6. Complete the two-column proof.
Given: 3x + 4 = 10 Prove: x = 2
3x + 4 = 10 __?___
3x = 6 __?___
x = 2 ___?___

Answer:

Step-by-step explanation:

around 24 meters

Answer:

To solve this system of equations using substitution, we first need to solve one of the equations for one of the variables in terms of the other.

Let's solve the first equation for x:

6x - 5y = 10

x = (10 + 5y)/6

Now let's substitute this expression for x into the second equation:

x - 3y = -20

(10 + 5y)/6 - 3y = -20

10/6 + 5y/6 - 3y = -20

(10/6 - 3y) + (5y/6) = -20

(10 - 3y)/6 + 5y/6 = -20

10/6 - 3y/6 + 5y/6 = -20

(-3y + 5y)/6 = -20

2y/6 = -20

y = -10

Now that we have found the value of y, we can substitute it back into one of the original equations to find the value of x. Let's use the first equation:

6x - 5y = 10

6x - 5(-10) = 10

6x + 50 = 10

6x = -40

x = -40/6

x = -40/6

x = -6.666666666666667

So the solution to the system of equations is (x, y) = (-6.666666666666667, -10).

The measure of the given angle is given as 1.25 π radian the measure of the shaded portion is 40 π radian

The given figure is a circle to find the shaded portion we use the formula for area of sector in radian

Area of sector = (1/2) × r²θ

1.25 π radian = θ

8 = radius, r

Area of sector = (1 / 2) × 8² * 1.25 π radian

Area of sector =  (1 / 2) × 64 * 1.25 π radian

Area of sector =  (1 / 2) × 80 π radian

Area of sector =  40 π radian

The area of the sector the shaded portion is 40 π radian

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Answer:

[tex]c=47[/tex]

Step-by-step explanation:

[tex]c-14=33[/tex]

Step 1: Add 14 to both sides.

[tex]c-14+14=33+14[/tex]

[tex]c=47[/tex]

Step-by-step explanation:

First you simplify the fractions

17/12 × 1/2 × 3

17×1×3=51 and 12×2=24

51/24

Further simplifying after dividing the two numbers by 3 you get

17/8

Answer:

Step-by-step explanation:

If you are looking for an exact form, it will be 17/8

If you are looking for a decimal form, it will be 2.125.

If you are looking for a mixed number form, it will be 2 1/8

I hope this helps.

The rocket will be at a height more than 100 feet for 10 seconds.

Quadratic equations are second degree equations involving a variable of the form [tex]ax^{2} + bx+c = 0[/tex].

We have the equation relating height and time as,

h(t) = -[tex]\frac{1}{2}[/tex] t² + 15 t

When height is 100 feet, equation becomes,

100 = -[tex]\frac{1}{2}[/tex] t² + 15 t

-[tex]\frac{1}{2}[/tex] t² + 15 t - 100 = 0

Multiplying throughout with -2, we get,

t² - 30t + 200 = 0

(t - 10) (t - 20) = 0

t - 10 = 0 or t - 20 = 0

t = 10 or t = 20

When rocket is at maximum height of 112.5 feet,

-[tex]\frac{1}{2}[/tex] t² + 15 t - 112.5 = 0

Multiplying throughout with -2, we get,

t² - 30t + 225 = 0

(t - 15) (t - 15) = 0

t - 15 = 0

t = 15

Hence the rocket reached at a height of 100 feet at t = 10 seconds and after 5 seconds it reached it's maximum height at t = 15 seconds and then the rocket goes down and again reached at a height of 100 m at t = 20 seconds. So the rocket is at height more than 100 m for 10 seconds.

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To find the linear distance needed for the staircase, you can multiply the number of steps by the width of each step. The length of the railing that would be attached to these stairs would be 7.6m.

3.6 m / (18 cm / 100 cm/m) = 20 steps

20 steps * 28 cm/step = 560 cm = 5.6 m

5.6 m + 2 m = 7.6 m.

The distance between the two specified sites or objects is the linear measurement. Therefore, we may define length as "the total distance measured between an object's outermost left and right ends in the aforementioned system of units." utilizing tape to gauge a banana's length. The length is around 5 inches. Children can measure nearby things such as a pencil, a shoe, a remote control, a toy vehicle, and a spoon using tools like a ruler and tape.

The example that follows shows how to measure the length of a screwdriver by placing a ruler against it. Place the item and ruler such that the ruler's tip is at "0" to get the length. To determine the object's overall length, make a mark with a number at the end.

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The mathematical result applied to prove sin²Ф + cos² = 1 , is the Pythagorean theorem.

Any right triangle has an area equal to the sum of a sides of the squares which sides are the two legs (remember, the hypotenuse is the side opposite the right angle).

Consider the given triangle ABC, right angled at C.

Then,

AC² + BC² = AB²

Divide the equation by AB².

AC²/ AB² + BC²/ AB² = AB²/ AB²

(opposite/ hypotenuse)² + (adjacent/ hypotenuse)² = 1

Thus,

sin²Ф  = (opposite/ hypotenuse)²

cos² =  (adjacent/ hypotenuse)²

Then,

sin²Ф + cos² = 1

Thus, the mathematical result applied to prove sin²Ф + cos² = 1 , is the Pythagorean theorem.

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The measure of the ∠ RST if, In Δ QRS, QS is extended through point S to point T, a measure of the ∠ QRS = (x + 8)°, m ∠ RST = (4x + 11)°, and m ∠ SQR = (x + 13)°, is 31°.

One of the fundamental shapes in geometry is represented by the symbol Δ and consists of three connected vertices, known as a triangle.

Given:

In Δ QRS, QS is extended through point S to point T, a measure of the ∠ QRS = (x + 8)°, m ∠ RST = (4x + 11)°, and m ∠ SQR = (x + 13)°,

Calculate the value of x as shown below,

∠ RST = ∠ SQR + ∠ QRS   (Using exterior angle theorem)

4x + 11 = x + 13 + x + 8

4x - x - x = 13 + 8 - 11

2x = 10

x = 5

Calculate the measure of ∠ RST as shown below,

∠ RST = 4 × 5 + 11

∠ RST = 20 + 11

∠ RST = 31°

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The width of the rectangle if The Area of the rectangle is 5/6 decimal 96 m, and the length is 16 m, is 1 / 2 decimal 6 m.

An enclosed 2-D shape called a rectangle has four sides, four corners, and four right angles (90°). A rectangle has equal and parallel, opposite sides. Since a rectangle is a two-dimensional form, it has two dimensions: length and width. The rectangle's longer side is its length, while its shorter side is its breadth.

Given;

The Area of a rectangle is 5/6 decimal 96 m,

the length and 16 m,

Area = 5 / 6 + 96

Area = 0.83 + 96

Area = 96.83

Area = 16 × width

96.83 = 16 × width

width = 96.83 / 16

width = 6.05

Thus, the width of the rectangle is 6.05 m or 6 and 1 / 2 m.

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The inequality that can be used to find out what grade Melissa needs in the third trimester is x ≥ 90. She needs at least 90.

Inequalities are simply created through the connection of two expressions. In this case, it should be noted that the expressions in an inequality are not always equal. Inequalities implies that the expressions are not equal. Here, they are denoted by the symbols ≥ < > ≤.

Let the grade be represented by x. This will be:

= (72 + 93 + x) / 3 ≥ 85

165 + x / 3 ≥ 85

165 + x ≥ 255

Collect the like terms

x ≥ 255 - 165

x ≥ 90

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Answer:

1/2

Step-by-step explanation:

If you look at the numbers , you will see    y = 1/2x      Answer: 1/2

Answer:

1/2

Step-by-step explanation:

y2-y1 3-2

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

x2-x1 6-4 =1/2

Answer:

Part A: From the table, it is clear that as the number of workers increases, the number of units produced also increases. This indicates a positive correlation between the number of workers and the number of units produced. As the number of workers increases by 10, the number of units produced also increases by 50. This pattern is consistent throughout the data, which supports the conclusion that there is a positive correlation between the number of workers and the number of units produced.

Part B: A function that best fits the data is y = 5x + 2, where y is the number of units produced and x is the number of workers. This function can be determined by using the slope-intercept form of a linear equation (y = mx + b) and substituting the values for the slope and y-intercept from the data. The slope (m) of the function is 5 and the y-intercept (b) is 2.

Part C: The slope of the plot (5) represents the rate of change in the number of units produced for every 1 unit change in the number of workers. In this case, for every 1 additional worker, the number of units produced increases by 5. The y-intercept (2) represents the point at which the line intercepts the y-axis when x = 0. In this case, when there are no workers, 2 units are produced.

The limits of the functions (a) 1, (b) does not exist (c) 2, (d) does not exist, (e) -4 and (f) 3.

Limit of a function is defined as a value that the function tends to the output for the given input values.

(a) [tex]\lim_{x \to 2} [f(x)+g(x)][/tex] =  [tex]\lim_{x \to 2} f(x)[/tex] +  [tex]\lim_{x \to 2} g(x)[/tex]

[tex]\lim_{x \to 2} f(x)[/tex] is the value of y when x is approaching 2 from left and right.

[tex]\lim_{x \to 2} f(x)[/tex] = -1

Similarly,  [tex]\lim_{x \to 2} g(x)[/tex] = 2

[tex]\lim_{x \to 2} [f(x)+g(x)][/tex] = -1 + 2 = 1

(b)  [tex]\lim_{x \to 0} [f(x)-g(x)][/tex] =  [tex]\lim_{x \to 0} f(x)[/tex] -  [tex]\lim_{x \to 0} g(x)[/tex]

[tex]\lim_{x \to 0} f(x)[/tex] = 2

But  [tex]\lim_{x \to 0} g(x)[/tex] does not exist as the function g(x) has different limits 3 and 1 from left and right respectively as x approaches 0.

So [tex]\lim_{x \to 0} [f(x)-g(x)][/tex] does not exist.

(c)  [tex]\lim_{x \to -1} [f(x)g(x)][/tex] =  [tex]\lim_{x \to -1} f(x)[/tex] × [tex]\lim_{x \to -1} g(x)[/tex]

[tex]\lim_{x \to -1} f(x)[/tex] = 1

[tex]\lim_{x \to -1} g(x)[/tex] = 2

[tex]\lim_{x \to -1} [f(x)g(x)][/tex] = 1 × 2 = 2

(d)  [tex]\lim_{x \to 3} [f(x)/g(x)][/tex] =  [tex]\lim_{x \to 3} f(x)[/tex] /  [tex]\lim_{x \to 3} g(x)[/tex]

[tex]\lim_{x \to 3} f(x)[/tex] = 1

[tex]\lim_{x \to 3} g(x)[/tex] = 0

 [tex]\lim_{x \to 3} [f(x)/g(x)][/tex] = 1 / 0, not defined. So does not exist.

(e)  [tex]\lim_{x \to 2}[x^{2} f(x)][/tex] =  [tex]x^{2} \lim_{x \to 2} f(x)[/tex]

[tex]\lim_{x \to 2} f(x)[/tex] = -1

[tex]\lim_{x \to 2}[x^{2} f(x)][/tex] = (2)² × -1 = -4

(f) f(-1) = 1

[tex]\lim_{x \to -1} g(x)[/tex] = 2

f(-1) + [tex]\lim_{x \to -1} g(x)[/tex] = 1 + 2 = 3

Hence the values of the limits are given.

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The probability that[tex]$QR \leq 100$[/tex] is equal to the ratio of the area of the region in which[tex]$QR \leq 100$[/tex] to the area of the region in which [tex]$0 \leq a \leq 75$ and $0 \leq b \leq 75$[/tex]. Therefore,[tex]$m + n$[/tex] is equal to the area of the region in which [tex]$0 \leq a \leq 75$ and $0 \leq b \leq 75$.[/tex]

The probability that

[tex]$QR \leq 100$[/tex]

is equal to the ratio of the area of the region in which

[tex]$QR \leq 100$[/tex]

to the area of the region in which

[tex]$0 \leq a \leq 75$[/tex]

[tex]$0 \leq b \leq 75$[/tex]

. To find this probability, we first need to find the area of the region in which

[tex]$0 \leq a \leq 75$ and $0 \leq b \leq 75$[/tex]

This region is a rectangle with sides of length[tex]$75$,[/tex] so its area is

[tex]$75^2$[/tex]. To find the area of the region in which

[tex]$QR \leq 100$[/tex],

we need to calculate the area of the triangle in which[tex]$QR$[/tex] is the hypotenuse.

This triangle has base [tex]$75$[/tex]and height [tex]$200$[/tex], so its area is [tex]$3750$[/tex]. Therefore, the desired probability is

[tex]$\frac{3750}{75^2}$, and $m + n[/tex]

[tex]= 75^2$.[/tex]

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If the Q and R are on the same side of the line OP and angle OQP and angle ORP are right triangles then the probability that QR ≤ 100 is 16/25 and the value of m+n is 41 .

The angles OQP and angle ORP are given as right angles that means

∠OQP = ∠ORP = 90° ,

In order to find the required probability we draw a semicircle having the diameter OP and points Q and R on the same side of the semicircle .

given that diameter  OP = 200 ,

So , the radius of the circle will be = 100 ,

Let QR = x ≤ 100 , From the figure ,

we can write ,

[tex]OQ = 200 \times Cos(a)[/tex] and [tex]PQ = 200 \times Sin(a)[/tex]

[tex]OR = 200 \times Cos(b)[/tex]  and  [tex]PR = 200 \times Sin(b)[/tex]

From the figure given below , we can conclude that the quadrilateral OQRP is a cyclic quadrilateral .

So , [tex]200x + (200 Cosa)\times(200 Sinb) = (200 Sina)\times(200 Cosb)[/tex]

we get , [tex]x = |200 Sin(a-b)| \leq 100[/tex] ,

| Sin(a-b)| ≤ 1/2 ;

which implies that |a-b| ≤ 30 ,

that means , -30 ≤ (a-b) ≤ 30 ,  

it is given that a and b are chosen independently and uniformly at random in interval (0,75).

which means , 0<a, b <75 .

The Probability that x ≤ 100 is = [tex]\frac{Favorable Area}{Total Area}[/tex]

=  [tex]\frac{75\times75-45\times45}{75 \times75}[/tex] = 1 - [tex]\frac{9}{25}[/tex] ,

On simplifying ,

we get

= 16/25 ,

and it is given that the probability is = m/n ,

So on comparing , we get that , m = 16 and n = 25 ,

and the value of m+n = 16+25 = 41 .

Therefore , the probability is 16/25 and the value of m+n is 41 .

The given question is incomplete , the complete question is

Two real numbers a and b are chosen independently and uniformly at random in the interval (0,75). Let O and P be two points on the plane with OP = 200. Let Q and R be on the same side of line OP such that the degree measures of angle POQ and angle POR are a and b respectively, and angle OQP and angle ORP are both right angles. The probability that QR ≤ 100 is equal to m/n , where m and n are relatively prime positive integers. Find m + n.

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The solution is Option B.

The percentage increment in the preference for  Enterprise Content Management is given by the equation A = 366 %

The difference between an exact value and an approximation to it is the approximation error in a data value. Either an absolute error or a relative error might be used to describe this error.

Percentage error is the difference between the measured value and the true value , as a percentage of the true value

Percentage Error = [ ( | Measured Value - True Value | ) / True Value ]x 100

Given data ,

Let the percentage increment be represented as A

Now , the equation will be

The number of 41+ year olds having a preference for Enterprise Content Management = 3,215

The number of 18-25 year olds having a preference for Enterprise Content Management = 690

So , the difference in number of preference = 3,215 - 690

On simplifying the equation , we get

The difference in number of preference = 2,525

And , the percentage increase A = ( difference in number of preference / number of 18-25 year olds having a preference for Enterprise Content Management ) x 100

Substituting the values in the equation , we get

The percentage increase A = ( 2525/690 ) x 100

The percentage increase A = 3.659 x 100

The percentage increase A = 366 %

Therefore , the value of A is 366 %

Hence , the percentage increment is 366 %

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The line segment AB in the square pyramid is parallel to the base side

A square pyramid is a three-dimensional shape with a square base and four triangular faces that meet at an apex. It is a type of polyhedron with five faces, a square base, and four triangular faces that meet at the top.

In this problem, the line segment AB is only parallel to the base side.

The base side of a square pyramid is the length of one side of the square base of the pyramid in which they are all equal to one another.

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The value of x using trigonometric functions will be 5.66.

What are trigonometric functions?

Trigonometric functions are also known as Circular Functions can be simply defined as the functions of an angle of a triangle. It means that the relationship between the angles and sides of a triangle are given by these trigonometric functions. The basic trigonometric functions are sine, cosine, tangent, cotangent, secant and cosecant.

The angles of sine, cosine, and tangent are the primary classification of functions of trigonometry. And the three functions which are cotangent, secant and cosecant can be derived from the primary functions

Now,

In given triangle, 3rd angle will be 54 degree.

                                         (Sum of all angle in triangle=180degree)

Therefore

               sin 54=P/H  (P=Perpendicular, H=Hypotenuse)

                0.809=x/7

               x=7*0.809

              x=5.66

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a. She had in her savings to begin with $212.5

b. She has $127.5 left in her savings after buying the bicycle.

Answer:

a) Molly had $212.50 in her account to begin with.

b) She has $127.50 in her account after buying the bicycle.

Step-by-step explanation:

40% of what number is 85.

.4x = 85  Divide both sides by .4

x = 212.50

212.50 - 85 = 127.50

Another way to look at it is if she spent 40%, she still have 60% left in her account.

.6 x 212.50 = 127.50

Answer:

The y intercept of the line is 2

Step-by-step explanation:

y intercept means where the line crosses the y axis which is at the value 2

The equation of line j in the form of slope intercept is " y = -2/3x - 24".

The definition of a line's slope commonly referred to as its gradient is the degree of steepness or the direction of a line in a coordinate plane. Given the equation of a line or the coordinates of points along the line, there are various ways to determine slope.

Given, line h passes through the point (3,3) and this is perpendicular to the graph of a y=3/2x-4 line

the slope of the line which is perpendicular to another line = -1/m

thus the equation of the line h is y = -2/3x + c

Since this line goes through Coordinate (3,3) this will satisfy the equation

Substituting the values in the equation, c = 15

thus the equation of line h is y = -2/3x +15

also given, line j is parallel to line h and passes through the point (-6,-3)

the slope of the line which is parallel to another line = m

thus the equation of the line j is y = -2/3x + c

Since this line goes through Coordinate (-6,-3) this will satisfy the equation

Substituting the values in the equation, c = -24

thus the equation of line h is y = -2/3x -24

Therefore, "Y = -2/3x - 44" is the slope-intercept version of the equation for line j.

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Answer:

Step-by-step explanation:

Answer: C and D respectively

Step-by-step explanation:

To find the sum "n"  consecutive even/odd integers you need to do x + (x+2) + (x + 4) ... for n times. You can just add them and solve.

To find the sum of "n" consecutive integers you need to do x + (x + 1) + (x + 2) ... for n times. In this problem we do x + (x + 1) + (x + 2) = 96; 3x +3 = 96; x = 31. Then we can find the other numbers by plugging in 31 for x. 31, 32, 33.

By using the table for the function f(x), we will see that:

f(4) = 74

f(6) = 40

Here we have a table for the function f(x), first we want to find f(4).

So we need to find x = 4 on the table (the first row of the table) and the value right below will be the value of f(4), there we can see that the value is:

f(4) = 74

Now we want to find:

f(x) = 40

So we want to find the value of x such that when we evaluate f(x) on it, it is equal to 40.

So we need to find 40 on the second row, and the value that is above it will be the value of x.

We can see that for x = 6:

f(6) = 40

Then here the answer is x = 6.

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Answer:

The ratio for the first print job is 150/30 = 5 pages per second.

The ratio for the second print job is 130/20 = 6.5 pages per second.

Since the ratios are different, the data do not represent a proportional relationship. In a proportional relationship, the ratio between two variables remains constant.

Step-by-step explanation:

Each dimension increased by 2.86 ft.

Given that A rectangular box has dimensions 5 ft by 3 ft by 3 ft

The volume of the rectangular box = l × w × h , where l , w , h are its dimensions

∵ l = 5 ft , w = 3 ft , h = 3 ft

∴ Its volume = 5 × 3 × 3 = 45 ft³

Since Each dimension of the box is increasing by the same amount to yield a new box

Let each dimension will increase by x ft

∴ The new dimensions are l = (5 + x) , w = (3 + x) , h = (3 + x)

The volume of the new box is six times the old

The volume of the old box is 45 ft³

∴ The volume of the new box = 6 × 45 = 270 ft³

∵ The volume of new box = (5 + x)(3 + x)(3 + x)

∴ (5 + x)(3 + x)(3 + x) = 270

x³+11x²+39x+45=270

Subtract 270 from both sides.

x³+11x²+39x+45−270=270−270

x³+11x²+39x−225=0

Use cubic formula.

x=2.860717

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Answer:

Step-by-step explanation:

x² + 81= x² + 20x + 100 · Polynomial.

The equation is x^2-x+71=0

The consequences of this congruence are AB is parallel to CD( property of parallelogram)

∠ABD is congruent to ∠CDB( Angles opposite of equal sides are congruent)

That quadrilateral in which opposite sides are parallel is called a parallelogram.

Thus, a parallelogram is always a quadrilateral but a quadrilateral can or cannot be a parallelogram.

We are given that ABCD is a parallelogram and [DB] is one of its diagonals.

In  Δs ABD and CDB

The Lines AB and CD are parallel

Therefore Angle B and Angle D are equal ; both angles are equal as they are alternate interior angles.

So An angle is always congruent to itself (Reflexive property of congruence)

Hence, the proved that Δs ABD and CDB are congruent.

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Answer:

The summation of 5+5+6+7 is 23.