Show your work and explain please!

put 62  it does have instructions right

The height of the building is approximately 102.86 meters.

Given that from a certain distance, the angle of elevation to the top of a building is 17 degrees. At a point 50 meters closer to the building, the angle of elevation is 31 degrees. To find the height of the building.In the given figure, AB is a building and C and D are two different points at a certain distance from the building. So that, angle BCA = 17° and angle BDA = 31°.We have to find the height of the building AB.

Now, let's find the distance between the building and point C and D. Let, BC = x50 meters closer to the building, so BD = x + 50 metersWe know, tangent of an angle = Perpendicular / BaseFor the angle 17°, we have to find AC, so;tan(17°) = AB / ACAB = AC tan(17°) ---(1)For the angle 31°, we have to find AD, so;tan(31°) = AB / ADAB = AD tan(31°) ---(2)Now, find the difference between (1) and (2).AC tan(17°) - AD tan(31°) = 0AC / AD = tan(31°) / tan(17°) [Divide both sides by AD tan(17°)]AC / (AD + 50) = tan(31°) / tan(17°) ----(3)We know the value of (AC + AD) is the actual distance between the building and the point.

So,(AC + AD) = x + (x + 50) = 2x + 50 metersNow we can use the value of AC / (AD + 50) in equation (3) and solve for AB.AB = AC tan(17°) = (2x + 50) tan(17°) tan(31°) / (tan(31°) - tan(17°))Substitute the value of AB from equation (1), so;AC tan(17°) = (2x + 50) tan(17°) tan(31°) / (tan(31°) - tan(17°))AC = (2x + 50) tan(31°) / (tan(31°) - tan(17°))Now, let's use the value of AC in equation (1),AB = AC tan(17°) = (2x + 50) tan(17°) tan(31°) / (tan(31°) - tan(17°)) = 102.86 meters (approx.)So, the height of the building is AB = 102.86 meters (approx.)

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Answer is 17, 60, -8, 4.25 in the given BODMAS

Brackets of Division, Multiplication, Addition, and Subtraction, or BODMAS, is an acronym. It is a method for resolving mathematical issues.

Brackets are resolved first in accordance with the BODMAS guidelines. Parentheses are also solved first in that case, then curly brackets, and finally square brackets.

It always produces the right outcomes when applied correctly. Uncertain questions that are poorly phrased may result in inaccurate answers.

Let's utilise BODMAS to answer a question:

W= 85 / 5

W = 17

W= 85 - 25

W = 60

W = 85/ 5- 25

W = -8

W= 85/ 25 - 5

W = 85/20

W = 4.25

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Answer:

C John should order 6 pizzas because if he orders 5 pizzas, he will be short 2 slices.

Step-by-step explanation:

14 people need 3 slices each.  We need to multiply.

14*3 = 42

Each pizza has 8 slices so divide by 8

42 /8 =  5   remainder 2

We round up to make sure to have enough.

We need at least 6 pizzas to have enough.  He is short 2 slices.

Answer:

C)

Step-by-step explanation:

John wants to order enough pizzas for 14 people to have 3 slices each. This means that he would need at least 42 slices of pizza:

[tex]14 * 3 = 42[/tex]

It is given that pizza has 8 slices. Divide 8 from 42:

[tex]\frac{42}{8} = 5.25[/tex]

Note that he cannot order .25 of a pizza, and so he will have to order 6.

C) John should order 6 pizzas because if he orders 5 pizza, he will be short 2 slices.

Note: The 2 slices is calculated by multiplying the total slices for a pizza (8) with .25:

[tex]8 * .25 = 2[/tex]

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The hypotenuse is √[(√2)² + (√3)²] = √(2+3) = √5 units.

The Iongest side of a right-angIed triangIe, or the side opposite the right angIe, is known as the hypotenuse in geometry. The Pythagorean theorem, which states that the square of the Iength of the hypotenuse equaIs the sum of the squares of the Iengths of the other two sides, can be used to determine the Iength of the hypotenuse.

The Iength of the Iegs are 1 unit, √2 units.

The hypotenuse is √[(1)² + (√2)²] = √(1+2) = √3 units.

The Iength of the Iegs are √5 units, √2 units.

The hypotenuse is √[(√5)² + (√2)²] = √(5+2) = √7 units.

The Iength of the Iegs are √5 units, √3 units.

The hypotenuse is √[(√5)² + (√3)²] = √(5+3) = √8 units.

The Iength of the Iegs are √5 units, 1 unit.

The hypotenuse is √[(√5)² + (1)²] = √(5+1) = √6 units.

The Iength of the Iegs are √2 units, √3 units.

The hypotenuse is √[(√2)² + (√3)²] = √(2+3) = √5 units.

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Therefore option b is the right option: After translated it in 6units: we get

A'(-2, 1), B'(-6, 1), C'(-10, 5)

The picture's coordinates are:

A’(-2+6, 1) = (-4, 1)

B’(6+6, 1) = (12, 1)

C’(2+6, 5) = (8, 5)

Define coordinates of a triangle?

A location on the plane that is defined by an ordered pair of numbers is called a coordinate in a triangle (x,y). The y-coordinate and x-coordinate of a point indicate its vertical and horizontal positions, respectively.

Triangles can be described using a variety of coordinate systems in the Euclidean plane. Triangular coordinates is a name for one of these systems. It is a particular instance of barycentric coordinates for a triangle, and in that case it is sometimes referred to as a ternary plot or as areal coordinates

A barycentric coordinate system in geometry is a coordinate system where a point's location is determined by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.) . A point's barycentric coordinates can be thought of as the center of mass (or barycenter) of masses that are positioned at the vertices of a simplex. These masses can all be positive or negative, but only if the point is contained within the simplex.

Many branches of mathematics and physics, such as computer graphics, finite element analysis, and fluid dynamics, make use of barycentric coordinates.

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The null hypothesis would be, "sleep deprivation does not affect reaction time."

A null hypothesis is an assumption in which researchers claim that there is no relationship between two factors. The hypothesis is used to examine whether the research hypothesis is true or false. For instance, in this study, the researcher would hypothesize that sleep deprivation affects reaction time.

The null hypothesis would be, "sleep deprivation does not affect reaction time."

Hence, the null hypothesis for the researcher who is interested in the effects of sleep deprivation on reaction time, and who randomly assigns participants to either a sleep deprivation group or a control group is "Sleep deprivation does not affect reaction time."

This would be tested using statistical tests like t-test and ANOVA.

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Answer:

An exponential function has the form `f(x) = ab^x`, where `a` and `b` are constants. To find the exponential function that passes through the points `(-2,1)` and `(-1,2)`, we can use these points to set up a system of equations to solve for the values of `a` and `b`.

Substituting the coordinates of the first point into the equation for an exponential function gives us:

`1 = ab^(-2)`

Substituting the coordinates of the second point into the equation for an exponential function gives us:

`2 = ab^(-1)`

Dividing these two equations gives us:

`(2)/(1) = (ab^(-1))/(ab^(-2))`

`2 = b`

Now that we know that `b=2`, we can substitute this value back into either equation to solve for `a`. Substituting into the first equation gives us:

`1 = a(2)^(-2)`

`1 = a(1/4)`

`a = 4`

So, the exponential function that passes through the points `(-2, 1)` and `(-1, 2)` is given by:

`f(x) = 4 * 2^x`.

To find the exponential function for the graph passing through (-2,1) and (-1,2), we need to use the general form of an exponential function, which is:

y = a * b^x

where:

a is the initial value or the y-intercept of the function

b is the base of the exponential function

x is the variable or the exponent

To determine the values of a and b, we need to use the two given points and solve for the corresponding equations. Substituting the first point (-2,1), we get:

1 = a * b^(-2)

Substituting the second point (-1,2), we get:

2 = a * b^(-1)

Now, we can solve for a and b by eliminating one variable. Dividing the second equation by the first equation, we get:

2/1 = a * b^(-1) / (a * b^(-2))

2 = b

Substituting this value of b into the first equation, we get:

1 = a * 2^(-2)

a = 4

Therefore, the exponential function that passes through (-2,1) and (-1,2) is:

y = 4 * 2^x

or

f(x) = 4 * 2^x

The translation rule for these circles is C): (x-1),(y+4), with a scale factor of 1/4. This means that when transforming Circle A to Circle B, each x-coordinate is decreased by 1 and each y-coordinate is increased by 4.

Circle is a two-dimensional shape with no sides or corners. Its defining feature is its curved line, which forms a continuous loop. The circle has no beginning or end, which symbolizes eternity, completeness and unity. Circles are often used in various designs, logos and artwork, as well as in mathematics, science, engineering and architecture.

The translation rule for these circles is C): (x-1),(y+4). We can calculate the scale factor by determining the difference between the coordinates of the two circles.

The coordinates of Circle A are (x,y). The coordinates of Circle B are (x-1, y+4). Therefore, the difference between the two is (1,4).

The scale factor is the ratio of the differences. Therefore, the scale factor is 1/4.

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Therefore, the translation rule for these circles is C): (x-1),(y+4), and the scale factor is 1/4.

In conclusion, the translation rule for these circles is C): (x-1),(y+4), with a scale factor of 1/4. This means that when transforming Circle A to Circle B, each x-coordinate is decreased by 1 and each y-coordinate is increased by 4.

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Using the area formula, we can find the area of the region to be 24.5 square units.

The total area of a three-dimensional object's faces represents the object's surface. The idea of surface areas has practical implications in wrapping, painting, and ultimately creating things to create the best possible design.

To determine the area of the territory in the above question, we must first calculate the areas of the three shapes.

Triangle 1: The height is equal to the distance between the x-(which axis's is 0) and the (3,6)-axis's (which is 6) y-coordinates.

The triangle's dimensions are as follows:

= 1/2 × base × height

= 1/2 × 3 × 6

= 9

Triangle 2's base is the area between the x-axis and the point where the line y=10-x intersects it (10,0).

Triangle 2:

= 1/2 × base × height

= 1/2 × 4 × 1

= 2

Trapezoid:

The areas of the three forms added together are:

= 9 + 2 + 13.5

= 24.5

The x-axis, lines x=2, x=6, lines y=x+3, and y=10-x surround the region, which is 24.5 square units in size.

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Based on the information, it should be noted that the theorem that is not applicable and doesn't prove the relationship is A. MCA = MCO, CAM = COM

The triangles are congruent if their two sides and included angles are identical to each other, according to the side-angle-side (SAS) theorem.

You cannot use any theorem along with option A. For B, you can use SAS. For option C, you can use SAS. For option D, you can use SSS. For option D, you can use ASA

Therefore, based on the information, the correct option is A.

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If the slopes are equal, the reflected line segments are parallel.

When a line segment is reflected in the y-axis, its image is formed by reflecting each point on the segment across the y-axis.

This means that the x-coordinate of each point is negated, while the y-coordinate remains the same.
To draw the images of Wayne Street and George Street, we need to know their coordinates.

Let's assume that Wayne Street runs from (2, 3) to (6, 5), while George Street runs from (-1, -2) to (-5, -4).
When we reflect Wayne Street in the y-axis, we negate the x-coordinates and keep the y-coordinates the same.

This gives us the image of Wayne Street, which runs from (-2, 3) to (-6, 5).

Similarly, when we reflect George Street in the y-axis, we get the image of George Street, which runs from (1, -2) to (5, -4).
We need to consider the slopes of the original segments and their images.

The slope of Wayne Street is (5-3)/(6-2) = 1/2, while the slope of George Street is (-4-(-2))/(-5-(-1)) = -1/2.
When we reflect a line segment in the y-axis, its slope is negated.

So,

The slope of the image of Wayne Street is -1/2, while the slope of the image of George Street is 1/2.
Therefore, the images of Wayne Street and George Street are not parallel, since their slopes are opposite.

In fact, they are perpendicular to each other, since the product of their slopes is (-1/2) x (1/2) = -1/4, which is equal to -1 (the negative reciprocal of each other).

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The appropriate graph displaying the solution set for 2x 3>-9 is:.

      o====================>

 -2  -1   0   1   2   3   4   5   6

A number line is a straight line that represents the set of real numbers.   It is a graphical representation of numbers in which each point on the line corresponds to a unique number.

To graph the solution set for 2x+3>-9, we need to solve for x and graph the inequality on a number line.

Starting with the given inequality:

2x + 3 > -9

Subtracting 3 from both sides gives:

2x > -12

Dividing both sides by 2 gives:

x > -6

Any value of x greater than -6 will therefore satisfy the inequality.

To graph this on a number line, we can start by marking -6 with an open circle (since x is not equal to -6).

Then, we shade to the right of -6, indicating all values of x that are greater than -6.

where the open circle at -6 indicates that x is not equal to -6, and the arrow pointing to the right indicates that all values of x greater than -6 satisfy the inequality.

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Answer:  To get to 1, divide 116 by __116___. From 1, multiply by ___32___ to get to 32. OR, in a single step, multiply 116 by __.3___ to get to 32.

Step-by-step explanation:

Therefore, the simplified expression is: [tex]1x+46[/tex].

A fraction is a way of representing a part of a whole or a quantity that is not a whole number. It is expressed in the form of a numerator and a denominator separated by a line, called a fraction bar or a vinculum. The numerator is the top number in a fraction, and the denominator is the bottom number. The numerator represents the number of equal parts of a whole, and the denominator represents the total number of equal parts into which the whole is divided. For example, the fraction 3/4 represents three equal parts of a whole that is divided into four equal parts. Fractions can be used in many mathematical operations, such as addition, subtraction, multiplication, and division, and are an essential concept in mathematics.

Expanding the left-hand side, we get:

[tex]4(-8x + 5) - (-33x - 26)[/tex]

[tex]= -32x + 20 + 33x + 26[/tex]

Simplifying, we can combine like terms:

[tex]-32x + 33x + 20 + 26[/tex]

[tex]= 1x + 46[/tex]

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Answer: The composite figure consists of two square pyramids, so we can find the total surface area by adding the surface area of each pyramid. The surface area of a square pyramid is given by:

S = B + (1/2)Pl

where B is the area of the base, P is the perimeter of the base, l is the slant height, and S is the total surface area.

For each pyramid, we have:

B = 24^2 = 576 cm^2

P = 4(24) = 96 cm

l = sqrt(24^2 + 5^2) = 25 cm

So the surface area of each pyramid is:

S = 576 + (1/2)(96)(25)

S = 576 + 1200

S = 1776 cm^2

Therefore, the total surface area of the composite figure is:

2S = 2(1776)

= 3532 cm^2

The expression that represents the total surface area, in square centimeters, of the figure is:

(24) (24) + 8 (one-half (24) (5)) + 8 (one-half (24) (13)) = 576 + 480 + 624 = 1680 cm^2

This expression only accounts for the surface area of the base and the triangular faces, but it does not include the surface area of the pyramid faces. So it is not correct.

The correct answer is:

8 (one-half (24) (5)) + 8 (one-half (24) (13)) = 480 + 624 = 1104 cm^2

which only represents the surface area of the triangular faces of both pyramids. To get the total surface area, we need to add the surface area of the base, which gives:

1104 + 2(576) = 2256 + 1104 = 3360 cm^2

Therefore, the total surface area of the composite figure is 3360 square centimeters.

Step-by-step explanation:

Answer:$1,331

Step-by-step explanation: Add each pairs earnings then subtract and you'll get the answer.

The water level is rising at a rate of 0.1 cm/min when the water is 50 cm deep

Let's begin by finding the volume of water in the trough when the water is 50 cm deep. We know that the trough has an isosceles trapezoid cross-section, so we can find the area of the cross-section using the formula for the area of a trapezoid

A = ((b1 + b2)/2) × h

where b1 and b2 are the lengths of the parallel sides of the trapezoid, and h is the height of the trapezoid.

In this case, b1 = 20 cm, b2 = 80 cm, and h = 60 cm. So the area of the cross-section is

A = ((20 + 80)/2) × 60

A = 3000 cm^2

To find the volume of water in the trough when the water is 50 cm deep, we need to multiply the cross-sectional area by the length of the trough and the depth of the water

V = A × L × d

V = 3000 cm^2 × 10 m × 0.5 m

V = 15000 L

Now that we know the volume of water in the trough, we can find how fast the water level is rising by taking the derivative of the volume with respect to time

dV/dt = 0.3 m^3/min

We need to convert this rate to units of L/min, so we can multiply by 1000 to convert m^3 to L

dV/dt = 0.3 m^3/min × 1000 L/m^3

dV/dt = 300 L/min

Finally, we can use the formula

dV/dt = A × dh/dt

to find how fast the water level is rising when the water is 50 cm deep. We know A = 3000 cm^2, and we want to find dh/dt when d = 50 cm. We can rearrange the formula to solve for dh/dt

dh/dt = dV/dt / A

dh/dt = 300 L/min / 3000 cm^2

dh/dt = 0.1 cm/min

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The given question is incomplete, the complete question is:

A water trough is 10 m long and has a cross-section in the shape of an isosceles trapezoid that is 20 cm wide at the bottom, 80 cm wide at the top, and has height 60 cm. If the trough is being filled with water at the rate of 0.3 m3/min how fast is the water level rising when the water is 50 cm deep?

In conclusion  62 inches is equal to 157.48 centimeters (rounded to the nearest tenth).

How to solve?

Part 1 of 2:

The first method to convert 62 inches to centimeters involves using the conversion factor of 1 inch equals 2.54 centimeters. We can set up a proportion to solve for the number of centimeters in 62 inches:

1 in. x cm

----- = -----

2.54 62 in.

To solve for x, we can cross-multiply and simplify:

1 × x = 2.54 × 62

x = 157.48

Therefore, 62 inches is equal to 157.48 centimeters (rounded to the nearest tenth).

So the first method gives us 157.5 cm as the answer.

Part 2 of 2:

The second method to convert 62 inches to centimeters involves multiplying 62 by the conversion factor of 2.54 centimeters per inch:

62 in. × 2.54 cm/in = 157.48 cm

Therefore, using this method, we also get 157.5 cm as the answer (rounded to the nearest tenth).

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cone’s radius r in terms of y is  r = 2(y + 5).

The quantity of space a cone encloses is referred to as its volume. It is the measure of the three-dimensional space occupied by the cone.

volume of a cone with a height of 9 inches as:

V = 3πy² + 30πy + 75π

Also, we are aware that the formula for a cone's volume with height hand radius r is

V = 1/3πr²h

Since we are looking for the radius of the cone in terms of y, we need to express h in terms of y. We can do this by noting that the height of the cone is 9 inches, and that it is made up of two parts: a smaller cone with height y and a larger frustum with height (9-y). The radius of the smaller cone is r/y times the radius of the larger frustum, so we can write:

h = y + (9 - y) = 9

r/y = r/(r + 3y)

As a result, the cone's volume may be represented as:

V = 1/3πr²(y + (9 - y))

= 1/3πr²(9)

Equating this to the given volume, we get:

1/3πr²(9) = 3πy² + 30πy + 75π

Simplifying and solving for r, we get:

r = √(4(y² + 10y + 25))

r = 2√(y² + 10y + 25)

Therefore, the cone's radius in terms of y is:

r = 2(y + 5)

So the answer is r = 2(y + 5).

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Based on the given graph, we can create a matrix to represent the graph as follows:

[tex]\left[\begin{array}{ccccc}A&B&C&D&F\\0&1&0&0&1\\1&0&1&1&0\\0&1&0&0&1\\0&1&0&0&0\\1&0&1&0&0\end{array}\right][/tex]

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are commonly used in mathematics, physics, engineering, computer science, and many other fields to represent and manipulate data, equations, and systems of equations. A matrix can be represented using parentheses or brackets, with the entries separated by commas or spaces.

Here,

The rows and columns of the matrix represent the vertices of the graph, and the entries in the matrix represent the edges. If there is an edge between two vertices, the corresponding entry in the matrix is 1, otherwise it is 0. For example, there is an edge from vertex A to vertex B, so the entry in row A and column B is 1. Similarly, there is an edge from vertex B to vertex A, so the entry in row B and column A is also 1.

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Answer: 4

Step-by-step explanation:

Answer: 4

Step-by-step explanation:

24/6

= 4

The area of the regular decagon with centre O and apothem 14 units is found as 637 square units

What is an apothem?

An apothem is like a radius for any polygon. It is the perpendicular distance from the centre of any given polygon to any of the opposite sides of that polygon. And polygon is a shape that is a closed two dimensional figure made up of straight lines or line segments. based on the number of sides we have different polygons such as triangle, quadrilateral,pentagon etc.

Here the regular polygon is decagon.

As decagin has 10 sides, number of sides N=10

Given the length of apothem(A) =14 units

Using trigonometric ratios we can find the length of side:

We know that the total angle at the centre=360

angle subtended by each side of decagon at centre=360÷10

                                                                                       =36°

angle bisected by the apothem= 36 ÷ 2

                                                    =18°

Tan18 = [tex]\frac{side opposite to the angle}{side adjacent to the angle}[/tex]

         =   [tex]\frac{half length of side}{apothem}[/tex]

          =[tex]\frac{0.5L}{14}[/tex]

0.325  =[tex]\frac{0.5L}{14}[/tex]

       L = (0.325 x 14)÷ 0.5

       L=9.1 units

The area of a regular polygon using apothem is given by

Area (A) = [number of sides×length of side×apothem] ÷2

             =[tex]\frac{N L A}{2}[/tex]

             =[tex]\frac{10 x 14 x 9.1}{2}[/tex]

             =637 sq. units

The area of given decagon is 637 sq. units

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Refer to the attachement for complete question.

After answering the provided question, we can conclude that Therefore, coordinates   the length of the midsegment of the trapezoid is [tex]\sqrt(13)[/tex]units.

In mathematics, a coordinate is a number or permutation of integers that indicates the location of a point or object in space. Coordinates are frequently used to define a point or object's position in a specific reference frame, such as the Euclidean or antarctic coordinate systems. In the Cartesian coordinate system, a point is located by its distance from the origin along the x-axis and its distance first from origin along the y-axis. These distances are represented by two numbers, its x-coordinate and the y-coordinate. The point's position in the plane is specified by the two coordinates.

To find the length of the midsegment of a trapezoid,

[tex]E = ( (x1 + x2)/2 , (y1 + y2)/2 )\\So, E = ( (0+2)/2 , (3+5)/2 ) = (1,4)\\F = ( (x3 + x4)/2 , (y3 + y4)/2 ) = ( (6+2)/2 , (4+0)/2 ) = (4,2)\\d = \sqrt( (x2 - x1)^2 + (y2 - y1)^2 )\\d =\sqrt( (4-1)^2 + (2-4)^2 ) = \sqrt(9+4) = \sqrt(13)\\[/tex]

Therefore, the length of the midsegment of the trapezoid is [tex]\sqrt(13)[/tex]units.

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The difference when -6c + 4 is subtracted from -8 + 7c is -2c - 11. This can be simplified to -2c - 11, which is the answer.

To calculate the difference when -6c + 4 is subtracted from -8 + 7c, we must first expand the terms. The first term, -6c, can be broken down into -6 x c, which simplifies to -6c. The second term, 4, can remain as is. The third term, -8, can also remain as is. The fourth term, 7c, can be broken down into 7 x c, which simplifies to 7c.

We can then calculate the difference by performing the subtraction. We begin by subtracting the second term, 4, from the first term, -6c. We can do this by adding -4 to -6c, which results in -6c - 4. We then subtract the fourth term, 7c, from the third term, -8. We can do this by adding 8 to -7c, which results in -7c + 8.

We can then combine the two terms together. We do this by adding -6c - 4 to -7c + 8. This results in -13c + 4. We can simplify this to -2c - 11, which is the final answer.

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Using the conversion factor, Joselyn bought 2700 grams of chocolate candy

A conversion factor is a numerical ratio used to convert a measurement from one unit to another. It is a mathematical expression that is used to convert a quantity expressed in one unit of measure into an equivalent quantity expressed in another unit of measure.

To convert 2.7 kg to grams, Joselyn would use a conversion factor between kilograms and grams. The conversion factor is 1000 grams per 1 kilogram, which means there are 1000 grams in one kilogram.

So, to convert 2.7 kg to grams, Joselyn would multiply 2.7 kg by the conversion factor

2.7 kg x 1000 g/kg = 2700 g

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I don't think anyone can, that's the wrong pic

Answer:

(1 - 5x + y)(1 + 5x - y).

(9a - c + 3b)(9a + c - 3b).

(bc - b - c - 1)(bc + b + c - 1).

Step-by-step explanation:

1 - 25x^2 + 10xy - y^2

= 1 - (25x^2 - 10xy + y^2)

= 1 - (5x - y)^2

This is the difference of 2 squares, so the answer is

(1 - (5x - y))(1 + (5x - y))

= (1 - 5x + y)(1 + 5x - y).

81a^2 + 6bc - 9b^2 - c^2

= 81a^2 - (c^2 - 6bc + 9b^2)

= 81a^2 - (c - 3b)^2

= (9a - (c - 3b))(9a  + (c - 3b))

= (9a - c + 3b)(9a + c - 3b)

b^2c^2 - 4bc - b^2 - c^2 + 1

= b^2c^2 - 4bc - b^2 - (c^2 - 1)

= (bc - b)  - (c + 1))((bc + b) + (c - 1)

= (bc - b - c - 1)(bc + b + c - 1).