Q 9-1 Pythagorean Theorem and Trig

A girl is 20 ft. east of her school and her bike is 21 ft. south of the school. How far is she from her bike? Round to the nearest hundredth.

x(t) = [tex]c_1e^1^4^t+c_2e^-^t[/tex]  , y(t) = [tex]2c_1e^1^4^t-c_2e^-^t[/tex]   is the general solution of the sytem of differention eqution dx/dt = 4x+ 5y , dy/dt = 10x + 9y .

given  dx/dt = 4x+ 5y  and dy/dt = 10x + 9y

X'(t) = [tex]\left[\begin{array}{ccc}4&5\\10&9\end{array}\right][/tex]  X

where X = [tex]\left[\begin{array}{ccc}x\\y\\\end{array}\right][/tex]

so A = [tex]\left[\begin{array}{ccc}4&5\\10&9\end{array}\right][/tex]

now we need to find eigen value of the matrix and the corresponding eigen vector.

| A- λI | = 0

so after equation the value to zero

λ = 14 and λ = -1

now for the  λ = 14 corresponding eigen vector = [tex]\left[\begin{array}{ccc}1\\2\\\end{array}\right][/tex]

for λ = -1 corresponding eigen vector =  [tex]\left[\begin{array}{ccc}1\\-1\\\end{array}\right][/tex]

So general equation is given by:

X(t) = [tex]c_1e^1^4^t[/tex]  [tex]\left[\begin{array}{ccc}1\\2\\\end{array}\right][/tex]  +  [tex]c_2e^-^t[/tex]   [tex]\left[\begin{array}{ccc}1\\-1\\\end{array}\right][/tex]

after solving this

x(t) = [tex]c_1e^1^4^t+c_2e^-^t[/tex]

y(t) = [tex]2c_1e^1^4^t-c_2e^-^t[/tex]

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Answer:

The area of the dog pen is 522 [tex]ft^2[/tex].    (feet squared)

The perimeter of the dog pen is 94 feet

Step-by-step explanation:

Area

Here is the formula for the area of a rectangle.

[tex]A=lw[/tex]

We are given

[tex]l=29\\w=18[/tex]

Lets evaluate [tex]A[/tex].

[tex]A=29*18\\A=522[/tex]

Perimeter

Here is the formula for the perimeter of a rectangle.

[tex]P=2w+2l[/tex]

We are given

[tex]l=29\\w=18[/tex]

Lets evaluate [tex]P[/tex].

[tex]P=2l+2w\\P=(2*29)+(2*18)\\P=58+(2*18)\\P=58+36\\P=94[/tex]

Therefore , the solution of gross pay problem is 62% is this of her gross wage.

Hannah's take home pay of percentage 785 is 62.8% of her gross wage of 1250. This can be calculated by dividing 785 by 1250 and multiplying the answer by 100.

Here,

62.8% is the same as 0.628.

This can be calculated by dividing 785 by 1250.

To get the answer to the nearest whole percentage, we round up 0.628 to 0.63 which is 63%.

This means that Hannah's take home pay of 785 is 63% of her gross wage of 1250.

63% is the answer to the question: what percentage is Hannah's take home pay of her gross wage? This is correct to the nearest whole percentage.

To summarise, Hannah's take home pay of 785 is 63% of her gross wage of 1250, correct to the nearest whole percentage.

Hannah's take-home pay is 62.8% of her gross wage. This percentage is the result of taxes and other deductions taken by her employer.

Knowing this percentage can help Hannah better understand how much money she is taking home each week and how much money is being withheld from her paychecks.

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Answer:

  a.  y = 2.5x

  b.  markup: $30; retail price: $42

  c.  75%

  d.  y = 0.75x

Step-by-step explanation:

You want equations relating markup, retail price, and sale price on a phone that costs $12, is marked up 250%, then put on sale for $31.50.

The problem statement tells you the markup is 250% of the cost. You could write the equation as ...

  y = (250%)x

Or you can simplify it to a decimal number:

  y = 2.5x

The markup amount is ...

  y = 2.5(12) = 30

The markup amount is $30.

That is added to the cost to get the retail price:

  retail = $12 +30 = $42

The retail price of the phone is $42.

The sale price as a percent of the retail price is ...

  percent = (sale price)/(retail price) × 100%

  percent = 31.50/42 × 100% = 75%

The sale price is 75% of the retail price.

As in part (a), we could write this as ...

  y = (75%)x

or, in decimal, as ...

  y = 0.75x

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Coordinate (0,1) of y-intercept implies that a line or curve cross the Y axis

at 1.

The location of a point on the cartesian coordinate plane where a straight line or curve line as well as a plane crosses the Y axis is called Y-intercept. The coordinate of y-intercept is expressed as (0, y). This y value is y-intercept. The X coordinate of y intercept is zero.

In case of an equation of line while writing in slope-intercept form it is denoted by b such as y = mx+ b

in the above equation, b is y-intercept and m is the slope.

The y -intercept given by coordinate (0, 1) defines that a straight-line or curve line cross the Y axis at 1. The x coordinate of y -intercept is always zero because over the Y axis there is no value of X.

The y-intercept can have negative or positive value.

Suppose we have an equation of line 4x + 8y + 10 = 0

                                we will write it in a slope-intercept form

                                     8y = -4x - 10

                                     y = - 4x/8 - 10/8

                                    y = -1/2x - 5/4

In the above equation - 5/4 or -1.25 is the y-intercept that means the line crosses the Y-axis at -1.25

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Answer:

It should be 6 because 6x6 is 36

Answer:

Step-by-step explanation:

fractions are division so just divide 36 by 6 and you will get 6 which is your answer

36/6 =

36 : 6 =

6

Answer:

m = 2

Step-by-step explanation:

Slope = rise/run or (y2 - y1) / (x2 - x1)

We see the y increase by 4 and the x increase by 2, so the slope is

m = 4/2 = 2

So, m = 2

Answer: Zasha spent $6 on packages of gum, which means that she bought 6 / $1.20 = 5 packages of gum.

If she has a $20 bill, she has $20 - $6 = $14 left to spend on gum.

Zasha can buy an additional 14 / $1.20 = 11.6666666666666666666666667 packages of gum with this money.

Rounded down, Zasha can buy an additional 11 packages of gum.

Therefore, Zasha can buy a total of 5 + 11 = 16 packages of gum.

Step-by-step explanation:

On solving the provided question, we can say that - on doing the fraction we have  [tex]\frac{1}{10} ; \frac{1}{100} ; \frac{1}{10000}[/tex]

Any number of equal portions, or fractions, can be used to represent a whole. Fractions in standard English indicate how many units of a certain size there are. 8, 3/4. A whole includes fractions. The ratio of the numerator to the denominator is how numbers are expressed in mathematics. Each of these is an integer in simple fractions. In the numerator or denominator of a complex fraction is a fraction. True fractions have numerators that are less than their denominators. A fraction is a sum that constitutes a portion of a total. By breaking the entire up into smaller bits, you can evaluate it. Half of a full number or item, for instance, is represented as 12.

here we have,

0.1 = 1/10

0.01 = 1/100

0.0001 = 1/10000

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Answer:  sqrt(2)

Step-by-step explanation:

Remember that sec x = 1/cos x.

That means that sec 7pi/4 = 1/cos(7pi/4)

cos(7pi/4) = -cos(7pi/4 - pi)

                 = -cos(3pi/4)

cos(3pi/4) is a special angle which evaluates to -sqrt(2)/2.

Hence, -cos(3pi/4) evaluates to simply sqrt(2)/2

Since cos(7pi/4) is equal to -cos(3pi/4),

cos(7pi/4) is equal to sqrt(2)/2.

Lastly, sec 7pi/4 = 1/cos(7pi/4) , which is the reciprocal of sqrt(2)/2, that is 2/sqrt(2).

Rationalizing the answer 2/sqrt(2) gives sqrt(2)

The 3 linear equations are

1. y = mx + b

2. ax + by = c

3. x/a + y/b = 1

1. y = mx + b is a linear equation in two variables, x and y. The equation is in the slope-intercept form, where m is the slope of the line and b is the y-intercept.

2. ax + by = c is a linear equation in two variables, x and y. The equation is in the standard form, where a and b are constants and c is the result of the equation.

3. x/a + y/b = 1 is a linear equation in two variables, x and y. The equation is in the general form, where a and b are constants and 1 is the result of the equation.

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David's deposit of $220 compounded continuously at a rate of 1.7% will become $260 after 10 years.

Here we have been given that David invested $220.

The interest rate was 1.7%

Since the amount was compounded continuously, there is no fixed date of when it was compounded, the whole process is taken as an exponential distribution.

There were no withdrawals or additional deposits to this, so our Principal is constant. Hence we can write the formula

Amount = Principal X eᵃˣ

where a = rate of interest/100

x = time

Therefore, we can say that

Ampunt after 10 years = 220 X e¹⁰ ˣ ⁰°⁰¹⁷

= $260.76707

rounding off to the nearest 10 dollars we get

= $260

Hence David's deposit of $220 compounded continuously at a rate of 1.7% will become $260 after 10 years.

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The difference between the polynomials 9x² + 8x and 2x² + 3x is 7x² + 5x.

A polynomial is formed mathematically by combining one or more algebraic terms (monomials). The roots of "polynomial" are poly- (many) and -nomial (terms). A polynomial can have exponents (powers), constants (numbers), and variables (letters), but not negative exponents or variable division.

Polynomials are typically expressed in standard form. Furthermore, because there are no similar terms among the exponents, they are arranged in descending order (largest to smallest).

To find the difference between

(9x² + 8x) - (2x² + 3x)

We do calculation

= 9x² + 8x - 2x² - 3x

= 9x² - 2x² + 8x - 3x

= 7x² + 5x

Thus, the difference between the polynomials is 7x² + 5x.

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Therefore, the multiple of the given equations is 10x^2yz + 6xyz^2, 5x^3y^2 + 14x^3y^3, and -10x^5 + 15x^4 - 30x^3 + 40x^2.

The process of multiplying two or more polynomials together is known as polynomial multiplication. Applying the distributive property to the multiplication of polynomials allows us to conduct polynomial multiplication.

Multiplication in mathematics is essentially just adding a number repeatedly in relation to another number. For instance, if we multiply 2 by 3, then implies that 3 gets multiplied by itself twice, meaning 3 + 3 = 6. Children can multiply numbers easily with the help of this method.

1. (5xy+3yz) * 2xz = 10x^2yz + 6xyz^2

2. (6/7x2y) * (35/6xy + 14/3 xy2) = 5x^3y^2 + 14x^3y^3

3. (2x3-3x2+6x-8) * (-5x2) = -10x^5 + 15x^4 - 30x^3 + 40x^2

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The height of an acute triangle can be calculated using the Pythagorean Theorem. The formula for this theorem is a2 + b2 = c2, where a and b are the two legs of a right triangle and c is the hypotenuse.

To find the height of an acute triangle, we must first identify the two legs of the triangle. The length of the two legs can be determined by measuring the lengths of the three sides of the triangle. Once we have the lengths of the two legs, we can apply the Pythagorean Theorem to calculate the length of the hypotenuse.

Once the length of the hypotenuse is determined, we can use the formula h2 = c2 - a2 - b2 to calculate the height of the triangle. This formula states that the square of the height of the triangle is equal to the square of the hypotenuse minus the square of the two legs. So, to find the height of an acute triangle, we first measure the lengths of the three sides, then use the Pythagorean Theorem to calculate the hypotenuse, and finally use the formula h2 = c2 - a2 - b2 to calculate the height of the triangle.The height of an acute triangle can be calculated using the Pythagorean Theorem. The formula for this theorem is a2 + b2 = c2, where a and b are the two legs of a right triangle and c is the hypotenuse. To find the height of an acute triangle, we must first identify the two legs of the triangle. The length of the two legs can be determined by measuring the lengths of the three sides of the triangle. Once we have the lengths of the two legs, we can apply the Pythagorean Theorem to calculate the length of the hypotenuse.

Once the length of the hypotenuse is determined, we can use the formula h2 = c2 - a2 - b2 to calculate the height of the triangle. This formula states that the square of the height of the triangle is equal to the square of the hypotenuse minus the square of the two legs. So, to find the height of an acute triangle, we first measure the lengths of the three sides, then use the Pythagorean Theorem to calculate the hypotenuse, and finally use the formula h2 = c2 - a2 - b2 to calculate the height of the triangle.

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Answer:

the one that is not equivalent to the others would be 116-34

Step-by-step explanation:

because its 116 and the others dont have anything to do with 116 :)

Answer: The statement that is not equivalent should be D. |16-34

The number of full test tubes Russel fill with water must be discrete number and not rational. 36 test tubes is the answer.

Volume of a cylinder can be known either by the use of formula (V = πr²h) or by measuring with the volume of a known liquid.

Given that Russel is doing an experiment for a science fair project. He fills a beaker with water three times. He puts 16.5 milliliters of water in each test tube. How many full test tubes can Russel fill with water?

1. Describe one way to estimate the solution to this problem.

By dividing the amount of milliliters of water in the baker by 16.5 ml

2. Russel calculates correctly and says the answer to the problem is 36.36 test tubes. Is Russel's answer reasonable?

No! The reason is based on the fact that the answer must be a whole number and not a decimal number since the question centered on number of full test tubes filled with water.

Therefore, the correct answer should be 36 test tubes.

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Oscar Wilde creates a hilarious tone by using puns, situational irony, dramatic irony, satire, and epigrams.

Oscar Fingal O'Flahertie Wills, an Irish poet, and playwright, lived from 16 October 1854 to 30 November 1900.

He wrote in a variety of genres throughout the 1880s before becoming one of London's most well-known playwrights in the early 1890s.

The Picture of Dorian Gray, his plays and epigrams, as well as the circumstances surrounding his meningitis-related early death at age 46 and criminal conviction for gross indecency for consensual homosexual activities in "one of the earliest celebrity trials," is what people will remember him for most.

Using puns, situational irony, dramatic irony, satire, and epigrams, Oscar Wilde establishes a humorous tone.

Therefore, Oscar Wilde creates a hilarious tone by using puns, situational irony, dramatic irony, satire, and epigrams.

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A grocer makes a display of cans in which there are 10 rows.

The top row has one can and each lower row has two more cans than the row above it.

We can write it as sequence:

1, 3, 5, 7, . . .

We can observe that it's an arithmetic sequence: 1, 3, 5, 7, . . . , 2n-1

We know that the formula for the sum of n-terms of arithmetic sequence:

S(n) = (n/2)(1 + 2n-1)

Here, Sn = 100

So, (n/2)[1 + 2n-1] = 100

n(1 + 2n + 1)

n(2n) = 200

2n^2 = 200

n^2 = 100

n = 10

Hence, the number of rows = 10

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Avogadro's number is a unitless object. It represents a number.

What is unit?

The mass of a particular piece of metal or the length of a particular rod are examples of quantities (specific properties) of objects that have historically been used to define measurement units. An object that has no units is known as unitless.

The proportionality factor that connects the number of constituent particles (often molecules, atoms, or ions) in a sample with the amount of material in that sample is called the Avogadro constant, also known as NA or L. It has a precise value of  6.02214076 × 10²³ reciprocal moles and serves as a SI defining constant.

The value of Avogadro's number is 6.02214076 × 10²³.

One mole of the chemical compound consists of 6.02214076 × 10²³ atoms.

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Consequently, it would take polynomials roughly 690,64 years for the disintegrations to disappear.

A polynomial is a mathematical statement made up of coefficients and uncertainty that exclusively uses additions, subtractions, multiplications, and positive integer powers of variables. A single indeterminate x polynomial is represented by the expression x2 4x + 7. A polynomial is an expression in mathematics that consists of variables (sometimes referred to as indeterminates) and coefficients that may be added, subtracted, multiplied, and raised to negative integer powers of non-variables. A polynomial is an algebraic expression made up of coefficients and variables. The only operations that an expression can contain are addition, subtraction, multiplication, and non-negative integer exponents. These expressions are known as polynomials.

The formula for this is Activity = decay constant x atoms.

1238 = 2.64 x 1010 times the decay constant

decay constant is equal to 4.76 x 10-8 min-1.

Currently, the disintegrations are being slowed to 809 disintegrations per minute.

Ln(809) = Ln(2.64 x 1010) - Ln(4.76 x 10-8 x t)

t=3.63x108min = 690.64y

Consequently, it would take roughly 690,64 years for the disintegrations to disappear.

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The increase was 250% over the original price.

percentage, a relative value indicating hundredth parts of any quantity.

let 'x' represent the original price, then 'x+0.25x' represents the first increase of 25%

six times greater, converted to a percentage:

(x+0.25x)×2×100 a second increase of six-fold

1.25x×2×100

250

The increase was 250% over the original price.

Hence, the increase was 250% over the original price.

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The equation that is modeled to the situation is x - 2y + 2 = 0 and the slope of the line is 1/2.

A line's steepness and direction are measured by the line's slope. Without actually using a compass, determining the slope of lines in a coordinate plane can assist in determining if the lines are parallel, perpendicular, or none at all.      

The slope intercept form of a line is y = mx + c              

Where, c - y-intercept and m - the slope of the line

Given that

The cost of using the internet at an internet cafe is equal to a fee plus a certain rate per minute.

4 minutes of internet costs $ 3

8 minutes of internet costs $ 4  

[ We can represent the given situation as given below ]      

And the coordinates formed are (4, 3) and (8, 4)

The slope of the line formed = (4 - 3)/(8 - 4) = 1/4

Let y = mx + c be the equation to the model situation

=> y = (1/2)x + c

=> 2y = x + c ---- (1)

As we know (1) will pass through (4, 3)

=> 2(3) = 4 + c

=> 6 = 4 + c

=> c = 6 - 4

=> c = 2

=> 2y = x + 2

=> x - 2y + 2 = 0

Therefore,

The equation that is modeled to the situation is x - 2y + 2 = 0 and the slope of the line is 1/2.

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Nate's car was filled with 16 gallons of gas.

The concept is used in this is equations. An equation is a mathematical statement with an 'equal to' symbol between two expressions that have equal values. An equation is the set of variables, constants, and mathematical operations like addition, subtraction, multiplication, or division balanced by an equal sign. For example, 3x + 5 = 15. There are different types of equations like linear, quadratic, cubic, etc.

As gas is $2. 09 per gallon at the gas station.

$2. 09 → per gallon

and Nate filled up his car and spent $33. 44.

$33. 44 for fully filled → how many gallons ?

So,

33.44 / 2.09 = 16 gallons.

Therefore, Nate filled 16 gallons gas in his car.

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Answer:

Step-by-step explanation:

55 times 25, making Angel attempt 1375 free throws

Answer:

Step-by-step explanation:

The data of the rainfall amount in inches to the locations are used to plot the attached line plot created with MS Excel.

A line plot is a graph is the graphical display of data as check marks or points above a number line.

A line plot shows the frequency of the values in the data on the number line, using dots or check marks. The frequency of the data is shown above the specified locations on the graph of a line plot.

The line plot is therefore, indicative of how frequent particular values in the dataset are.

The possible data in the question are presented as follows;

Rainfall Amount; 1/8,[tex]{}[/tex] 3/8,  3/4,  3/4,     1/4,    1 1/4, 1/8,     1/4,    1,     1/8

Location;     [tex]{}[/tex]          1,    2,     3,      4,        5,       6,     7,       8,      9,    10

The frequency of the data in a tabular form are;

Rainfall        [tex]{}[/tex]  Frequency

1/8        [tex]{}[/tex]          3 locations

3/8        [tex]{}[/tex]         1 location

3/4        [tex]{}[/tex]         2 locations

1/4       [tex]{}[/tex]           2 locations

1 1/4       [tex]{}[/tex]         1 location

1        [tex]{}[/tex]              1 location

The above frequency values can be used to create the attached line plot created with MS Excel

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B = (2a + 11)/5 is an expression for b when expressed in terms of a.

What exactly are slopes and lines?

We are aware that lines might have several kinds of equations; the typical form is

The equation of a line in slope-intercept form is Ax + By + c = 0, and

y = mx + b.

m is the slope, and b is the y-intercept.

The y-intercept, or (0,b), is the point where the line crosses the y-axis when x = 0. The slope represents the rate of change of the y-axis relative to the x-axis.

Given:

Coordinates of point Q = (2,3)

Coordinates of point R = (a,b)

The equation of the line is 5x + 2y = 9 ⇒ 2y = - 5x + 9

Therefore, y = (-5/2)x + 9/2.

We know slope m = (y2 - y1) / (x2 - x1)

∴ m = (b - 3)/(a - 2).

∴ (2/5) = (b - 3).(a - 2).

2(a - 2) = 5(b - 3).

2a - 4 = 5b - 15.

2a + 11 = 5b.

b = (2a + 11)/5.

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Answer:

1/ 0.8 kilograms

2/ 800 grams

Step-by-step explanation:

We know

1/

Baseball bat = 28 ounces

1 kilogram = 35 ounces

To find weight in kg we take

28 / 35 = 0.8 kg

2/

We know

1 kg = 1000 g

0.8 kg = 800 grams

So, the baseball bat weights 0.8kg or 800 grams