Find the value of x for the following

The distance between the line and the point is 2 units

A perpendicular line will give the shortest distance between a point and a line.

From the equation of the line y = x+1. Therefore the slope is 1

using the equation (y-y1) = m(x-x1)

equation of the line joining the point and the line

= y-(-6) = -1(x - (-5)

= y+6 = -x-5

y = -x-11

x-1 = -x-11

2x = -10

x = -5

y = -5+1

y = -4

(x,y) = (-5,-4)

d = √ -5-(-5)²+ -4-(-6)²

d = √ 0+ 4

d = √4

d = 2 unit

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The number of blue marbles in the representative sample which contains 2 green marbles is 8.

What is meant by a representative sample?

A subset of a population that aims to correctly reflect the traits of the larger group is called a representative sample. Because the results accurately mirror those you would get from interviewing the full population, it is known as a representative sample.

Because samples comprise more manageable, smaller representations of the broader group, they are helpful in statistical analysis when population numbers are huge.

Given the total number of green and blue marbles is 8 and 32 respectively.

So the ratio of green marbles to blue marbles = 8:32 = 1:4

Now, this ratio is maintained when we form a representative sample.

Given there are 2 green marbles in a representative sample.

Then the number of blue marbles x is:

2 : x = 1 : 4

2 : x = 2 : 8

x = 8

Therefore the number of blue marbles in the representative sample is 8.

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The trigonometric identities can be proved as follows;

32. Using the substitution, tan(x) = sin(x)/cos(x), and cot(x) = cos(x)/sin(x), we get;

cot(x) - tan(x) = (1 - 2·sin²(x))/(sin(x)·cos(x)) = sec(x)·(csc(x) - 2·sin(x))

36. tan(θ)/(1 + sec(θ)) = sin(θ)/(1 + cos(θ)) = (1 - cos(θ))/(sin(θ)) = -cot(θ) + csc(θ)

Trigonometric identities are equations involving trigonometric functions which are valid for the values of the input variable.

as follows;

cot(x) - tan(x) = sec(x)·(csc(x) - 2·sin(x))

The left hand side of the equation can be expressed using sin(x) and cos(x) as follows;

cot(x) = cos(x)/sin(x)

tan(x) = sin(x)/cos(x)

Therefore;

cot(x) - tan(x) = cos(x)/sin(x) - sin(x)/cos(x) = (cos²(x) - sin²(x))/(sin(x)·cos(x))

cos²(x) = 1 - sin²(x), therefore

(cos²(x) - sin²(x))/(sin(x)·cos(x)) = (1 - sin²(x) - sin²(x))/(sin(x)·cos(x))

(1 - sin²(x) - sin²(x))/(sin(x)·cos(x)) = (1 - 2·sin²(x))/(sin(x)·cos(x))

(1 - 2·sin²(x))/(sin(x)·cos(x)) = csc(x)·sec(x)·(1 - 2·sin²(x))

csc(x)·sec(x)·(1 - 2·sin²(x)) = sec(x)·(csc(x) - 2·sin(x))

Therefore;

cot(x) - tan(x) = csc(x)·sec(x)·(1 - 2·sin²(x)) = sec(x)·(csc(x) - 2·sin(x))

36. tan(θ)/(1 + sec(θ)) = -cot(θ) + csc(θ)

The left hand side can be manipulated as follows;

tan(θ) = sin(θ)/cos(θ)

sec(θ) = 1/cos(θ)

Therefore; tan(θ)/(1 + sec(θ)) = (sin(θ)/cos(θ))/(1 + (1/cos(θ)))

(sin(θ)/cos(θ))/(1 + (1/cos(θ))) = (sin(θ)/cos(θ))/((cos(θ) + 1)/(cos(θ)))

((sin(θ)/cos(θ))×cos(θ))/((cos(θ) + 1)/(cos(θ)) × cos(θ))) = sin(θ)/(cos(θ) + 1)

tan(θ)/(1 + sec(θ)) = sin(θ)/(cos(θ) + 1)

sin(θ)/(cos(θ) + 1) = sin(θ)/(1 + cos(θ)) × ((1 - cos(θ))/(1 - cos(θ)))

sin(θ)/(1 + cos(θ)) × ((1 - cos(θ))/(1 - cos(θ))) = ((sin(θ)·(1 - cos(θ))/(1 - cos²(θ)))

((sin(θ)·(1 - cos(θ))/(1 - cos²(θ))) = ((sin(θ)·(1 - cos(θ))/(sin²(θ)) = (1 - cos(θ))/(sin(θ))

sin(θ)/(cos(θ) + 1) =  (1 - cos(θ))/(sin(θ)) = csc(θ) - cot(θ) = -cot(θ) + csc(θ)

tan(θ)/(1 + sec(θ)) = sin(θ)/(cos(θ) + 1) = -cot(θ) + csc(θ)

Therefore;

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For the linear equation 3x + y = 9 we have:

Here we want to find the two intercepts for the linear equation:

3x + y = 9

First, the x-intercept is the point that we get when we evaluate on y = 0.

3x + 0 = 9

3x = 9

x = 9/3

x = 3

Then we get: x-intercept (3, 0)

And for the y-intercept we need to evaluate in x = 0.

3*0 + y = 9

y = 9

Then the y-intercept is (0, 9).

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The y-coordinate for the solution to the system of equations is y = -3.

The substitution method is typically used in mathematics to solve an equation system. In this approach, you solve the equation for one variable first, then you enter its value into the other equation.

Simultaneous equations may usually be solved easily using the substitution method. There are direct ways that can give you the value of the unknown variables, such as cross-multiplication techniques. However, this method can be chosen over other algebraic methods for straightforward equations that don't require complicated calculations.

The system of equation is given as:

y = 1/2x - 4

y = -2x + 1

Substituting the value of y in equation 1 we have:

-2x + 1 = 1/2x - 4

-2x + 1 = x - 8 / 2

(-2x + 1)(2) = x - 8

-4x + 2 = x - 8

2 + 8 = x + 4x

10 = 5x

x = 2

Substituting the value of x in the second equation:

y = -2(2) + 1

y = -4 + 1

y = -3

Hence, the y-coordinate for the solution to the system of equations is y = -3.

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Answer:

The original price of the shoes was $53.

The discount percentage was 20%.

The discount amount was 53 * 0.2 = 10.6.

The sale price after the discount was 53 - 10.6 = 42.4.

The processing fee percentage was 20%.

The processing fee amount was 42.4 * 0.2 = 8.48.

The final price after the processing fee was 42.4 + 8.48 = 50.88.

Therefore, the discount was $10.60, and the final price was $50.88.

Step-by-step explanation:

Answer:

-5

Step-by-step explanation:

The equation is in slope intercept form,

y = mx+b

where m is the slope and b is the y intercept.

y = -4x-5

The slope is -4 and the y intercept is -5.

Answer:

a=1000.10

Step-by-step explanation:

If 60 dogs are registered in Castroville, we can find the number of Golden Retrievers and Boxers by using the given percentages.

Number of Golden Retrievers = 0.45 x 60 = 27

Number of Boxers = 0.55 x 60 = 33

To find how many more Boxers there are than Golden Retrievers, we can subtract the number of Golden Retrievers from the number of Boxers:

Boxers - Golden Retrievers = 33 - 27 = 6.

Therefore, there are 6 more Boxers than Golden Retrievers in the group of 60 registered dogs in Castroville.

The given information only provides the percentages of Golden Retrievers and Boxers among registered dogs in Castroville, so we don't have enough information to directly calculate the percentage of all other dog breeds. However, we know that the total percentage of all dog breeds must add up to 100%. Since Golden Retrievers and Boxers make up 45% and 55% of the registered dogs, respectively, the percentage of all other dog breeds must be 100% - 45% - 55% = 0%. This means that there are no other registered dog breeds in Castroville, and all registered dogs belong to either the Golden Retriever or Boxer breed.

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Answer:

Step-by-step explanation:

The decay rate of 80% per year means that after each year, the value of the quantity decreases by 80/100 * 3600 = 2880.

To find the value of the quantity after 0.2 decades, we need to multiply the number of years by 10, since a decade is equal to 10 years. In this case, 0.2 decades is equal to 0.2 * 10 = 2 years.

The value of the quantity after 2 years can be found using the formula:

3600 * (1 - 0.8)^2

Plugging in the values, we get:

3600 * (1 - 0.8)^2 = 3600 * (0.2)^2 = 3600 * 0.04 = 144

So, the value of the quantity after 0.2 decades, to the nearest hundredth, is 144.

Answer: 726.83

Step-by-step explanation:

Answer:

Step-by-step explanation:

Let the smaller integer be x, then the larger integer is x + 3.

The sum of the reciprocal of the smaller and twice the reciprocal of the larger is 9/10:

1/x + 2/(x + 3) = 9/10

Expanding both sides:

10/x + 20/(x + 3) = 90/10

Combining like terms on the left side:

(10 + 20)/(x(x + 3)) = 90/10

30/(x(x + 3)) = 9/10

Cross multiplying both sides:

30 * 10 = 9 * (x(x + 3))

300 = 9x(x + 3)

Expanding the right side:

300 = 9x^2 + 27x

Subtracting 27x from both sides:

273 = 9x^2 + 27x - 27x

273 = 9x^2

Taking the square root of both sides:

√273 = √(9x^2)

√273 = 3x

Dividing both sides by 3:

√273/3 = x

The smaller integer is √273/3 and the larger integer is √273/3 + 3.