12. y. y. y
What is the variable?

The length of fencing needed around her triangular garden is 41 feet.

A given shape which has three sides and three measures of internal angles which add up to 180^o is said to be a triangle.

For Mary to build a fence around her triangular garden, the amount of fencing needed can be determined by adding the length of each side of the garden.

So that;

A + B + C = 180^o

A + 49 + 40 = 180

A = 180 - 89

   = 91

A = 91^o

Applying the sine rule, we have;

a/Sin A = b/Sin B = c/Sin C

a/Sin A = b/Sin B

a/Sin 91 = 13/ Sin 49

aSin 49 = 13*Sin 91

             = 12.998

a = 12.998/ 0.7547

  = 17.22

a = 17 ft

Also,

b/Sin B = c/Sin C

13/Sin 49 = c/ Sin 40

cSin 49 = 13*Sin 40

             = 8.3562

c = 8.3562/ 0.7547

  = 11.0722

c = 11 feet

Thus the amount of fencing required = 13 + 17 + 11

                                                    = 41

The fencing required is 41 feet length.

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The actual length of the bus is given as follows:

45 feet.

The actual length of the bus is obtained applying the proportions in the context of the problem.

The scale drawing is of 0.5 inches to 5 feet, hence the length represented by each inch of drawing is given as follows:

5/0.5 = 10 feet.

The length of the drawing is given as follows:

4.5 inches.

Hence the actual length of the bus is given as follows:

4.5 x 10 = 45 feet.

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Answer:

Step-by-step explanation:

Let's factor the expression w^2(w+8)-5(w+8):

First, we can see that (w+8) is a common factor in both terms of the expression.

So, we can factor out (w+8) from both terms:

(w+8)(w^2 - 5)

Now, the expression is factored as (w+8)(w^2 - 5).

Answer:

The angle in the first problem measures 67°.

x=16, y=40

Step-by-step explanation:

1st problem: draw the line r in figure (red) parallel to the two given one and containing the vertex of the angle. The two green angles are congruent since alternate angles from the two parallels t and r when crossed with the upper side of the angle. Same with the two blue angles created by the parallels r and b. The measure of the angle you need is the sum of the two, or 36+31= 67°.

Second problem. The two angles in red are congruent since corresponding angles generated by the parallels m and b and the transversal r. we can then write [tex]4x-5=3x+11\\[/tex]. Solving for x we find x= 16. Now the two angles marked in blue are conjugates when looking at the parallels l and r cut by the transversal m, thus they add up to 180° We know how much the angle with double marks measures (since we found x earlier) so we can write

[tex](3y+1) + (4\times16-5) = 180\\3y+ 1 + 59 = 180\\3y=120 \implies y=40[/tex]

The area of the semicircle is 47.5 square centimeters

We have to find the area of the semi circle

The diameter of the semicircle is 11 cm

Radius is half of the diameter

Radius = 5.5 cm

Let us find the area of semicircle  by formula 1/2(πr²)

Radius = 1/2×3.14×(5.5)²

= 1/2×3.14×30.25

=47.4925 square centimeters

Hence,  the area of the semicircle is 47.5 square centimeters

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The function that is the inverse of the exponential function y = 4ˣ is lnx/ln4

The inverse of a function f is denoted by [tex]f^{-1[/tex] and it exists only when f is both one-one and onto function. Note that  [tex]f^{-1[/tex]  is NOT the reciprocal of f. The composition of the function f and the reciprocal function  [tex]f^{-1[/tex]  gives the domain value of x.

(f o  [tex]f^{-1[/tex] ) (x) = ( [tex]f^{-1[/tex]  o f) (x) = x

For a function 'f' to be considered an inverse function, each element in the range y ∈ Y has been mapped from some element x ∈ X in the domain set, and such a relation is called a one-one relation or an injunction relation.

The standard exponential function is given as y = abˣ

Given the function,

y = 4ˣ

Replace y with x

[tex]x = 4^y[/tex]

Make y the subject of the formula,

[tex]lnx = ln4^y\\\\lnx = yln4[/tex]

Divide both sides by ln4

lnx/ln4 = y

Hence the function that is the inverse of the exponential function y = 4ˣ is lnx/ln4

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Answer:

FACTOR

(−1)×2×3×5

2 or    -150

Step-by-step explanation:

The area of the given figure is 27.5 square centimeter which has a rectangle and triangles

The given figure has a rectangle and triangles

The area of rectangle is length times width

Area of rectangle =5×4

=20 square centimeter

Area of triangle =1/2×base×height

=1/2×2.5×3

=7.5/2

=3.75 square centimeter

As there are two triangle = 2(3.75)

=7.5 square centimeter

Total area = 7.5+ 20

=27.5 square centimeter

Hence, the area of the given figure is 27.5 square centimeter

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As all sides of a Rhombus are of equal length, the length of the missing sides are 82

The property of a Rhombus required for answering this question is associated with the length of it's sides. A Rhombus has four sides with the length of all these sides being equal.

This means that with the length of one of it's sides being 82 , the length of the remaining 3 sides would be 82 as well.

The length of sides x = y = z = 82. Hence, x, y and z are 82

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Answer:

1/12 gallons

Step-by-step explanation:

We can determine how much more hot chocolate than apple cider die Joe serve by subtracting 1/12 from 1/6.

Step 1:  Since 1/12 has the bigger denominator and 6 * 2 = 12, let's give 1/6 the same denominator as 1/12 by multiplying the entire fraction by 2/2

(1/6) * (2/2) = 2/12

Step 2:  Now we can subtract 1/12 from 2/12:

2/12 - 1/12 = 1/12

Thus, Joel served 1/12 more gallons of hot chocolate than apple cider.

The outside border of the garden would have walked around approximately  58%.

Without more information about the shape and dimensions of the garden, it's impossible to give an exact answer. However, if we assume that the garden is a rectangle, we can use the formula for the perimeter of a rectangle to estimate the percentage of the garden's border that would be walked around.

Let's say that the length of the garden is L and the width of the garden is W. The perimeter of the rectangle is then:

P = 2L + 2W

If the person walks from point A to point B along the outside of the garden, they are essentially walking along two sides of the rectangle. Let's call these sides S1 and S2. Depending on the location of A and B, S1 and S2 may be two adjacent sides, two opposite sides, or one side and one diagonal.

To estimate the percentage of the garden's border that the person would walk around, we can calculate the length of S1 and S2 and divide by the total perimeter of the rectangle:

Percentage walked = (S1 + S2) / P * 100%

Again, without more information about the shape and dimensions of the garden, we can't give an exact answer. However, if we assume that the person walks along two adjacent sides of the rectangle, the percentage of the garden's border that they would walk around would be:

Percentage walked = (2L + W) / (2L + 2W) * 100%

Simplifying this expression, we get:

Percentage walked = (2L + W) / (2(L + W)) * 100%

Assuming that L and W are measured in the same units (e.g. meters), we can simplify further:

Percentage walked = (2 + W/L) / (2 + 2W/L) * 100%

For example, if the length of the garden is 10 meters and the width of the garden is 5 meters, then the percentage of the garden's border that the person would walk around if they walked along two adjacent sides would be:

Percentage walked = (2 + 5/10) / (2 + 2*5/10) * 100%

= 7/12 * 100%

= 58.3%

So the outside border of the garden would have walked around approximately  58%.

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Answer: 0

Step-by-step explanation: It is furthest away from the other numbers with a difference of at least 5. If plotted in a table it would visually be seen to be further away from the general trend.

Answer: x=25 degrees

Step-by-step explanation:

25 degrees (blue) and x degrees (orange) are vertical angles. Vertical angles always equal each other.

(80 degrees {green} is just there to throw you off)

[tex]y \geq -3x + 2\\\\y \leq x + 3[/tex]

is the equation of the given system.

Equation of the line using the coordinates (-3, 0) and (0, 3), we get:

slope = (3 - 0) / (0 - (-3)) = 1

Now we have the slope of the line. Next, we can use the point-slope form of the equation of a line, which is:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is a point on the line.

Using the point (-3, 0) and the slope we just found (which is also the slope of the line passing through the point (0, 3)), we get:

y - 0 = 1(x - (-3))

Simplifying, we get:

y = x + 3

Therefore, the equation of the line passing through (-3, 0) and (0, 3) is y=x+3.

Similarly, the equation of the line passing through (1, -1) and (0, -4) is y = -3x + 2.

Thus, from the graph the equation of the system is,

[tex]y \geq -3x + 2\\\\y \leq x + 3[/tex]

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Answer:

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Step-by-step explanation:

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Answer:

  10.  x = 4

  11. UT = 44

Step-by-step explanation:

You want the value of x and the length of segment UT that bisects chord PR at point U into halves marked UP = (9x -3) and UR = (7x +5). The radius of circle T is 55 units.

Point U is the midpoint of chord PR, so ...

  UP = UR

  9x -3 = 7x +5

  2x = 8 . . . . . . . . . add 3-7x

  x = 4 . . . . . . . . divide by 2

Segments UP, UT and TP form a right triangle with one leg ...

  UP = 9(4) -3 = 33

and hypotenuse 55.

We observe that the ratio of these measures being 33/55 indicates triangle PUT is a 3-4-5 right triangle, with scale factor 11. That means UT is 4·11 = 44 units.

  UT = 44 units

__

Additional comment

Recognizing triangles made from Pythagorean triples, such as {3, 4,5}, {5, 12, 13}, {7, 24, 25}, or {8, 15, 17} can save a lot of work. If you don't recognize these, you can find the missing side length from ...

  TP² = UP² +UT²

  UT = √(TP² -UP²) = √(55² -33²) = √1936 = 44

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The equation of the line is: x = 1 + 3t,, y = 10, z = 3 + 3t.

To determine the equation of the line passing through points M(1, 10, 3), N(4, 10, 6), and P(7, 10, 9), we need to find the direction vector of the line.

Let's denote the direction vector as D = (a, b, c). The line can be represented by the parametric equations:

x = 1 + at,

y = 10 + bt,

z = 3 + ct,

To find the direction vector, we can use two points on the line, for example, M and N. The direction vector can be obtained by subtracting the coordinates of M from the coordinates of N:

D = N - M = (4 - 1, 10 - 10, 6 - 3) = (3, 0, 3).

So, the equation of the line passing through M(1, 10, 3) and having the direction vector D = (3, 0, 3) is:

x = 1 + 3t,

y = 10 + 0t = 10,

z = 3 + 3t.

Therefore, the equation of the line is:

x = 1 + 3t,

y = 10,

z = 3 + 3t.

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With a triangle, the sum of any two side lengths must be greater than the third side length. If this is not true, then the side lengths cannot make a triangle. Let's go through each set of side lengths and determine which would and wouldn't work.

a. 3, 4, 8 - will not make a triangle

3 + 4 = 7 > 8 = false

3 + 8 = 11 > 4 = true

4 + 8 = 12 > 3 = true

b. 7, 6, 12 - will make a triangle

7 + 6 = 13 > 12 = true

7 + 12 = 19 > 6 = true

6 + 12 = 18 > 7 = true

c. 5, 11, 13 - will make a triangle

5 + 11 = 16 > 13 = true

5 + 13 = 18 > 11 = true

11 + 13 = 24 > 5 = true

d. 4, 6, 12 - will not make a triangle

4 + 6 = 10 > 12 = false

4 + 12 = 16 > 6 = true

6 + 12 = 18 > 4 = true

e. 4, 6, 10 - will not make a triangle

4 + 6 = 10 > 10 = false

4 + 10 = 14 > 6 = true

6 + 10 = 16 > 4 = true

Hope this helps!

We have that Linda's net and expression incorrect

With the Expression of the surface area of attic

45 (40 + 25 + 25) + 1/2 (40 x 15)

Hence, Surface area is,

X = 4050 + 600

X = 4650 feet²

Hence, The answer is No, Because Her expression for the triangles are incorrect;

1/2(40 x 15)

Instead of,

1/2 x 40 x 15

Hence, Linda's net and expression incorrect,

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Linda is adding padding to all of the surfaces inside her attic for extra warmth in the winter.

She needs to find the approximate surface area of the attic, including the walls, floor, and

ceiling. The attic is in the shape of a triangular prism. Linda draws the net and writes

the expression below to represent the surface area of the attic. Are Linda's net and

expression correct?

Answer:

Approx. 0.2079

Step-by-step explanation:

To find the value of sin 78°, we can use a scientific calculator or a trigonometric table. However, I can provide an approximate value using the trigonometric identity:

sin(90° - θ) = cos(θ)

In this case, we can use the identity sin(90° - 78°) = cos(78°). Since the cosine of 78° can be determined more easily, we can calculate that instead.

cos 78° ≈ 0.2079

Therefore, sin 78° is approximately equal to 0.2079.

The amount of wrapping paper which Elijah used is equal to 812 m².

In Mathematics and Geometry, the surface area of a rectangular prism can be calculated and determined by using this mathematical equation or formula:

Surface area of a rectangular prism = 2(LH + LW + WH)

Where:

By substituting the given side lengths into the formula for the surface area of a right rectangular prism, we have the following;

Surface area of a right rectangular prism = 2[3 × 10 + (10 × 12 × 2) + (13 × 12 × 2)]

Surface area of a right rectangular prism = 812 m².

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Answer:

8

Step-by-step explanation:

3x - 10 = x + 2

2x = 12

x = 6

3x - 10 = 3(6) - 10 = 8

The length of each leg of the triangle is 8.

Answer: 187 apples

Step-by-step explanation:

Substitute "year 5" into the equation. y = 15.2(5) +111.

y = 76 + 111

y = 187

The given text presents a map of South Africa with various locations marked. However, important information such as the scale of the map and the number of provinces is missing. Without the scale, we cannot accurately determine distances or calculate probabilities. Similarly, the absence of information regarding the provinces prevents us from calculating the probability of selecting a specific province. Therefore, based on the given text, we can only answer the first two questions, which involve the type of scale and the general direction between two locations.

1. The type of scale given in the diagram is a ratio scale, specifically a linear scale. It represents the relationship between map distance and actual distance in a straight line.

Interpretation of the given scale:

The scale on the map is not clearly visible in the provided text. However, it is typically written as a fraction or ratio, where the numerator represents the map distance and the denominator represents the corresponding actual distance on the ground. For example, a scale of 1:1,000,000 means that 1 unit on the map represents 1,000,000 units in the real world.

2. The general direction of Badplaas from Colesburg is to the northeast.

3. To determine the actual straight-line distance between East London and Bloemfontein, we would need the scale from the map. However, the scale is not provided in the given text, making it impossible to calculate the distance accurately.

4. Stellenbosch is located in the Western Cape province on the map.

5. To calculate the probability of randomly selecting the Western Cape province in South Africa, we would need the total number of provinces in South Africa and the number of provinces that belong to the Western Cape. However, this information is not provided in the given text, so it is not possible to calculate the probability.

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Answer:

95 × 10^3 = 95000

95 × 10^4 = 950000

95 × 10^5 = 9500000

The required solution of the random variable X that follows an exponential distribution is shown below.

To find the expected value, standard deviation, and probabilities associated with the exponential distribution, we'll use the given parameter λ = 1/9.848.

Expected Value (Mean):

The expected value of an exponential distribution is equal to the reciprocal of the rate parameter λ. Therefore, the expected value is given by:

E(X) = 1 / λ

E(X) = 1 / (1/9.848)

E(X) = 9.848

So, the expected value of the random variable X is 9.848.

Standard Deviation:

The standard deviation of an exponential distribution is equal to the reciprocal of the rate parameter λ. Therefore, the standard deviation is given by:

σ = 1 / λ

σ = 1 / (1/9.848)

σ = 9.848

So, the standard deviation of the random variable X is also 9.848.

P(X < 12):

To find the probability that X is less than 12, we can use the cumulative distribution function (CDF) of the exponential distribution. The CDF is given by:

[tex]F(x) = 1 - e^{(-\lambda x)}[/tex]

Substituting the given λ = 1/9.848 and x = 12, we have:

[tex]P(X < 12) = F(12) = 1 - e^{-(1/9.848) * 12)}[/tex]

Calculating this expression, we find:

P(X < 12) ≈ 0.7043

Therefore, the probability that X is less than 12 is approximately 0.7737.

P(8 < X < 12):

To find the probability that X lies between 8 and 12, we subtract the cumulative probability at 8 from the cumulative probability at 12:

P(8 < X < 12) = F(12) - F(8)
                     = [tex]1 - e^{(-(1/9.848) * 12)]} - [1 - e^{(-(1/9.848) * 8)]}[/tex]

Calculating this expression, we find:

P(8 < X < 12) ≈ 0.2025

Therefore, the probability that X lies between 8 and 12 is approximately 0.2025.

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The average rate of change for this problem is given as follows:

2.4 inches per hour.

The average rate of change of a function is given by the change in the output of the function divided by the change in the input of the function.

The change in the output is given as follows:

8.1 - 3.3 = 4.8.

The change in the input is given as follows:

2 - 0 = 2.

Hence the average rate of change is given as follows:

4.8/2 = 2.4 inches per hour.

The table is given by the image presented at the end of the answer.

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All the equations which are equivalents are,

⇒ 2 + x = 5

⇒ x + 1 = 4

⇒ Negative 5 + x = negative 2

We have to given that;

All expressions are,

2 + x = 5

x + 1 = 4

9 + x = 6

x + (-4) = 7

-5 + x = -2

Now, We can simplify all the expressions as;

⇒ 2 + x = 5

⇒ x = 5 - 2

⇒ x = 3

⇒ x + 1 = 4

⇒ x = 4 - 1

⇒ x = 3

⇒ 9 + x = 6

⇒ x = 6 - 9

⇒ x = - 3

⇒ x + (-4) = 7

⇒ x = 7 + 4

⇒ x = 11

⇒ -5 + x = -2

⇒x = - 2 + 5

⇒ x = 3

Thus, All the equations which are equivalents are,

⇒ 2 + x = 5

⇒ x + 1 = 4

⇒ Negative 5 + x = negative 2

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Answer:

A: Function 1
B: Function 4
C: Function 2

Explanation A:

In order to solve this problem, you will have to find the slope for each function.

1st Function: For the first function, you will need to find 2 points, let's use (0, -4) and (1, 1). Remember slope is rise over run, so you would count the spaces from the first point to the second, it rises (y-axis) 5 points and runs (x-axis) 1 point. Which would give you 5/1 = 5. Function 1's slope is 5.

2nd Function: Same equation, rise over run, or y over x, so we need to find the pattern on each side. For the x-values, we can see starting from the first point that they increase by one, -2 to -1 to 0, and for the whole x-values, so 1 will be our run, since it is the x-value. For the rise, or y-values, we can see a pattern of -1 starting from the top going towards the bottom. Now, we can rise over run, -1/1 = -1

3rd Function: Slope is -4, as the equation is y = mx + b, and m stands for slope.

4th Function: Says slope is 2.

Now that we know all the slopes, we can see that Function 1's slope is the greatest.

Explanation B:
For this problem, we will have to find the y-intercept for all the functions, this means where the x-value is 0, so that the y-value will be right on the y-axis.

Function 1: We can see that the function intercepts the y-axis at (0, -4), it's y-intercept being -4.

Function 2: Where the x-value is 0, we can find the y-intercept, which is (0, 1). Making the y-intercept 1.

Function 3: Following the y = mx + b format, the b is the y-intercept, so it is -2.

Function 4: Gives you the y-intercept as 5.

In the end, the only function with a y-intercept greater than 4 is Function 4.

Explanation C:

Since we already know the y-intercepts from the previous problem, we only need to see which one is closest to 0. In this case, it would be Function 2 with a y-intercept at -1, only 1 unit off of 0, making it the closest.

Answer:

Step-by-step explanation:

Total cost function, C = 15+9x-6x²+x

By differentiation,

dc/dx= 9-12x²+1

but

dc/dx= 0

9-12x²+1 = 0

-12x = -10

x=10/12

x=5/6