Work out the equation of the line which has a gradient of 3 and passes through the point (-1,3).

The center and radius of a circle given by the equation x² - 10x + y² + 6y - 30 = 0  is (5,-3) and√64 = 8 units.

When finding the center and radius of a circle given by the equation x² - 10x + y² + 6y - 30 = 0, one can use the following steps:

The first step is to rearrange the equation into the standard form, (x - a)² + (y - b)² = r². This is done by completing the square for both the x and y terms in the equation.

x² - 10x + y² + 6y - 30 = 0x² - 10x + 25 + y² + 6y + 9 - 30 = 25 + 9(x - 5)² + (y + 3)² = 64 Therefore, the center of the circle is (5,-3), and the radius is √64 = 8 units.

Explanation:  Given the equation x² - 10x + y² + 6y - 30 = 0, we want to find the center and radius of the circle. The standard form of the equation of a circle with center (a,b) and radius r is (x - a)² + (y - b)² = r². We will begin by completing the square for the x terms and the y terms separately: For the x terms: x² - 10x= x² - 10x + 25 - 25= (x - 5)² - 25 For the y terms: y² + 6y= y² + 6y + 9 - 9= (y + 3)² - 9 Now we can substitute these expressions back into the original equation and simplify: x² - 10x + y² + 6y - 30 = 0(x - 5)² - 25 + (y + 3)² - 9 - 30 = 0(x - 5)² + (y + 3)² = 64 The equation is now in standard form,

which means that the center of the circle is (5,-3) and the radius is √64 = 8 units.

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Answer:

58 m

Step-by-step explanation:

The correct answer is 58.

To get the perimeter you add all the sides of the image, however, you are missing 2 values from the image.

If you make the shape into a square where all opposite sides are the same length, then you will see that one missing length is 6 m =(10m-4m).

The other missing number is 12 m which is base of 19 m - 7m that you are given on the top.

So you add the measurements (going clockwise starting at the top, 7+6+12+4+19+10=58 m

Answer:

  58 m

Step-by-step explanation:

You want the perimeter of the L-shaped figure shown.

The perimeter is the sum of the side lengths. Here, a couple of lengths are missing from the diagram, but that doesn't prevent us finding the perimeter.

The horizontal lengths at the top have the same total length as the length at the bottom marked 19 m. This means the sum of all of the horizontal lengths is ...

  2 × 19 m = 38 m

The vertical lengths at the right side have the same total length as the vertical length at the left side, marked 10 m. This means the sum of all of the vertical lengths is ...

  2 × 10 m = 20 m

The perimeter is the sum of the horizontal and vertical lengths:

  P = 38 m + 20 m = 58 m

The perimeter of the figure is 58 m.

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a: y = 6x²

b: y = 8x² + 64x + 128

c: y = -23x² + 322x - 1056

d: y = 64x² - 64x + 8

e: y = -9x² - 81

f: y = -4x² + 64

g: y = -2x² + 8x

h: y = 1.4x² - 25.2x + 118.4

i: y = -18x² + 180x - 680

Answer:

The correct answer is enlargement scale factor of 1.5.

Step-by-step explanation:

the reason for this is that if you divide the D' numbers by the D numbers you get 1.5

so 8×1.5=12

6×1.5=9

any scale factor 0-1 is a reduction. Greater than 1 (like this case here) is an enlargement. as you can see the after image D' is bigger than the pre image D

I hope this helps :)

Answer:

19

Step-by-step explanation:

20+5+13 / 2 = 19

Answer:

4π (approx. 12.57) inches

Step-by-step explanation:

Equation to find circumference of a circle is πd, where d is the diameter of the circle.

d = 2r (radius)

d = 4

Circumference = πd

Circumference = 4π (inches)

Circumference ≈ 12.57 (inches)

Using the definition of cosecant and cotangent, the expression can be rewritten as 1/sine (x) + 1/cosine (x).

1/sine (x) + 1/cosine (x) = 1 confirms that the Pythagorean identity.

The cosecant of an angle is defined as 1/sine (x), and the cotangent of an angle is defined as 1/cosine(x).

Using the definition of cosecant and cotangent, the expression can be rewritten as 1/sine (x) + 1/cosine (x).

Using the fundamental Pythagorean identity, which states that

sine²(x) + cosine² (x) = 1, the expression can be further simplified to

sine²(x) + cosine² (x) + 1/sine (x) + 1/cosine (x) = 1.

To verify the identity, we can substitute sine²(x) + cosine² (x) with 1, leaving us with 1 + 1/sine (x) + 1/cosine (x) = 1.

Simplifying further, we get 1/sine (x) + 1/cosine (x) = 1, which is the original expression. This confirms that the Pythagorean identity is true for the given expression.

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All the quadratic functions but f(x) = 3x^2-24x+46 and f(x) = 2x^2+8x+5 have no real solutions

Function 7

Given that

f(x) = -2x^2 + 12x - 22

Using the discriminant D = b^2 - 4ac, we have

D = 12^2 - 4 * -2 * -22

D = -32

This is less than 0

The equation has no real solution

Function 8

Given that

f(x) = x^2-8x+20

Using the discriminant D = b^2 - 4ac, we have

D = -8^2 - 4 * 1 * 20

D = -16

This is less than 0

The equation has no real solution

Function 9

Given that

f(x) = 3x^2-24x+46

By the use of graph, we have

x = 3.184 and x = 4.816

Function 10

Given that

f(x) = x^2+2x+2

Using the discriminant D = b^2 - 4ac, we have

D = 1^2 - 4 * 2 * 2

D = -15

This is less than 0

The equation has no real solution

Function 11

Given that

f(x) = -1/2x^2+4x-10

Using the discriminant D = b^2 - 4ac, we have

D = 4^2 - 4 * -1/2 * -10

D = -4

This is less than 0

The equation has no real solution

Function 12

Given that

f(x) = 2x^2+8x+5

By the use of graph, we have

x = -3.225 and x = -0.775

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The two solutions for the given equation in the interval are:

x = 0°

x = 159.1°

Here we have the equation:

|1 + 3sin(2x)| = 1

Breaking the absoulte value part, we will get two equations, these are:

1 + 3sin(2x) = 1

1 + 3sin(2x) = -1

Now we need to solve these two, the first one gives:

3sin(2x) = 1 - 1

3sin(2x) = 0

Then we know that:

2x = 0°

x = 0°/2 = 0

the other equation gives:

1 + 3sin(2x) = -1

3sin(2x) = -1 - 1

3sin(2x) =-2

sin(2x) = -2/3

2x = Asin(-2/3)

2x = 318°

x = 318.2°/2 = 159.1°

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Answer:

Area of the shape = 62.135 (units^2)

Step-by-step explanation:

To start, divide the shape into simpler parts.

A triangle (4 by 6)

A Rectangle (6 by 6)

A half Circle (Radius of 3) - take the height (6) and divide it by 2 (= 3)

First get the area of the Triangle. Base x Height / 2

    4 x 6 = 24; 24 / 2 = 12;

         Area of the Triangle is 12

Second get the area of the Rectangle. Length x Width

    6 x 6 = 36

         Area of the Rectangle is 36

Third get the area of the circle Pie x Radius ^2 (squared)

   3.14 x (3 ^2) = 28.27

         Now take the area of the whole circle and divide it by 2 to get the      half circle

              28.27 / 2 = 14.135; Area of the half Circle is 14.135

Add up all the areas to get the total for your shape.

    12 + 36 + 14.135 = 62.135

Therefore, y = (1/3) sin ((1/2)x²) is the solution of the initial value problem y′=(2/3)x√(1−9y²); y(0) = 0.

To solve the differential equation y′=(2/3)x√(1−9y²)

The differential equation to be solved is: y′=(2/3)x√(1−9y²).

Here, we need to find y.

For this, we will separate the variables and integrate both sides. Integration gives us:

`∫1/(√(1−9y²))dy=∫(2/3)x dx`

.On integrating the left side, we will use u-substitution.

u = 3y → du = 3 dy

dy = (1/3) du → y = (1/3) u.

Now the equation becomes `∫du/(√(1−u²))=(2/3)∫xdx`.

Now, substituting u = sin t in the left integral, we have: `

∫du/(√(1−u²))

=∫cos(t)dt

=[sin⁻¹(u)]+C`.

So, the left-hand side is `

[sin⁻¹(u)]+C

= [sin⁻¹(3y)] + C`

Now, the right-hand side will be:

∫xdx=(1/2)x²+D`

On combining both sides, we get the solution to the differential equation as: `

[sin⁻¹(3y)]+C=(1/2)x²+D`

On solving for y, we get:

y = (1/3) sin ((1/2)x² + D' ) or  y = (1/3) sin ((1/2)x²)

since we can choose D' = C.

To solve the initial value problem

y′=(2/3)x√(1−9y2); y(0) = 0

To solve the initial value problem

y′=(2/3)x√(1−9y2)

y(0) = 0

we will substitute x = 0, y = 0 in the general solution that we obtained in part .

y = (1/3) sin ((1/2)x²)

y = (1/3) sin ((1/2)0²) = 0.

So the required solution is y = (1/3) sin ((1/2)x²).

Therefore, y = (1/3) sin ((1/2)x²) is the solution of the initial value problem y′=(2/3)x√(1−9y²); y(0) = 0.

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The hypotenuse of the triangle is x = 10.

We can see that we have a right triangle, then we can use the trigonometric relation:

cos(a) = (adjacent cathetus)/hypotenuse

Replacing the known values:

cos(60°) =5/x

Solving for x we will get:

x = 5/cos(60°) = 10

That is the value of x.

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The volume of the prism is 0.343 cubic inches.

What is Volume ?

Volume is a measure of the amount of space occupied by a three-dimensional object. It is the quantity of space that a solid object occupies in three dimensions. Volume is often expressed in cubic units, such as cubic meters, cubic centimeters, or cubic inches , depending on the system of measurement used.

To find the volume of a rectangular prism, we multiply its length, width, and height.

In this case, the length, width, and height of the prism are all 0.7 inch. Therefore, the volume of the prism is:

Volume = (length) x (width) x (height)

= (0.7 x (0.7) x (0.7)

= 0.343 cubic inches

Therefore, the volume of the prism is 0.343 cubic inches.

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The type of error depends on the level of significance and whether the test is one-tailed or two-tailed.

The type of error occurs if you fail to reject H0 when, in fact, it is not true.

The type of error that occurs when one fails to reject H0 when, in reality, it is false is type II error.

Type II error is an error in which one accepts a null hypothesis that should be rejected.

This type of error is the opposite of type I error, where one rejects a null hypothesis that is true.

A type II error is a serious issue because it suggests that a significant difference exists, but the statistical test fails to detect it.

There are two types of errors associated with hypothesis testing, type I and type II errors, depending on the significance level of the test.

A type I error occurs when the null hypothesis is rejected when it is true.

A type II error happens when the null hypothesis is not rejected when it is false.

A type I error, also known as an alpha error, is caused by rejecting the null hypothesis when it is true.

It occurs when the significance level is set too high or when a statistical test is not performed correctly.

A type I error occurs when the observed value lies in the rejection region of the null hypothesis, and we reject the null hypothesis even though it is true.

In conclusion, when one fails to reject H0 when it is false, a type II error occurs. On the other hand, when one rejects H0 when it is true, a type I error occurs.

The type of error depends on the level of significance and whether the test is one-tailed or two-tailed.

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There are 4 red jellybeans in a jar of 20 total. The probability that you will choose a red jellybean from the jar is represented by the fraction 4/20.

The probability of picking a red jellybean out of the jar is the number of red jellybeans in the jar divided by the total number of jellybeans in the jar.

So, P(red jellybean) = number of red jellybeans / total number of jellybeans

= 4/20

= 1/5

Therefore, the fraction that represents the probability of picking a red jellybean out of the jar is 4/20.

The complete question is:-

You have a jar of 20 jellybeans, and 4 are red. Which fraction represents the probability that you will pick a red jellybean out of the jar?

A) 4/20

B) 16/20

C) 20/4

D) 20/16

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Answer:

You end at the coordinates (4, 2)

Answer:

1)  a = 1, b = -8, c = 17;  Vertex:   (4, 1)

2)  a = -1, b = -2. c = -2;  Vertex:   (-1, -1)

3)  a = -1, b = 6, c = -8;  Vertex:   (3, 1)

4)  a = -3, b = 6, c = 0;  Vertex:   (1, 3)

5)  a = -2, b = -16, c = -31;  Vertex:   (-4, 1)

6)  a = -1/2 or -0.5, b = -4, c = -6;  Vertex:   (-4, 2)

Step-by-step explanation:

The quadratic functions listed are all in standard form:

y = ax² + bx + c

where a, b, and c, are coefficients for each of the terms.

Vertex

To find the vertex of a parabolic equation in standard form. Calculate -b/2a. This will be your x-coordinate. Then substitute this back into f(x) to obtain the y-coordinate; The calculated point is your vertex.

1)   x = - b / 2a = - (-8) / 2 (1) = 8 / 2 = 4

    f(4) = 4² - 8 (4) + 17 = 16 - 32 + 17 = 1

   Vertex:   (4, 1)

2)  x = -b / 2a = - (-2) / 2 (-1) = 2 / (-2) = -1

    f(-1) = - (-1)² - 2 (-1) - 2 = -1 + 2 - 2 = -1

    Vertex:   (-1, -1)

3)  x = - b / 2a = - (6) / 2 (-1) = -6 / -2 = 3

    f(3) = - (3)² + 6 (3) -8 = -9 + 18 - 8 = 1

    Vertex:   (3, 1)

4)  x = - b / 2a = - (6) / 2 (-3) = -6 / -6 = 1

     f(1) = -3 (1)² + 6 (1) = -3 + 6 = 3

     Vertex:   (1, 3)

5)  x = - b / 2a = - (-16) / 2(-2) = 16 / -4 = -4

    f(-4) = -2 (-4)² - 16 (-4) - 31 = -32 + 64 - 31 = 1

    Vertex:   (-4, 1)

6)  x = - b / 2a = - (-4) / 2 (-0.5) = 4 / -1 = -4

    f (-4) = (-0.5) (-4)² - 4 (-4) - 6 = -8 + 16 - 6 = 2

    Vertex:   (-4, 2)

Answer:

Step-by-step explanation:

The percentage of calories that comes from saturated fat in each serving of these crackers is 0.4%

To calculate the percentage of calories that come from saturated fat, we need to first determine how many calories come from saturated fat in one serving of crackers.

We know that each serving of crackers provides 120 calories, and 0.5 grams of saturated fat. We can convert the amount of saturated fat from grams to calories by multiplying it by 9 (since 1 gram of fat provides 9 calories).

0.5 grams of saturated fat x 9 calories per gram = 4.5 calories from saturated fat

Therefore, out of the 120 total calories in one serving of crackers, 4.5 calories come from saturated fat.

To find the percentage of calories that come from saturated fat, we can divide the number of calories from saturated fat by the total number of calories in one serving of crackers, and then multiply by 100.

(4.5 calories from saturated fat / 120 total calories) x 100 = 0.0375 x 100 = 0.4%

Therefore, each serving of crackers provides 0.4% of calories from saturated fat.

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The probability that a randomly selected car had automatic transmission or air conditioning or both is 0.92517.

Total number of cars sold, n = 147

Let A denotes the car is air conditioning.

And B denotes the car is automatic message transmission.

A = 81

B = 82

Number of cars that neither of these extras = 11

Only A = 54

Only B = 55

Now,

P(A ∩ B') = A/n

P(A ∩ B') = 81/147

P(A ∩ B') = 0.551

P(A' ∩ B') = 11/147

P(A' ∩ B') = 0.07483

The probability that a randomly selected car had automatic transmission or air conditioning or both is:

P(A ∪ B) = 1 - P(A' ∩ B')

P(A ∪ B) = 1 - 0.07483

P(A ∪ B) = 0.92517

The likelihood that an automobile chosen at random has either an automatic gearbox, air conditioning, or both is 0.92517.

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Thus, the value of x in obtained in the simplest form for the unit circle is: x = -√5/3.

A circle with a radius of one unit and a centre at the origin is referred to as a unit circle just on Cartesian Plane (0, 0). When working with trigonometric functions including angle measurements, the unit circle is a useful tool that makes reference much simpler.

The equation of unit circle, radius r = 1 unit. :

x² + y² = 1.

For the given point P (x, 2/3), we can solve for x by substituting each quantity into the equation.

x² + (2/3)² = 1

x² + 4/9= 1.

Subtract 4/9 from both side.

x² = 5/9.  

Taking square root on both side.

x = √5/3

x = -√5/3

Thus, the value of x in obtained in the simplest form for the unit circle is: x = -√5/3.

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Complete question:

the point P=(x,2/3) lies on the unit circle shown below . what is the value of x in simplest form?

The diagram is attached.

Answer:

x=3, y=7

Step-by-step explanation:

Substituting the first equation y=x+4 into the second:

3x+(x+4)=16

Simplifying:

3x+x+4=16

4x+4=16

Subtracting 4 from both sides:

4x=12

Dividing both sides by 4:

x=3

We can now substitute x=3 into our first equation, y=x+4.

Substituting:

y=3+4

y=7
So, x=3, y=7

The value of FG( SAY X) = x=131.

An angle is the result of the intersection of two lines.

An "angle" is the length of the "opening" between these two beams.

Angles are commonly measured in degrees and radians, a measurement of circularity or rotation.

In geometry, an angle can be created by joining the extremities of two rays. These rays are intended to represent the angle's sides or limbs.

The two primary components of an angle are the limbs and the vertex.

The joint vertex is the common terminal of the two beams.

According to our question-

35 - 3x + 2x + 14= 180

49-x=180

-x=180-49

x=131

The value of FG( SAY X) = x=131.

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Thus, the surface area with the dimensions of a rectangular prism as 6m by 4 m by 15 m is found as: S = 348 sq. m.

A rectangular prism is a 3 solid that is surrounded by 6 rectangular faces, 2 of which are the bases (the top face and bottom face), and the remaining 4 are lateral faces. It likewise has 12 edges and 8 vertices.

A rectangular prism is sometimes known as a cuboid because of its shape. A shoe box, an ice cream bar, or a matchbox are some instances of rectangular prisms in everyday objects.

Dimensions of rectangular prism :

surface area of a rectangular prism:

S = 2(lw + wh + hl)

S = 2(6*4 + 4*15 + 15*6)

S = 348 sq. m.

Thus, the surface area with the dimensions of a rectangular prism as 6m by 4 m by 15 m is found as: S = 348 sq. m.

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Answer: 50

Step-by-step explanation:

2*6+2*4+2*15=12+8+30=50

The probability of having a boy is 0.51, and the probability of having a girl is 1 - 0.51 = 0.49. To find the probability of having three boys, we use the binomial probability formula: P(X = k) = (n choose k) * pk * (1-p)(n-k).

Probability is an estimate of how likely an event is to occur. It is a value ranging from zero to one, with 0 indicating an impossible event and 1 indicating a certain event. The higher the likelihood, the more probable the event will occur, and the lower the probability, the less likely the event will occur.

The probability of having a boy is 0.51, which means that the probability of having a girl is 1 - 0.51 = 0.49.

We want to find the probability that a family of five children has exactly three boys. We can use the binomial probability formula:

P(X = k) = (n choose k) * [tex]p^k * (1-p)^(n-k)[/tex]

where:

P(X = k) is the probability of getting k successes (in this case, having k boys)

n is the number of trials (in this case, the number of children born)

p is the probability of success (in this case, the probability of having a boy)

(n choose k) is the binomial coefficient, which represents the number of ways to choose k successes from n trials

So, for this problem:

n = 5

k = 3

p = 0.51

P(X = 3) = (5 choose 3) * [tex]0.51^3 * 0.49^(5-3)[/tex]

= (10) * [tex]0.51^3 * 0.49^2[/tex]

= 0.234

Therefore, the probability that a family of five children has exactly three boys is 0.234, or about 23.4%.

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The statement "the data point 5 lies outside the range of the data" is not true about the data shown by the box-and-whisker plot below.

To understand why the statement is not true, we need to interpret the box-and-whisker plot. The box represents the middle 50% of the data, with the bottom and top of the box indicating the 25th and 75th percentiles, respectively. The line inside the box represents the median. The whiskers represent the range of the data, with the endpoints of the whiskers indicating the minimum and maximum values, unless there are outliers.

Looking at the plot, we can see that the minimum value is 5, which is within the whisker range. Therefore, the statement "the data point 5 lies outside the range of the data" is not true. The statement "half the data lies between 37 and 51" is true, as the bottom and top of the box represent the 25th and 75th percentiles, respectively. The statement "the range is 57" is true, as the distance between the minimum and maximum values is 57. The statement "one fourth of the data is greater than 51" is also true, as the top of the box represents the 75th percentile.

Therefore, the correct statement is "the data point 5 lies within the range of the data."

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Answer:

Step-by-step explanation:

Answer: Let x be the number of weeks since Robert started his diet, and let y be his weight in pounds. We know that he is losing weight at a rate of 2 pounds per week, so the slope of the line is -2 (negative because he is losing weight). We also know that after 6 weeks, his weight is 205 pounds, so we have the point (6, 205).

Using the point-slope form of a linear equation, we can write the equation of the line as:

y - 205 = -2(x - 6)

Simplifying this equation gives:

y - 205 = -2x + 12

y = -2x + 217

Therefore, the equation that models Robert's weight loss is y = -2x + 217.

To find how much weight Robert will lose after 8 weeks, we substitute x = 8 into the equation:

y = -2(8) + 217

y = 201

Therefore, Robert will weigh 201 pounds after 8 weeks on his diet.

To check that this answer is reasonable, we can use the information that Robert is losing weight at a rate of 2 pounds per week. In 8 weeks, he would have lost:

2 pounds/week x 8 weeks = 16 pounds

205 pounds - 16 pounds = 189 pounds

Since 201 pounds is more than 189 pounds, our answer of 201 pounds after 8 weeks is reasonable.

So the completed work is:

Let x be the number of weeks since Robert started his diet, and let y be his weight in pounds.

We know that he is losing weight at a rate of 2 pounds per week, so the slope of the line is -2 (negative because he is losing weight).

We also know that after 6 weeks, his weight is 205 pounds, so we have the point (6, 205).

Using the point-slope form of a linear equation, we can write the equation of the line as:

y - 205 = -2(x - 6)

Simplifying this equation gives:

y - 205 = -2x + 12

y = -2x + 217

Therefore, the equation that models Robert's weight loss is y = -2x + 217.

To find how much weight Robert will lose after 8 weeks, we substitute x = 8 into the equation:

y = -2(8) + 217

y = 201

Therefore, Robert will weigh 201 pounds after 8 weeks on his diet.

Step-by-step explanation:

Answer:  Gross income is the total amount of money earned before any deductions or taxes are taken out. Net income, on the other hand, is the amount of money left after all deductions and taxes have been taken out. In other words, net income is what you actually take home after all expenses have been accounted for.

The apprentice can complete the job alone in approximately 11.43 hours (rounded to two decimal places). We can calculate it in the following manner.

Let's assume that the apprentice can complete the job alone in "x" hours.

In 2 hours, the roofer completes a fraction of the job which is equivalent to:

(2/8) = 1/4 of the job.

This means that the remaining fraction of the job that the apprentice has to complete is:

1 - 1/4 = 3/4 of the job.

The apprentice completes this remaining fraction of the job in 10 hours, so the rate at which he works is:

(3/4) of the job / 10 hours = 3/40 of the job per hour.

Since we know that the apprentice can complete the entire job alone in "x" hours, we can set up the equation:

1 job / x hours = 3/40 of the job per hour * (x - 10) hours

Simplifying this equation, we get:

x = 80/7

Therefore, the apprentice can complete the job alone in approximately 11.43 hours (rounded to two decimal places).

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