive at conclusions and make connections between mathematical concepts. It involves making logical deductions and inferences based on mathematical statements, definitions, postulates, and theorems.

Let p, q, and r be the propositions

p: Grizzly bears have been seen in the area.

q: Hiking is safe on the trail.

r: Berries are ripe along the trail.

a) Berries are ripe along the trail, but grizzly bears have not been seen in the area.

Solution:

r ∧ ~p

b) Grizzly bears have not been seen in the area and hiking on the trail is safe, but berries are ripe along the trail.

Solution:

~p∧q∧r

c) If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area

Solution:

r → (q ↔ ~p)

d) It is not safe to hike on the trail, but grizzly bears have not been seen in the area and the berries along the trail are ripe.

Solution:

~q ∧ ~p ∧ r

e) For hiking on the trail to be safe, it is necessary but not sufficient that berries were not ripe along the trail and for grizzly bears not to have been seen in the area.

Solution:

q → (~r ∧ ~p)

f) Hiking is not safe on the trail whenever grizzly bears have been seen in the area and berries are ripe along the trail.

Solution:

(f) (p ∧ r) → ~q

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Answer:horizontal stretch of 3

Step-by-step explanation:

Audrey is 244 kilometers far from its starting point, as of the given conditions.

In a right-angled triangle, its sides, such as hypotenuse, perpendicular, and base are Pythagorean triplets.

Here,
Audrey walks 79 kilometers due west, then 240 kilometers due north, and finally 123 kilometers due east.
The distance between starting and final point is given as,
= √[240² + [123-73]²]
= √59536
= 244 kilometers

Thus, Audrey is 244 kilometers far from its starting point, as of the given conditions.

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I think the question should be tan theta = 1/√3, but I have given the solutions of both.

Hope it helps.

If you have any query, feel free to ask.

There are 17280 ways 5 letter words can be made from "math is fun" if each word must have 2 consonants and 3 vowels.

Permutation and Combination:

In mathematics, permutation refers to the process of putting all members of a set into a particular order or order. In other words, if the set is already ordered, the permutation of its elements is called permutation. Substitutions occur in more or less noticeable ways in almost every field of mathematics. They often occur when different orders are considered in a particular finite set.

Combining is a way of selecting items from a collection, and (unlike permutation) the order of selection does not matter. In smaller cases you can count the number of combinations. Combining refers to combining n things, taken k times at once, without repetition.

In the word " MATH IS FUN"

The number of Vowels = 03

The number of Consonant = 06

Total Number of words = 09

and, total number of words in even place = 04

The Number of ways = ⁴C₃ = 4 × 3 × 2 ×1/ 3×2×1

                                             = 4

Now,

The vowels can be arranged among themselves in 3! = 6 ways and 5 consonants can fill the remaining 6 places in 5! ,

i.e., 6× 5 × 4× 3× 2× 1 = 720

Therefore,

Total Number of ways = 4 × 6× 720

                                     = 17280 ways

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It takes Mika about 1 1/8 minutes to read per page.

Now, According to the question:

Let's know:

What is Simplification?

Simplify simply means to make it simple. In mathematics, simply or simplification is reducing the expression/fraction/problem in a simpler form. It makes the problem easy with calculations and solving.

Mika read a 405-page book in 6 hours.

We convert pages per hours (405/6.) to pages per minute [60 minutes per hour]:

[tex]\frac{405}{6}[/tex] × [tex]\frac{1}{60}[/tex]

To simplify the above expression:

= 1.125

So, it takes Mika about 1 1/8 minutes to read per page.

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The price of the most expensive house sold is 3.112 millions of dollars.

What is algebra ?

Algebra is a branch of mathematics in which letters and symbols are used to represent numbers and quantities in equations and formulas. Algebraic concepts include operations such as addition, subtraction, multiplication, and division.

The stem-and-leaf plot shows house sale prices over the last week in Tacoma. The stem is in hundreds of thousands of dollars and the leaf is in tens of thousands of dollars.

The most expensive house is represented by the number 3|0001122.

The stem is 3 which means the house price is in hundreds of thousands of dollars and the leaf is 001122, which means the house price is 1,122 thousands of dollars.

So the price of the most expensive house sold is ,

3 (hundreds of thousands of dollars) + 1,122 (tens of thousands of dollars) = 3.112 (millions of dollars)

So the price of the most expensive house sold is 3.112 millions of dollars.

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Answer:

B And D

Step-by-step explanation:

X has two of the same numbers and the both have different outputs

On solving the provided question we can say that by null hypothesis There is insufficient proof to conclude that student attendance has changed.

A null hypothesis is a kind of statistical hypothesis that asserts that a specific set of observations has no statistical significance. Using sample data, hypotheses are tested to determine their viability. Sometimes known as "zero" and symbolized by H0. Researchers start off with the presumption that there is a link between the variables. In contrast, the null hypothesis claims that there is no such association. Although the null hypothesis may not appear noteworthy, it is a crucial component of research.

This is a z-test for a proportion.

[tex]H0:p=.34 H1:p not=.34\\\alpha=0.01 ... this is a\\2-tailed test\\critical values are +2.58, -2.58\\p=.34 (given) q=1-p=.66\\p^=.33\\z=(p^-p)/sqrt(pq/n)\\z=(.33-.34)/sqrt(.34*.66/8302)\\z=-.01/.0051961\\z=-1.92[/tex]

1.96 does not fall in the reject region

There is insufficient proof to conclude that student attendance has changed.

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Answer:

$32

$22

Step-by-step explanation:

We know

Membership cost per month = $20

Price of a slice of pizza for members = $2

How much would Alang have to pay the pizza shop if he bought 17 slices of pizza this month?

The total cost of Alang on 17 slices of pizza = membership cost per month + prices of 17 discounted pizza

$20 + $2(6) = $32

What would be the monthly cost for a slice of pizza

The monthly cost of Alang a slice of pizza = membership cost per month + price of 1 discounted pizza

$20 + 2 = $22

So,

Alang would have to pay $32 if he bought 17 slices of pizza this month.

The cost of a slice of pizza is $22

Answer:

y = (7/4)x - 4

Step-by-step explanation:

Rewrite the equation in standard format of y=mx+b, where m is the slope and b is the y-intercept (the value of y when x=0).

4x - 7y=9

- 7y=9-4x

y=(-4x + 9)/(-7)

y = -(4/7)x - (9/7)

This line has a slope of -(4/7).  A line perpendicular to this line will have a slope that is the negative inverse of the reference line.  The prependicular line will have a slope of (7/4) [the negative inverse of -(4/7)].

We can now write the new line as

  y = (7/4)x + b

We need a value of b that forces the line to go through point (4,3).  This can be di=one by entering (4,3) into the perpendicular equation (y = (7/4)x + b) and solve for b:

y = (7/4)x + b

3 = (7/4)*4 + b  for point (4,3)

3 = 7 + b

b = -4

The equation is y = (7/4)x - 4

See the attached graph.

Yes, it is possible that both Emilio and Jim solved the problem correctly using addition and subtraction, respectively.

For example, let us consider the problem to find the solution for

(4/5) + (3/5)

Here (4/5) and (3/5) are rational numbers.

Emilio would use addition and get the solution of

(4/5) + (3/5)

= ( 4 + 3)/ 5

= 7/5

Jim could have used subtraction and found the solution of

(4/5) + (3/5)

= (4/5) - (-3/5)

= (4 - (-3))/5

= 7/5

In this example, both Emilio and Jim would have arrived at the correct solution of 7/5 though one person uses addition and other person uses subtraction.

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Answer: (a^2 - 4)/(x^2 - 9) + (a^2 - 2a + (2-y))/(xy + 3y) - (x-3)

Step-by-step explanation: First, you need to simplify the fraction in the first term:

(a^2 - 4)/(x^2 - 9)

Next, you simplify the fraction in the second term:

(a^2 - 2a + (2-y))/(xy + 3y)

Then you subtract (x-3) from the fractions obtained in step 1 and 2:

(a^2 - 4)/(x^2 - 9) + (a^2 - 2a + (2-y))/(xy + 3y) - (x-3)

This is the final result of the operation.

It is important to note that the answer is a simplified fraction and only valid if x does not equal 3. If x=3, the denominator x-3 would be equal to zero, which is not allowed in mathematics.

Check the picture below.

so if we peel off the cylinder, we end up with that, hmmm how long is the width?  well, we know the circular bases have a diameter of 9mm, so half that is its radius or 4.5mm, and if we peel off the circle, we'll end up with a long line whose length is its perimeter or namely its circumference, which oddly enough is the length of the width

[tex]\textit{Circumference of a circle}\\\\ C=2\pi r ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=4.5 \end{cases}\implies C=2\pi (4.5)\implies C=9\pi \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{\LARGE Lateral Area}}{\stackrel{width}{(9\pi )} \stackrel{height}{(17)}} \implies {\Large \begin{array}{llll} 153\pi \end{array}} ~mm^2[/tex]

Answer: Option b: 34

Explanation: The cosecant of an angle θ is defined as the reciprocal of the sine of that angle. Therefore, csc(θ)=1/sin(θ). Using a calculator, you can solve for sin(θ) by entering csc(θ) = 1.7894 and pressing the sin-1 button. This will give you the result of sin(θ) = 0.5930. You can then use a calculator to find the measure of θ in degrees by entering sin-1(0.5930) and pressing the Degrees button. This will give you the result of θ = 34°.