4. Evaluate the following trigonometric expressions e.
10

The solution of the given problem of expressions comes out to be  ⁶√x⁷*∛x² = [tex]x^{(11/6)[/tex].

Calculatiοns that cοmbine jοining, remοving, and randοm subdivisiοn must be dοne with ever-changing factοrs. They cοuld accοmplish the fοllοwing if they united: An prοgramme, sοme data, and a mathematical prοblem. Fοrmulas, cοmpοnents, and arithmetic οperatiοns like adds, subtractiοns, mistakes, and grοupings can all be fοund in a declaratiοn οf truth.

Here,

Using rational exponents, it is possible to formulate the expression 6x7*x2 more concisely:

=> ⁶√x⁷*∛x² = [tex]x^{(7/6)} \times x^{(2/3)[/tex]

Now, we can include the exponents to further simplify:

=> [tex]x^{(7/6)} * x^{2/3)} = x^{(7/6 + 2/3)[/tex]

We can add 3 and 6 to discover a common denominator of 18 for the numbers 6 and 3. Then, we could type:

=> [tex]x^{(7/6 + 2/3)} = x^{(21/18 + 12/18)}[/tex]

We can now multiply the exponents:

=> [tex]x^{(21/18 + 12/18)} = x^{(33/18)[/tex]

Further reducing the exponent, we can write:

=> [tex]x^{(33/18) }= x^{(11/6)[/tex]

Therefore, in simplified version, ⁶√x⁷*∛x² = [tex]x^{(11/6)[/tex].

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Answer:

percentage = (part/whole) x 100%

where "part" is the value we are trying to express as a percentage of "whole". In this case, "part" is 3984 and "whole" is 24.9.

So we can substitute these values into the formula:

percentage = (3984/24.9) x 100%

percentage = 159.838% (rounded to three decimal places)

Therefore, 3984 is approximately 159.838% of 24.9.

Answer:

3984 is 16,000% of 24.9

Step-by-step explanation:

percentage = (3984÷24.9) x 100%

percentage = 159.838%

or percentage ≈ 16,000%

Answer: 15x-2 (answer A)

Step-by-step explanation:

To find the total travel time, we need to add the time it takes to travel in the morning and the time it takes to travel in the evening:

Total travel time = (3x - 10) + (12x + 8)

Simplifying the expression by combining like terms, we get:

Total travel time = 15x - 2

Therefore, the total travel time is 15x - 2. So, the answer is (a) 15x - 2.

Answer:

Step-by-step explanation:

The expression 3x − 10 represents the time it takes a commuter to travel in the morning to work. The expression 12x + 8 represents the time it takes a commuter to travel in the evening from work. What is the total travel time?

15x − 2

9x − 2

15x + 2

9x + 2

 the answer is (a) 15x-2 hope that helps :)

Considering the definition of perimeter, the length and width of the rectangle is 85 in and 7 in respectively.

The perimeter of a two-dimensional figure is the distance around the figure. That is, the perimeter of a flat geometric figure is called the length of its contour.

The perimeter is the measurement obtained as a result of the sum of the sides of a flat geometric figure.

A rectangle is a geometric figure that has two pairs of sides of equal length. The expression to calculate the perimeter of a rectangle is:

Perimeter= 2× length + 2× width

In this case, you know

Replacing in the definition of the perimeter of the rectangle:

184= 2× (11×width + 8) + 2× width

Solving:

184= 2×11×width + 2×8 + 2× width

184= 22×width + 16 + 2× width

184 - 16= 22×width + 2× width

168= 24×width

168÷ 24= width

7 in= width

So, the length can be calculated as:

lenght= 11×7 in + 8 in

lenght= 85 in

Finally, the length of the rectangle is 85 in and the width of the rectangle is 7 in.

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There is a moderate negative linear relationship between x and y for the given points.

When determining if there is a linear relatiοnship between twο quantitative variables, Pearsοn's cοrrelatiοn is used. Just that—a linear relatiοnship between thοse variables—is the research hypοthesis.

[tex]\mathrm{r = (n \Sigma xy - \Sigma x\Sigma y) / \sqrt{(n\Sigma x^2- (\Sigma x)^{2)(n\Sigma y)}^2 - (\Sigma y)^2)}}[/tex]

where n is the number οf pοints, Σxy is the sum οf the prοducts οf x and y cοοrdinates, Σx is the sum οf x cοοrdinates, Σy is the sum οf y cοοrdinates, Σx² is the sum οf squares οf x cοοrdinates, and Σy² is the sum οf squares οf y cοοrdinates.

First, we need tο find these values frοm the given pοints. We can use a table tο οrganize οur calculatiοns:

x y xy x² y²

1 6 6 1 36

2 10 20 4 100

3 4 12 9 16

4 4 16 16 16

6 2 12 36 4

7    

The table continues:

|x ||y ||xy ||x² ||y² | |- |- |- |- |- | 1 6 6 1 36

|| || || || || || || || || ||

The table continues:

|x ||y ||xy ||x² ||y² | |- |- |- |- |- | 1 6 6 1 36

|| || || || ||

The table continues:

|x ||y ||xy ||x² ||y² | |- |- |- |- |- | 1 ||6 ||6 ||1 ||36 | 2 ||10 ||20 ||4 ||100 | 3 ||4 ||12 ||9 ||16 | 4 ||4 ||16 ||16 ||16 | 5||8||40||25||64| 6||2||12||36||4| 7||2||14||49||4|

Now we can add up each column to get:

Σx = 28 Σy = 36 Σxy = 120 Σx² = 140 Σy² = 240

Next, we can plug these values into the formula and simplify. Remember to round any intermediate calculations to no less than six decimal places.

r = (7(120) - (28)(36)) / √[(7(140) - [tex](28) ^{* * * * * * * * * *} (240) - (36)^ {* * *}[/tex]

r = (840 - 1008) / √[(980 - 784) (1680 - 1296)]

r = (-168) / √[196(384)]

r = (-168) / (√75264)

r = (-168) / (274.343596)

r ≈ (-0.612487)

Finally, we round our answer to three decimal places.

r ≈ -0.612

This means that there is a moderate negative linear relationship between x and y for the given points.

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The correct statement regarding the average rate of change of the functions f(x) and g(x) on the interval [-2,2] is given as follows:

Over the interval [-2,2], the function f is increasing at a faster rate than function g is decreasing.

The average rate of change of a function is given by the change in the output of the function divided by the change in the input of the function. Hence we must identify the change in the output, the change in the input, and then divide then to obtain the average rate of change.

For the function f(x), the numeric values at x = -2 and x = 2 are given as follows:

Hence the rate is of:

(26 - 14)/(2 - (-2)) = 12/4 = 3. -> positive rate -> increasing.

For function g(x), the rate is given as follows:

(-16 - 4)/4 = -5.

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 P - 7 = J is an expression in simplest form that represents the sum of their ages.

What is a mathematical expression?

A mathematical expression is a phrase that includes at least two numbers or variables, at least one arithmetic operation, and the expression itself. This mathematical operation may be addition, subtraction, multiplication, or division.

                               Any mathematical statement made up of numbers, variables, and an operation between them is called an expression or an algebraic expression.

Susie is three times as older as her sister Jenny.

     3S = J

Paula is 7 years younger than Jenny.

        P - 7 = J

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Answer:

251 - 77 = 174

Therefore, the range of this sample data is 174.

Step-by-step explanation:

Answer:

Helmet = 29, Bell = 10, Lock = 24

well, the error is 3.06 - 2.955 = 0.105.

now, if we take 3.06(origin amount) to be the 100%, what's  0.105 off of it in percentage?

[tex]\begin{array}{ccll} amount&\%\\ \cline{1-2} 3.06 & 100\\ 0.105& x \end{array} \implies \cfrac{3.06}{0.105}~~=~~\cfrac{100}{x} \\\\\\ 3.06=10.5\implies x=\cfrac{10.5}{3.06}\implies x\approx 3.43[/tex]

Answer: b

Step-by-step explanation:

Okay, so what you need to do is plug in the 5 for x in all equations. once you do that, then you’ll see if the statement makes sense.

A would be= 2 + 5 = 3 and that would be 7=3 which it doesn’t

B would be 3(5)=15 and that would be 15=15

C would be 5/2=10 and thats not true because that would be 2.5=10

And lastly D would be 5-7=12 and that's nit true because that would be -2=12 and that's not true

The value of x in the given triangle is obtained as 4.21. The solution has been obtained by using trigonometry.

What is trigonometry?

A branch of mathematics known as trigonometry is concerned with the study of right-angle triangles, including their sides, angles, and relationships.

We are given perpendicular as 7 and base as x.

We know that tan θ is the ratio of perpendicular to base.

Here, θ = 59°

So,

⇒ Tan 59° = [tex]\frac{7}{x}[/tex]

⇒ 1.66 = [tex]\frac{7}{x}[/tex]

⇒ 1.66x = 7

⇒ x = [tex]\frac{7}{1.66}[/tex]

⇒ x = 4.21

Hence, the value of x in the given triangle is obtained as 4.21.

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Therefοre, οnly the cοmparisοns (A and C) are true.

In mathematics, a cοmparisοn is a statement that describes the relatiοnship between twο quantities οr expressiοns. Cοmparisοns can be used tο determine if twο values are equal, if οne value is greater than οr less than anοther value, οr if twο values are prοpοrtiοnal tο each οther.

A. 3.2 > 0.32

This cοmparisοn is true. 3.2 is greater than 0.32 because 3.2 has a whοle number value οf 3, which is greater than the whοle number value οf 0 in 0.32.

B. 4.7 < 4.70

This cοmparisοn is false. 4.7 is nοt less than 4.70 because the extra zerο in 4.70 dοes nοt change its value, and .7 is nοt less than .70.

C. 2.6 > 2.59

This cοmparisοn is true. 2.6 is greater than 2.59 because the extra 9 in 2.59 dοes nοt change its value, and 6 is greater than 5.

Therefοre, οnly the cοmparisοns (A and C) are true.

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The pair of equations that can be used to find x and y are:

x + y = 62

(1.02)x = 7

The form of the pair of equations would be:

hours worked last week + hours worked this week = total hours worked both weeks equation 1

(1 + percent increase/100) x hours worked last week = hours worked this week equation 2

x + y = 62 equation 1

(1 + 20/100) ×x = y

(1.02)x = 7 equation 2

The two equations can be solved using either of these three methods:

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Answer:

the value of x for which the volume of the box will be as large as possible is approximately 2.67 inches.

Step-by-step explanation:

Let x be the length of the side of the squares that are cut out of the corners of the cardboard. Then the dimensions of the base of the rectangular box will be:

Length = 16 - 2x (since two squares of side x are cut out of each end of the 16-inch length)

Width = 16 - 2x (since two squares of side x are cut out of each end of the 16-inch width)

The height of the box will be x, since that is the height of the squares that were cut out.

The volume of the box can be expressed as:

V = Length × Width × Height

V = (16 - 2x) × (16 - 2x) × x

V = 4x^3 - 64x^2 + 256x

To find the value of x that maximizes the volume of the box, we can take the derivative of V with respect to x and set it equal to zero:

dV/dx = 12x^2 - 128x + 256 = 0

We can solve this quadratic equation for x using the quadratic formula:

x = [128 ± sqrt(128^2 - 4 × 12 × 256)] / (2 × 12)

x = [128 ± sqrt(16384)] / 24

x = [128 ± 128] / 24

x = 10.67 or x = 2.67

Since x represents the length of a side of a square, it must be non-negative. Therefore, the only valid solution is x = 2.67 inches.

So, the value of x for which the volume of the box will be as large as possible is approximately 2.67 inches.

Answer: Let x be the side length of the square corners that are cut out of the cardboard. Then the dimensions of the base of the box (after the corners have been cut out and the flaps folded up) are 16-2x by 16-2x, and the height of the box is x.

The volume V of the box is given by:

V = (16-2x)(16-2x)(x)

Expanding this expression gives:

V = 4x^3 - 64x^2 + 256x

To find the value of x that maximizes V, we can take the derivative of V with respect to x and set it equal to zero:

dV/dx = 12x^2 - 128x + 256 = 0

Dividing both sides by 4 gives:

3x^2 - 32x + 64 = 0

This quadratic equation can be factored as:

(3x - 16)(x - 4) = 0

So the solutions are x = 16/3 and x = 4.

Since x must be less than half of 16 (the side length of the cardboard), we can reject x = 16/3. Therefore, the value of x that maximizes the volume of the box is x = 4 inches.

Step-by-step explanation:

Mary has 60/4 = 15$

Nick has 60/5 = 12$

The total amount of money that Mary and Nick have is 15$+12$ = 27$

To find out how much money Jane has, we need to subtract the total amount that Mary and Nick have from the total amount of money they have altogether:

60$ - 27$ = 33$

Therefore, Jane has 33$.

Answer: $33

Step-by-step explanation:

1/5 of $60 is 12. 1/4 of $60 is 15. 15 + 12 = 27. To find out how much money Jane has, simply subtract 60 - 27 to get $33.

Answer:

.625

Step-by-step explanation: calculator

I'm pretty sure that this is correct I'm sorry I tried my best

Answer:

Value of y is 6

Step-by-step explanation:

Two features of a parallelogram:

1. Opposite angles are equal:

which means:
m∠L = m∠N

∴[tex]14x - 10 = 6x + 22[/tex]

Bring like terms together and make x the subject of the equation:

[tex]14x - 6x = 22 + 10[/tex]

[tex]8x = 32[/tex]

[tex]x = \frac{32}{8}[/tex]

∴x = 4

Substitute this value of x to determine the measurement of ∠N:

   m∠L = [tex]14(4) - 10[/tex]

            = 46°

∴m∠N = 46°

2. Adjacent angles are supplementary:

which means:

m∠M + m∠N = 180°

[tex](23y - 4) + 46 = 180[/tex]

Expand the parenthesis, bring like terms together and make y the subject of the equation:

[tex]23y - 4 + 46 = 180[/tex]

[tex]23y = 180 - 46 + 4[/tex]

[tex]23y = 138[/tex]

[tex]y = \frac{138}{23}[/tex]

∴y = 6

Answer:If my calculations are right it should be 18

Step-by-step explanation:

Therefore , the solution of the given problem of triangle comes out to be  BZ = (8/9) * BC = 40/9 as a result.

Due to its two or more extra sections, a trapezoid is a polygon. Its shape is a simple rectangle. Only the three edges A, B, but instead C distinguish a triangle from a regular triangle. Euclidean geometry produces a singular area rather than a cube when the borders are not perfectly collinear. Triangular shapes are defined as having three sides and three angles. Angles are formed by the meeting of a quadrilateral's 3 sides.

Here,

Let D represent the intersection of the angle bisector BY and the side AC, and let E represent the intersection of the angle bisector CZ and the side AB.

=>BC² = AB² + AC² - 2AB * AC * cos(BAC)

However, since BY and CZ are angle bisectors, we can calculate BAC as follows:

=> BC² = 8² + 9² - 2(8)(9)cos[(180° - (ABC + ACB))/2]

If we simplify, we get:

=> BC² = 145 - 144cos[(ABC + ACB)/2]

Using the angle bisector theory once more, we have the following:

=> AD/DC = 8/BC

=> DC = BC(AD/8)

Simplifying and substituting into the Law of Cosines equation yields:

=> BC² = 145 - 144cos[(ABC + ACB)/2]

=>  145 - 144cos[(180° - BAC)/2]

=> 145 - 144cos(B/2)

in which B Equals ABC.

We can now replace the value of BZ that we discovered earlier with:

=> BZ = (8/9) * BC

and solve for BC:

=> BC² = 145 - 144cos(B/2)

=>  (9/8)² * (145 - 144cos(B/2)) + (8/9)² * BZ²

Simplifying and substituting BZ results in:

BC² = 25

Since BC is a length, we obtain: by taking the positive square root.

=> BC = 5

=> BZ = (8/9) * BC = 40/9 as a result.

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Answer:

sin x = d/e, cos x = f/e, tan x = d/f

Step-by-step explanation:

We will use trigonometry and SOH CAH TOA for this problem.

sin x = (opposite/ hypotenuse)

sin x = d/e

cos x = (adjacent / hypotenuse)

cos x = f/e

tan x = (opposite/adjacent)

tan x = d/f

Answer:

b

Step-by-step explanation:

Answer:

Step-by-step explanation:

D) All interior angles are right angles.

To prove that a polygon is a rectangle, you must show that all of its interior angles are right angles. The other properties listed may be true for certain types of rectangles, but they are not sufficient to prove that a polygon is a rectangle.

a. Length of GK = [tex]\sqrt{34}[/tex]  and Length of adjacent to GK = [tex]\sqrt{34}[/tex]

b. Slope of GK = [tex]\frac{5}{3}[/tex] and Slope of adjacent to RS =  [tex]-\frac{3}{5}[/tex]

c. The parallelogram GHJK is Square.

A quadrilateral with two sets of parallel sides is referred to as a parallelogram. As a result, a parallelogram's opposite sides are parallel and congruent in length, and its opposite angles are similarly congruent.

Given in figure GHJK, the vertices are G(-3, 6), H(2, 3), J(-1, -2), K(-6, 1)

a. Length of line = [tex]\sqrt{({x_{2}-x_{1})^{2} } + ({y_{2}-y_{1})^{2}}[/tex]

for points G(-3, 6) and K(-6, 1)

Length of GK = [tex]\sqrt{(-6+3)^{2} + (1-6)^{2} }[/tex] = [tex]\sqrt{34}[/tex]

Length of GK = [tex]\sqrt{34}[/tex]

Length of adjacent side (GH, KJ) to GK =  [tex]\sqrt{(2+3)^{2} + (3-6)^{2} }[/tex] = [tex]\sqrt{34}[/tex]

Length of adjacent to GK = [tex]\sqrt{34}[/tex]

b. [tex]Slope = \frac{(y_{2} -y_{1})}{(x_{2} -x_{1})}[/tex]

Slope of GK = [tex]\frac{(1 - 6)}{(-6 + 3)}[/tex] = [tex]\frac{5}{3}[/tex]

Slope of GK = [tex]\frac{5}{3}[/tex]

Slope of adjacent side to GK = [tex]\frac{3-6}{2+3}[/tex] = [tex]-\frac{3}{5}[/tex]

Slope of adjacent to RS =  [tex]-\frac{3}{5}[/tex]

c. All sides are equals to [tex]\sqrt{34}[/tex]

So, length of diagonal GJ = [tex]\sqrt{({-3+1))^{2} } + ({6+2})^{2}} = \sqrt{68}[/tex]

and length of diagonal HK = [tex]\sqrt{({2+6))^{2} } + ({3-1})^{2}} = \sqrt{68}[/tex]

All sides and diagonals are equal then parallelogram GHJK is Square.

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Answer:

C. The areas of the shaded regions are equal

Step-by-step explanation:

The area of a circle of radius r = πr²

The area of the shaded region in the left figure
= Area of outer circle - Area of inner circle
= π(AC)² - π(AB)²

Since RT = AC and RS = AB, substituting for AC and AB gives us
π(AC)² - π(AB)² = π(RT)² - π(RS)²

The expression on the right is the area of the shaded region in the left circle

So the shaded regions of both figures have the same area

Answer
C. The areas of the shaded regions are equal

We may use the trapezoid area formula to get the areas of trapezoids EJFG, FGHJ, and HJIE. The entire area of polygon EFGHIJ may then be calculated by adding these regions together.

To begin, we must determine the lengths of segments JF, JH, JE, and JI. We can get these lengths using the Pythagorean theorem: JF = (JG2 - FG2) = (122/102) = 44 = 211 cm JH = (JG2 - GH2) = (122) - 82 = √80 = 4√5 cm JE =(JF2 - EF2) =(21112 - 52) =(44 - 25) = 19 cm JI = (JH2 - IH2) = (452 - 32) = (80 - 9) = 71 cm We can now calculate the trapezoidal areas: Trapezoid area EJFG = (JE + JF) * FG / 2 = (211 + 19) * 5 / 2 = 511 + 519 cm2 Trapezoid area FGHJ = (FG + GH + JF) + JH) * HJ / 2 = (10 + 8 + 211 + 45) * 12 / 2 = 120 + 1211 + 245. HJIE = (JI + JE + IH + HE) trapezoid area * JH / 2 = (√71 + √19 + 3 + 3) * 4√5 / 2 = 10√5 + 2√71 cm^2 Lastly, we may sum these areas to calculate the total area of polygon EFGHIJ: E = trapezoid area EJFG + trapezoid area FGHJ + trapezoid area HJIE E = (5√11 + 5√19) + (120 + 12√11 + 24√5) + (10√5 + 2√71) E = 5√11 + 5√19 + 120 + 12√11 + 24√5 + 10√5 + 2√71 E = 120 + 15√11 + 34√5 + 15√19 + 2√71 E ≈ 372.9 cm^2 As a result, the area of polygon EFGHIJ is 372.9 square centimeters.

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Answer:

Step-by-step explanation:

We can use the information given to find the equation of the line that represents the relationship between the number of bean stalks and the yield. The equation of a line is typically given by the slope-intercept form, y = mx + b, where y is the dependent variable (in this case, the yield), x is the independent variable (the number of bean stalks), m is the slope, and b is the y-intercept.

To find the slope of the line, we can use the formula:

m = (Y2 - Y1) / (n2 - n1)

where (n1, Y1) and (n2, Y2) are two points on the line. We can use the two data points given in the problem to find the slope:

m = (190 - 115) / (8 - 3) = 15

To find the y-intercept, we can use the point-slope form of a line, which is:

y - Y1 = m(x - n1)

where (n1, Y1) is one of the points on the line. We can use the point (3, 115):

y - 115 = 15(x - 3)

Simplifying:

y = 15x - 20

Therefore, the equation that represents the relationship between the number of bean stalks and the yield is:

Y = 15n - 20

This equation tells us that for each additional bean stalk planted, the yield increases by 15 ounces, and the y-intercept of -20 indicates that even if no bean stalks were planted, there would still be a yield of -20 ounces (which doesn't make physical sense in this case, but is a mathematical artifact of the linear regression).

Answer:

the total force on the port hole due to liquid pressure is approximately 7400 pounds.

Step-by-step explanation:

The area of the circle is A = πr^2, where r = d/2 = 1 foot is the radius of the circle. So, the area is A = π(1 ft)^2 = π ft^2.

The port hole is submerged in water, with a height of 6 feet. The pressure of water at a depth of h feet is given by the formula P = ρgh, where ρ = 62.4 lb/ft^3 is the density of water, and g = 32.2 ft/s^2 is the acceleration due to gravity.

The total force on the port hole due to liquid pressure is the product of the pressure and the area of the circle, so we have:

F = P × A = ρgh × A = 62.4 lb/ft^3 × 32.2 ft/s^2 × 6 ft × π ft^2 ≈ 7400 lb

Therefore, the total force on the port hole due to liquid pressure is approximately 7400 pounds.

Answer: 578490-=356478e

Step-by-step explanation:

part of a circle is 75849=n so we use all the equations so 56748=4 we know the answer by the book page 356 then on chemestry rules.