A quantity with an initial value of 3600 decays continuously at a rate of 80% per year.
What is the value of the quantity after 0.2 decades, to the nearest hundredth?

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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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.

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.

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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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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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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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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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:

The area is 510 in^2.

Step-by-step explanation:

You need to find two areas. The area of the rectangle and then the area of the triangle above.

The formula for a rectangle's area is base x height. The base is 20 inches, and the height is 15 inches as shown on the side. So, you multiply 20 x 15, and the product is 300.

The formula for a triangle's height is 1/2(base x height). The 3 ft on the side is equal to 36 inches. To find the height of the triangle alone, you take the 36 inches and subtract by 15 inches, resulting in a difference of 21 inches. The base of the triangle is 20 inches, so multiply 20 x 21 = 420. Then multiply 420 by 1/2 (or divide by 2) and you get 210.

Now that we have the areas of both shapes, 210 + 300 = 510.

The area of the composite figure is 510in².

Check attached image 1.

a) For the rectangle, we have that the base is 20 inches, and that the height is 15 inches.

b) For the triangle, we see that the base is also 20 inches long, and the height must be the difference between the height of the rectangle and the total height of the composite figure.

First, convert the 3 feet to inches, because we need to have the same unit in order to make the calculations.

We know that 12 inches is 1 feet, therefore, to covert 3 feet to inches we do the following operation:

[tex]3feet*\frac{12inches}{1feet}[/tex]

The feet unit cancel out and the 3 is multiplied by 12, leaving us with the following result:

[tex]36inches[/tex]

Check attached image 2.

So the height of the triangle must  given by the following expression:

[tex]36in-15in=21in[/tex]

a) For the rectangle:

[tex]A=b*h[/tex]; where "b" is the length of the base and "h" is the height.

[tex]A=(20in)(15in)=300in^{2}.[/tex]

b) For the triangle:

[tex]A=\frac{b*h}{2} =\frac{(20in)(21in)}{2} =210in^{2} .[/tex]

So if the figured is formed by a triangle and a rectangle, the sum of the area of the 2 shaped equals the area of the composite figure. Let's add up the areas and calculate:

[tex]300in^{2} +210in^{2} =510in^{2} .[/tex]

The area of the composite figure is 510in².

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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