Function f is defined by the equation f(x)=y?.
a. What is f(2)?

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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Answer:

[tex]2x + 3 + 2x = 90[/tex]

Step-by-step explanation:

Complementary angles are either of two angles that added together produce an angle of 90°.

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:

Correct option is C)

We know that

Final price = initial price(1+

100

rate

)

time

Here the final price = Rs. 8784, time =3 yrs, rate =−10%p.a.

The rate is negative since the price is depriciating.

Let the initial price = Rs. x.

∴x×(1−

100

10

)

3

=8748

⇒x×

10

9

×

10

9

×

10

9

=8748

⇒x=8748×

9

10

×

9

10

×

9

10

= Rs. 12000

So, the price of the machine 3 years back = Rs. 12000.

Ans- Option C.

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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The meaning of the statement g(1) = 200 include the following: A. after 1 hour the helicopter is 200 miles from its destination.

In Mathematics, a linear function can be defined as a type of function whose equation is graphically represented by a straight line on the cartesian coordinate.

Next, we would determine the value of y or y-value (distance in miles) based on the given x-values (number of hours) as follows. When x = 1 hour, the y-value (distance in miles) can be calculated as follows:

g(x) = 300 - 100x

g(1) = 300 - 100(1)

g(1) = 200

Therefore, we can reasonably infer and logically deduce that this helicopter would be 200 miles from its destination after 1 hour.

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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 values of RS and ST for the collinear points R, S and T are RS = 66 and ST = 49.

When points are collinear, it means they lie on the same straight line. In this case, we are given three collinear points R, S, and T, and the lengths of RS, ST, and RT.

Let's use the fact that the sum of the lengths of RS and ST equals the length of RT.

This gives us the equation:

RS + ST = RT

Substituting the given values, we get:

(8y + 2) + (5y + 9) = 115

Simplifying this equation, we get:

13y + 11 = 115

Solving for y, we get:

y = 8

Now that we know the value of y, we can find the values of RS and ST:

RS = 8y + 2 = 66

ST = 5y + 9 = 49

Therefore, RS = 66 and ST = 49.

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Complete question

Use the number line below, where RS=8y+2, ST=5y+9, and RT=115

Find the value of RS and ST,

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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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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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:

a) 8.5 kg

b) 14.2 N

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.

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 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.

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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Answer:

let m = 4x

n = 5x

therefore

4x+5x = 180⁰

x = 20⁰

n⁰ = p⁰

5×20⁰ = p⁰

100⁰ = p⁰

The center of the dilation is the same as the center of the original triangle. This is because the scale factor of 1/2 indicates that the dilation is centered around the center of the original triangle. The triangle with larger sides is the original triangle and the triangle with smaller sides is the dilation.

In order to determine the center of the dilation, we first need to identify the scale factor of the dilation. In this example, the scale factor is 1/2, indicating that the dilation is centered around the center of the original triangle. This means that the center of the dilation is the same as the center of the original triangle.

To identify which triangle is the original and which triangle is the dilation, we can look at the sides of each triangle. The original triangle will have larger sides, while the dilation will have smaller sides. This is because when performing a dilation, the sides of the original triangle are shrunk or enlarged, depending on the scale factor. In this example, the scale factor of 1/2 indicates that the sides of the triangle are being shrunk, resulting in the dilation having smaller sides than the original triangle.

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the answer is (D) 32.2

hopefully it help

Answer: d.) 32.2

Step-by-step explanation:

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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Answer:

x = 4.995

Step-by-step explanation:

Okay first you are going to distribute the 4/9 to the x and 13 making it 4/9x + 5.78 = 8

Then subtract 5.78 from each side to make it 4/9x = 2.22

Then divide 4/9 from both sides making x = 4.995

(this answer is rounded and not exact)

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:

[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]

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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D. ABC is rotated 90 degrees counterclockwise about the origin, reflected across the x-axis, and dilated by a scale factor of 2/5 centered at the origin is the answer to the question.

Rotation is the motion of turning a figure around a fixed point.

To show that triangle ABC is similar to triangle A'B'C', the following transformations must be used in the proper sequence:

rotation, reflection, and dilation.

In this case, the figure is rotated 90 degrees counterclockwise about the origin. In this case, the figure is reflected across the x-axis. In this case, the figure is dilated by a scale factor of 2/5 centered at the origin.

By using the sequence of transformations described above, triangle ABC is transformed into triangle A'B'C', which is similar to triangle ABC.

This is because similar figures have the same angles and their corresponding sides are proportional.

Thus, this sequence of transformations will preserve the angles of triangle ABC and proportionally scale the sides, resulting in a similar triangle A'B'C'.

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Answer:

The answer is D!!! Hopes this helps!!

Step-by-step explanation:

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

Answer:

46in²

Step-by-step explanation:

1/2 1 +5+3+1+3+14

46in²