What is the perimeter? 3 mi. Il mi.​

Answer:

Step-by-step explanation:

1. -(5/x^6)

2. 4398046511104

3. 65536

4. 1/100

5. 1/5764801

On Using Heron's formula, area of the triangle is 60.866 square units

Using the formula A = 1/2bh,  

area of the triangle: Area = 1/2 * √170 * 9.073 ≈ 60.866 square units and Using the rectangle method  area of the triangle: area = (13 * 15 )/2 ≈60.866 square units  .

Method 1: Heron's Formula

Heron's formula can be used to calculate the area of a triangle given the lengths of its three sides. The formula is as follows:

[tex]Area = \sqrt{(s(s-a)(s-b)(s-c))}[/tex], where a, b, and c are the lengths of the sides of the triangle, and s is the semi-perimeter of the triangle (s = (a+b+c)/2).

Using the coordinates of the vertices of the triangle, we can find the lengths of the sides using the distance formula:

[tex]AB = \sqrt{ ((1-(-8))^2+(-6-7)^2)} = \sqrt{170} \\AC = \sqrt{((5-(-8))^2+(9-7)^2) } = \sqrt{290}\\BC =\sqrt{((5-1)^2+(9-(-6))^2)} = \sqrt{370}[/tex]

The semi-perimeter, s, is then:

s = (AB + AC + BC)/2 = (√170 + √290 + √370)/2 ≈ 16.362

Using Heron's formula, we can then find the area of the triangle:

[tex]Area = \sqrt{s(s-AB)(s-AC)(s-BC))} \\= \sqrt{ (16.362(16.362-\sqrt{170} )(16.362-\sqrt{290} )(16.362-\sqrt{370} ))} \\= 60.866[/tex]

Method 2: Base and Altitude Method

Another way to find the area of the triangle is to use the formula A = 1/2bh, where b is the length of the base of the triangle and h is the height. We can find the length of the base by using the distance formula to calculate the distance between the points (-8,7) and (1,-6):

[tex]b =\sqrt{((1-(-8))^2+(-6-7)^2) }= \sqrt{170}[/tex]

To find the height, we need to find the equation of the line passing through (-8,7) and (1,-6), which is the same as the equation of the line passing through the midpoint of AB and C, which can be found using the midpoint formula:

Midpoint of AB: ((-8+1)/2, (7-6)/2) = (-3.5, 0.5)

Line passing through (-8,7) and (1,-6): y = (-13/9)x + (103/9)

To find the altitude, we need to find the distance from point (5,9) to this line. We can use the formula for the distance between a point and a line:

h = |(-13/9)(5) + 1(9) - (103/9)|/√((-13/9)² + 1²) ≈ 9.073

Using the formula A = 1/2bh, we can then find the area of the triangle:

Area = 1/2 * √170 * 9.073 ≈ 60.866 square units

Method 3: Rectangle Method

Another way to find the area of the triangle is to use the rectangle method. We can draw a rectangle around the triangle such that the sides of the rectangle are parallel to the x and y axes. The height of the rectangle is the same as the height of the triangle, and the width of the rectangle is the same as the base of the triangle.

We can find the dimensions of the rectangle by finding the differences between the x-coordinates and y-coordinates of the vertices:

Width: 5 - (-8) = 13

Height: 9 - (-6) =15

Using the rectangle method ,we can then find the area of the triangle:

area = (13 * 15 )/2 ≈60.866 square units

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Answer:

Step-by-step explanation:

To factor the equation u^2 - 4u + 4, we need to find two numbers that add up to -4 and multiply to 4.

The two numbers are -2 and -2 because:

-2 + (-2) = -4 and (-2)(-2) = 4

Using these numbers, we can factor the equation as:

u^2 - 4u + 4 = (u - 2)(u - 2) = (u - 2)^2

Therefore, the factored form of the equation is (u - 2)^2.

Answer:

Step-by-step explanation:

We can simplify the expression by first finding the common denominator for the fractions in each term, which is 45:

4/9 x 18/5 + 14/5 x 2/9 - 8/18

= (418)/(95) + (142)/(59) - 8/18 (multiplying to get the common denominator)

= 72/45 + 28/45 - 8/18 (simplifying each term)

= (72 + 28)/45 - 8/18 (adding the first two terms)

= 100/45 - 4/9 (simplifying)

= (1001)/(451) - (45)/(95) (multiplying by 1 in the form of 5/5)

= 100/45 - 20/45 (simplifying)

= 80/45 (final answer, which can be simplified to 16/9)

Therefore, 4/9 x 18/5 + 14/5 x 2/9 - 8/18 = 16/9.

the area of the equilateral triangle using the formula for the area of a regular polygon: 43.5 units squared

The area of a customary polygon is given as:

[tex]=\frac{1}{2} ap[/tex]

'a' is the apothem and 'p' is the edge.

The edge of the equilateral triangle is 3×10=30 units.

The apothem is given by:

[tex]a= \frac{8}{\sqrt[2]{3} }\\ \\a= \frac{10}{\sqrt[2]{3} } \\\\a=2.9[/tex]

Consequently, the area is

1/2 * 2.9 * 30 = 43.5

On the off chance that all the polygon sides and inside points are equivalent, they are known as ordinary polygons. Instances of standard polygons are square, equilateral triangle, and so forth. In standard polygons, not exclusively are the sides consistent however the points are as well. That means they are equiangular.

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the complete question is:

Bianca calculated the height of the equilateral triangle with side lengths of 10. tangent (30) = StartFraction 5 Over h EndFraction An equilateral triangle with side lengths of 10 is shown. A bisector is drawn to split the side into 2 equal parts and splits the angle into 2 30 degree segments. Then, she used the formula for area of a triangle to approximate its area, as shown below. A = one-half b h. = one-half (10) (8.7). = 43.5 units squared. Calculate the area of the equilateral triangle using the formula for area of a regular polygon, and compare it to Bianca's answer. The apothem, rounded to the nearest tenth, is units. The perimeter of the equilateral triangle is units. Therefore, the area of the equilateral triangle is , or approximately 43.5 units2. The calculated areas are .

[tex]\textit{volume of a cylinder}\\\\ V=\pi r^2 h~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ h=19\\ V=4100 \end{cases}\implies 4100=\pi r^2(19)\implies \cfrac{4100}{19\pi }=r^2 \\\\\\ \sqrt{\cfrac{4100}{19\pi }}=r\implies 8.29\approx r[/tex]

The requirements for a given year to be a leap year are the year must be divisible by 4, if the year is divisible by 100, it is not a leap year, unless it is also divisible by 400.

In the Gregorian calendar, a leap year is a year that has 366 days instead of the usual 365 days. The reason for this is to account for the fact that the Earth takes slightly longer than 365 days to orbit the Sun.

This system of leap years helps to keep our calendar in synchronize with the astronomical year and ensure that seasonal events, such as the solstices and equinoxes, occur at approximately the same time each year.

For example, 1900 was not a leap year because it is divisible by 100 but not by 400, while 2000 was a leap year because it is divisible by both 100 and 400.

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the ladder reach 14.13 feet high up the wall.

Anelys leans a 16-foot ladder against a wall so that it forms an angle of 61° with the ground.

We have to calculate how high up the wall does the ladder reach.

Solution

We have to calculate how high up the wall does the ladder reach.

Let's assume that the height of the wall is h, and the distance between the ladder's foot and the wall is d.

The hypotenuse of the right-angled triangle (the ladder) is 16 feet.

The angle formed by the ladder and the wall is 61°.

We know that sine function of an angle is the ratio of the opposite side to the hypotenuse:

Therefore: Now we can calculate the height of the wall:

We get that the ladder reaches 14.13 feet up the wall (rounded to the nearest hundredth of a foot).

So, the answer is 14.13 feet.

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The skier needs to go at least 65.4 miles/hour to beat the person in first place.

What is trigonometry?

It is the branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.

To solve this problem, we need to use trigonometry to find the horizontal distance the skier needs to cover in order to beat the person in first place, and then use this distance and the time it took the first-place skier to calculate the required speed.

A represents the skier's starting point, B represents the end of the slope where the person in first place finished, and C represents the top of the mountain. The angle of depression is labeled θ and is equal to 60 degrees.

Using trigonometry, we can calculate the horizontal distance AB using the tangent function:

tan θ = opposite/adjacent

where opposite = h = 10000 ft (the height of the mountain) and adjacent = AB (the horizontal distance we want to find).

Substituting θ = 60 degrees and h = 10000 ft, we get:

tan 60 = AB/10000

√3 = AB/10000

AB = 10000√3 ≈ 17321.17 ft

So the skier needs to cover a horizontal distance of about 17321.17 feet in order to beat the person in first place.

To find the required speed, we need to convert this distance to miles and divide by the time it took the first-place skier (3 minutes):

17321.17 ft = 17321.17/5280 miles ≈ 3.28 miles

Speed = distance/time = 3.28 miles / 3 minutes = 1.09 miles/minute

To convert miles/minute to miles/hour, we need to multiply by 60 (the number of minutes in an hour):

Speed = 1.09 miles/minute x 60 minutes/hour ≈ 65.4 miles/hour

Therefore, the skier needs to go at least 65.4 miles/hour to beat the person in first place.

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A grouped frequency table (grouped frequency conveyance) is an approach to coordinating an enormous arrangement of information into additional reasonable gatherings.

The grouped frequency table

________________________________________________________

 Width (in mm) C.F. Class Interval Frequency ________________________________________________________

 ≥ 20 50 20 - 30 50 - 44

= 6 ≥ 30 44 30 - 40 44 - 28

= 16

 ≥ 40 28 40 - 50 28 - 20

= 8

 ≥ 50 20 50 - 60 20 - 15

= 5

 ≥ 60 15 60 - 70 15 - 0

= 15

_____________________________________________________

                                                                                          ∑f = 50

The gatherings that we arrange the mathematical information into are called class intervals. They can have something similar or different class widths and should not cover.

The principal segment is for the gatherings where we compose the class intervals.

The center section is for the count marks.

The last segment is the frequency section where we can include the count marks.

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the complete question is:

The widths of 50 leaves of a plant were measured in mm and their cumulative frequency distribution is shown in the following table. Make frequency distribution table for this . CLASS FREQUENCY >20 50 >30 44 >40 28 >50 20 ​>60 15

The values of a=6 and b=22.5 by using property of similar triangles that ratio of similar sides is equal.

Similar triangles are triangles that have the same shape, but not necessarily the same size. This means that the corresponding angles of the two triangles are equal, and the corresponding sides are proportional to each other.

Since the triangles are similar, then the following equivalent ratios

[tex]\frac{4}{10}= \frac{9}{b} =\frac{a}{15}[/tex]

[tex]\frac{4}{10}= \frac{9}{b}[/tex]

[tex]4b =90[/tex]

[tex]b= 22.5[/tex]

Similarly, [tex]\frac{4}{10} = \frac{a}{15}[/tex]

[tex]10a= 60\\a= 6[/tex]

The values of a=6 and b=22.5 by using property of similar triangles that ratio of similar sides is equal.

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The complete question is :

These two triangles are similar. Find side lengths a and b. The two figures are not drawn to scale. Diagram is shown in attached image.

Answer:

[tex]\large\textsf{See below.}[/tex]

Step-by-step explanation:

[tex]\textsf{We are asked which angles are equal to 180}^{\circ} \textsf{- x.}[/tex]

[tex]\textsf{We should look at m} \angle 6 \ \textsf{as it's equal to x.}[/tex]

[tex]\textsf{Note that m} \tt \angle 6 \ \textsf{is formed by Perpendicular Lines.}[/tex]

[tex]\large\underline{\textsf{What are Perpendicular Lines?}}[/tex]

[tex]\textsf{Perpendicular Lines are \underline{2 lines} that are \underline{opposite slopes}; They \underline{intersect} each other.}[/tex]

[tex]\textsf{Because Perpendicular Lines are opposite, they form \underline{4} 90}^{\circ} \ \textsf{angles.}[/tex]

[tex]\large\underline{\textsf{We can identify that;}}[/tex]

[tex] \tt m \angle 6 = 90^{\circ}.[/tex]

[tex]\textsf{Remember that we are asked for angles equal to the difference of 180}^{\circ} \ \textsf{and} \ \tt m \angle 6.[/tex]

[tex]\large\underline{\textsf{Substitute;}}[/tex]

[tex]\tt 180^{\circ} - 90^{\circ} = 90^{\circ}.[/tex]

[tex]\textsf{We are asked for angles with the same measure.}[/tex]

[tex]\textsf{Remember that Perpendicular Angles equal 90}^{\circ}.[/tex]

[tex]\textsf{This means that Angles 5, 7, 8, 9, 10, 11, and 12 have the measure of 90}^{\circ}.[/tex]

[tex]\large\underline{\textsf{Hence;}}[/tex]

[tex]\Large\boxed{\tt \angle 5, \angle 7, \angle 8, \angle 9, \angle 10, \angle 11, and \ \angle 12 \ equal \ 180^{\circ} - m\angle 6.}[/tex]

These equatiοns represent the fact that the slοpes and y-intercepts οf the lines fοrming the sides οf the triangle are related, and that the lines intersect at the vertices οf the triangle.

An equatiοn is a mathematical statement that asserts that twο expressiοns are equal. It cοnsists οf twο sides, a left-hand side and a right-hand side, cοnnected by an equal sign "=". The expressiοns οn either side οf the equal sign can be numbers, variables, οr cοmbinatiοns οf bοth.

Here,

Part a: If the three sides οf the triangle are denοted by a, b, and c, then we can write the fοllοwing system οf equatiοns:

a + b = c

b - a = 3

These equatiοns represent the fact that the sum οf twο sides οf a triangle is equal tο the third side (a + b = c), and that the difference between twο sides is equal tο a given value (b - a = 3).

Part b: If the cοοrdinates οf the vertices οf the triangle are given by (x₁, y₁), (x₂, y₂), and (x₃, y₃), then we can write three equatiοns representing the three sides οf the triangle. Fοr example, the equatiοn οf the line passing thrοugh (x₁, y₁) and (x₂, y₂) is given by:

(y₂ - y₁)/(x₂ - x1) = (y - y₁)/(x - x₁)

where (x, y) are the cοοrdinates οf any pοint οn the line. Similarly, we can write equatiοns fοr the οther twο sides οf the triangle, and οbtain a system οf three linear equatiοns.

Part c: If the equatiοns οf the three sides οf the triangle are given by Ax + By + C = 0, Dx + Ey + F = 0, and Gx + Hy + I = 0, then we can write the fοllοwing system οf equatiοns:

A/Bx + C/B = -(A/By + C/B) (frοm the first equatiοn)

D/Ex + F/E = -(D/Ey + F/E) (frοm the secοnd equatiοn)

G/Hx + I/H = -(G/Hy + I/H) (frοm the third equatiοn)

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Complete question:

The lines that form the three sides of the triangle can be grouped into three different systems of two linear equations. Create 3 equations using different methods.

the answer is (D) 32.2

hopefully it help

Answer: d.) 32.2

Step-by-step explanation:

The slope of the ramp can also be expressed as -1/3.

What is the slope?

The slope of a line is a measure of its steepness. Mathematically, the slope is calculated as "rise over run" (change in y divided by change in x).

The slope of a ramp is a measure of its steepness or incline, expressed as the ratio of its vertical rise to its horizontal run.

In this case, the slope of the ramp is given as -7/21 feet.

This means that for every 21 feet of horizontal distance along the ramp, the ramp drops (or rises) 7 feet. The negative sign indicates that the ramp is descending, rather than ascending.

We can also simplify the fraction -7/21 by dividing both the numerator and denominator by their greatest common factor, which is 7:

-7/21 = (-1 * 7) / (3 * 7) = -1/3

So the slope of the ramp can also be expressed as -1/3. This means that for every 3 feet of horizontal distance along the ramp, the ramp drops (or rises) 1 foot. Again, the negative sign indicates that the ramp is descending.

Hence, the slope of the ramp can also be expressed as -1/3.

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Answer:

the star , the down arrow (3rd one), the stop sign

The area of a circle can be found using the formula:

A = πr^2

where A is the area and r is the radius of the circle.

Given that the radius of the circle is 6.13m, we can substitute this value into the formula and simplify as follows:

A = πr^2

A = π(6.13m)^2

A = 118.066129 m^2 (using a calculator)

Rounding this answer to 2 decimal places, we get:

A ≈ 118.07 m^2

Therefore, the area of the circle with radius 6.13m is approximately 118.07 square meters.

If two polygons are similar, the corresponding sides must be C. congruent.

A polygon is a two-dimensional geometric figure with at least three straight sides and angles. Triangles, squares, rectangles, and pentagons are all examples of polygons. Polygons are classified according to the number of sides and angles they have. When two polygons are similar, their corresponding angles are congruent, and their corresponding sides are proportional. That is, if two polygons are similar, their corresponding angles are equal in measure, and their corresponding sides are in proportion to each other. Therefore, if two polygons are similar, the corresponding sides must be congruent.

An explanation of the remaining answer options: A. Complementary refers to angles. Angles are complementary if their sum is 90 degrees. Therefore, this answer option is irrelevant to the question.B. Supplementary refers to angles. Angles are supplementary if their sum is 180 degrees. Therefore, this answer option is irrelevant to the question.D. Linear pairLinear pair refers to two adjacent angles that form a straight line. Therefore, this answer option is irrelevant to the question.

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Answer:

Step-by-step explanation:

(a) A slice through the center, parallel to a side:

To visualize this slice, imagine cutting the rectangle horizontally, exactly in the middle, so that you end up with two smaller rectangles stacked on top of each other. The resulting cross section will be a rectangle with the same width as the original rectangle, but only half of its length.

(b) A slice that cuts through the left top edge and the right bottom edge:

To visualize this slice, imagine cutting the rectangle diagonally from the top left corner to the bottom right corner. The resulting cross section will be a right triangle, with one of its legs being half of the original rectangle's width, and the other leg being half of its length.

(c) A slice that cuts across a corner through three sides of the rectangle:

To visualize this slice, imagine cutting the rectangle diagonally from one of the top corners to the opposite bottom corner, and then continuing the cut along one of the sides of the rectangle, until you reach the middle of that side. The resulting cross section will be a trapezoid, with one of its parallel sides being half of the original rectangle's width, and the other parallel side being half of its length. The two non-parallel sides will be angled.

The binomial is (a² + 4)

We can rewrite this as follows:

[tex]-a^2*(a^2 + 4) + (a^2 + 4)[/tex]

You can think of the a squared plus 4 thing as a common factor because it appears in both terms, then we can rewrite this as:

[tex]-a^2*(a^2 + 4) + (a^2 + 4)\\-a^2*( a^2 + 4 + 1)\\-a^2*(a^2 + 5)[/tex]

That is the expression factorized.

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Creating a rule y=f(x) by applying transformations to the fundamental function f(x) = √x  The rule that would produce the given graph is f(x) = 2√(x-1) + 1.

To transform the basic square root function f(x) = √x to the given graph, we can use the following transformations:

1. Horizontal shift to the right by 1 unit: f(x-1) = √(x-1)

This will move the point (1, 1) on the basic function to (2, 1) on the new graph.

2. Vertical stretch by a factor of 2: 2f(x-1) = 2√(x-1)

This will result in a two-fold vertical stretch of the graph.

3. Vertical shift upward by 1 unit: 2f(x-1) + 1 = 2√(x-1) + 1

This will move the entire graph upward by 1 unit.

Therefore, the rule that would produce the given graph is f(x) = 2√(x-1) + 1.

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The final function that represents the given transformations is:

f(x) = -log(x + 2) - 3

What is parent function?

In mathematics, a parent function is a basic function from which other functions can be derived through a set of transformations. It serves as a starting point for understanding more complex functions by showing the most basic characteristics of the function.

Starting with the parent function f(x) = log(x), the given transformations are:

Horizontal shift to the left by 2 units: The graph has been shifted 2 units to the left, which means that the argument of the logarithmic function should be (x + 2) instead of "x". So the function becomes:

f(x) = log(x + 2)

Reflection about the x-axis: The graph is the opposite of the parent function, which means it has been reflected about the x-axis. This changes the sign of the function, so we get:

f(x) = -log(x + 2)

Vertical shift down by 3 units: The graph has been shifted down by 3 units, which means that the entire function has been shifted downward by 3 units. So we get:

f(x) = -log(x + 2) - 3

Therefore, the final function that represents the given transformations is:

f(x) = -log(x + 2) - 3

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Answer:

the main points :

a) 100% - 21% is equal to 79% when expressed as a percentage.

b) To express 100% - 21% as a decimal, we can convert both percentages to decimals by dividing them by 100. Then, we can subtract the resulting decimals to get 0.79 as the decimal equivalent of 100% - 21%

Answer:

a) 8.5 kg

b) 14.2 N

the center of circle O is (12, -6).

To find the center of the circle , we need to find the intersection point of the perpendicular bisectors of any two sides of triangle .

The midpoint of BC is ((12+12)/2, (7+0)/2) = (12, 3.5), and the midpoint of AB is ((0+12)/2, (0+7)/2) = (6, 3.5).

The slope of BC is (7-0)/(12-12) = undefined, which means that the perpendicular bisector of BC is the vertical line x=12.

The slope of AB is (7-0)/(12-0) = 7/12,

the perpendicular bisector of AB has a slope of -12/7

passes through the midpoint (6, 3.5).

Therefore, its equation is:

y - 3.5 = (-12/7)(x - 6)

Simplifying:

y = (-12/7)x + 27

To find the intersection point of the two lines , x=12 into the second equation:

y = (-12/7)(12) + 27 = -6

Therefore, the center of circle O is (12, -6).

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a) Approximate the angles for quotient, Adjacent leg / hypotenuse = 0. 966 is equals to 15°.

b) Approximate the angles for quotient, Opposite leg / hypotenuse = 0. 469 is equals to 28°.

c) Approximate the angles for quotient, Adjacent leg / hypotenuse = 0. 309 is equals to 72°.

d) Approximate the angles for quotient, Opposite leg /adjacent leg = 1. 36 is equals to 54°.

In mathematics, the trigonometric functions are defined as periodic functions which relate an angle of right angled triangle to the ratio of the lengths of a right triangle to the lengths.

Now, we have provide some quotients and we have to calculate the approximate angles.

a) Adjacent leg / hypotenuse = 0. 966

from above definiation, Adjacent leg / hypotenuse = cosθ

so, cosθ = 0.966

=> θ = cos⁻¹(0.966) = 14.98° ~ 15°

b) Opposite leg /hypotenuse = 0. 469

from above definiation, Opposite leg / hypotenuse = sinθ

so, sinθ = 0.469

=> θ = sin⁻¹(0.469) = 27.79° ~ 28°

c) Adjacent leg / hypotenuse = 0.309

from above definiation, Adjacent leg / hypotenuse = cosθ

so, cosθ = 0.309

=> θ = cos⁻¹(0.309) = 72°

d)Opposite leg / adjacent leg = 1.36

from above definiation, Adjacent leg / hypotenuse = tanθ

so, tanθ = 1.36

=> θ = tan⁻¹(1.36) = 53.67° ~ 54°.

Hence, required angle is 54°.

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Complete question:

Approximate the angles that have following quotient

a)Adjacent leg / hypotenuse = 0. 966

b)Opposite leg / hypotenuse = 0. 469

c)Adjacent leg / hypotenuse = 0. 309

d)Opposite leg / adjacent leg = 1. 36

Answer:

x = 3

Step-by-step explanation:

5x = 61/4 - 1/4

5x = 60/4

5x = 15

x = 3

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Solution:

5x + 1/4 = 61/4

5x = 61/4 - 1/4

5x = 61-1/4

5x = 60/4

5x = 15

x = 15/5

x = 3

Hence x = 3

The order of the angles of the triangle from largest measure to smallest measure. will be ∠B, ∠A, and ∠C.

An angle measure is the measurement of the angle created by two rays or arms at a shared vertex in geometry. A protractor is used to measure angles in degrees (°).

Angle obtained from the triangle are:

[tex]\angle B=73.6^\circ[/tex]

[tex]\angle A= 55.345^\circ[/tex]

[tex]\angle C =50.977^\circ[/tex]

As a result, the arrangement of the triangle's angles is from biggest to smallest measure. ∠B, ∠A, and ∠C will be the results.

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The surface area of the cylinder is approximately 339.292 square feet.

What is cylinder?

A cylinder is a three-dimensional geometric shape that consists of two parallel, congruent circular bases and a curved lateral surface that connects the two bases.

The formula for the surface area of a cylinder is:

SA = 2πrh + 2π[tex]r^2[/tex]

where r is the radius of the base, h is the height of the cylinder, and π is the mathematical constant pi.

Substituting the given values, we get:

SA = 2π(6)(3) + 2π[tex](6)^2[/tex]

SA = 2π(18) + 2π(36)

SA = 36π + 72π

SA = 108π

To round the answer to the nearest thousandth, we need to evaluate 108π and approximate it to three decimal places using a calculator or by estimating π as 3.14159. Therefore:

SA ≈ 108π ≈ 339.292 [tex]ft^2[/tex]

Thus, the surface area of the cylinder is approximately 339.292 square feet.

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6.892 metres separate the pole's paper from the building's base.

The transition of the question is:

An 8 meter tall pole is leaning to a building at an angle of 60 degrees with respect to the ground, what is the distance from the paper of the pole to the base of the building?

The perpendicular is the side that is 8 metres long because this triangle is a right triangle.

The hypotenuse is the side that faces a 90 degree angle.

Also, we need the pole's height, which is the side that is "opposite" to the specified angle of 60 degrees.

What trigonometric ratio connects "opposite" and "hypotenuse" at this point?

That's SINE. Now that we have the answer, we may write it down as follows and solve it:

H = 8x1.723/2 Sinx = p/h Sin60 = 8/h H = 6.892

6.892 metres separate the pole's paper from the building's base.

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