Why are 30 60 90 and 45 45 90 triangles called special right triangles?.

Longitude and latitude are geographic coordinates that indicate a location on the Earth's surface. They are used to locate a specific point on a map or globe.

Longitude is the measure of a location's east-west position on the Earth's surface. It is measured in degrees east or west of the Prime Meridian, which is an imaginary line that runs from the North Pole to the South Pole through the Royal Observatory in Greenwich, England. Longitude values range from -180 degrees (West) to 180 degrees (East).

Latitude, on the other hand, is the measure of a location's north-south position on the Earth's surface. It is measured in degrees north or south of the Equator, which is an imaginary line that runs around the Earth's middle, dividing it into the Northern and Southern Hemispheres. Latitude values range from -90 degrees (South) to 90 degrees (North).

The diagram below illustrates how longitude and latitude work together to pinpoint a location on the Earth's surface. The intersection of the red line (longitude) and the blue line (latitude) represents the specific location.

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Challenge To travel on vacation, a family wants to rent a car. Company A bills $70.50 plus 17 cents each mile. Charges from Company B are $30.50 plus 8 per mile. For every x miles that Company A charges, Company B is (39.24x)

Before figuring out their difference for x miles, we must first determine their difference for one mile.

Difference per mile = ($70.50 + $0.14) - ($30.50 + $0.90)

= $70.64 - $31.40

= $39.24

Difference for x miles = $39.24 × x miles

= $(39.24x)

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The derivative of log 2x is 1/x. This is obtained by applying the formula of derivative of log x.

Let y is the given function = log 2x.

Differentiating with respect to x on both sides, we have,

d/dx(y) = d/dx(log 2x)

dy/dx = 1/2x × d/dx(2x)

dy/dx = 1/2x × 2

dy/dx = 1/x

Chain rule is applied in the above solution.

Its formula is given as, d/dx [f(g(x)] = d/dx [f(g(x)] × d/dx[g(x)]

where, f(x) and g(x) are a function of function.

In the above solution, f(x) is log x and g(x) is 2x.

Thus, the derivative of log 2x is 1/x.

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Limits are the rate of change of a function and also used in many other concepts such as continuity, differentiability and many other concepts in calculus.

In calculus, limits are used to study the behavior of a function as the input, or independent variable, approaches a specific value. The limit of a function at a particular point, say x = a, is defined as the value that the function approaches as the independent variable x gets arbitrarily close to a, but not necessarily equal to it.

The notation for a limit is usually represented as:

lim x->a f(x) = L

Which means as x approaches a, f(x) approaches L.

Properties of limits include:

1) The limit of a constant is equal to the constant.

2) The limit of a sum, difference, or product of functions is equal to the sum, difference, or product of their limits, respectively.

3) The limit of a function at a point is unique, if it exists.

Limits are an essential concept in calculus as they are used to define the derivative, which is the rate of change of a function and also used in many other concepts such as continuity, differentiability and many other concepts in calculus.

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The range of

The logarithmic function f(x) = log_b x is a decreasing function if and only if b > 1.

To see why, consider that for any positive real number x, log_b x increases as x increases. This can be seen by using the definition of logarithms: log_b x is the power to which b must be raised to produce x. As x increases, the power to which b must be raised to produce x must also increase, so log_b x must increase as well.

Therefore, if b > 1, then log_b x is a strictly decreasing function, because as x increases, the power to which b must be raised to produce x increases at an increasing rate, so log_b x decreases at an increasing rate. On the other hand, if 0 < b < 1, then log_b x is a strictly increasing function, because as x increases, the power to which b must be raised to produce x increases at a decreasing rate, so log_b x increases at a decreasing rate.

It is important to note that for b = 1, the logarithmic function is undefined, as log_1 x is not defined for any positive real number x.

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The percent error is a measure of how accurate a measurement is. It is calculated by taking the difference between the measured value and the true value, dividing that difference by the true value, and then multiplying by 100 to get a percentage.

In this case, the true value of the baseball's speed is 4.2 miles per hour and the measured value is 5 miles per hour. To find the percent error:

(5 - 4.2) / 4.2 * 100 = 0.190476 * 100 = 19.0476%

So the percent error in this measurement is 19.0476%. This means that the measurement was off by 19.0476% from the true value.

Alternatively, we can round it to 19% for ease of interpretation

Therefore, the percent error in this measurement is 19%.

The value of S, i.e., prependicular length in right angled triangle is twelve units.

We have, a right angled triangle ABC as see in above figure. From the above figure, we get the following information,

The base length of right angled ∆ ABC, BC = 9 units

The perpendicular length/height of

∆ ABC , AC = S units

The hypotenuse length of ∆ ABC, AB

= 25 units

We have to calculate the value of S i.e., the prependicular length. Using the Pythagoras threoem to solve the value of S,

(hypotenuse)²= (base)² + (prependicular)²

=> AB² = BC² + AC²

=> 15² = 9² + S²

=> 225 = 81 + S²

=> 225 - 81 = S²

=> S² = 144

=> S = ±√144

=> S = ± 12

but S, is represent the length and it's value is always positive. Therefore, the required value of S is 12 units.

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Complete question:

For the above figure, What is the value of S units?.

Using properties of Similar triangles, The measure of QP is 14.49.

Triangles with the same shape but different sizes are said to be similar triangles. Squares with any side length and all equilateral triangles are examples of related objects. In other words, if two triangles are similar, their corresponding sides are proportionately equal and their corresponding angles are congruent.

Similar triangles differ in size but have the same shape. Corresponding angles are identical in similar triangles. Similar triangles have corresponding sides that have the same ratio. The ratio of the square of any two of their corresponding sides to any identical triangle's area is the same.

Given two triangles,

ΔPQR and ΔPNO are Similar,

So, their sides are present in the ratio -

NO / QR = NP / PQ

We are given, QR = 10

NP = 21

On putting the values in the NO / QR = NP / PQ equation,

⇒ NO / 10 = 21 / QP

As NO = QP (given)

We can write as -

QP / 10 = 21 / QP

⇒ QP²  =  10 × 21

⇒ QP²  = 210

⇒ QP = 14.49

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A. The answer for part A is $460.

B. The answer for part B is 11,166.67 dollars

What are the Profit and Loss?

A. To find the commission earned on the first $4,000 in sales, we can multiply $4,000 by 4% which is (4,000 * 0.04) = 160 dollars

To find the commission earned on the remaining amount of sales above $4,000, we can subtract $4,000 from $9,000 to find the amount of sales above $4,000 which is (9,000 - 4,000) = 5,000

then we can multiply $5,000 by 6% which is (5,000 * 0.06) = 300 dollars

So the total commission earned is 160 + 300 = 460 dollars.

So the answer for part A is $460.

B. To find the amount of money in sales, we can use the commission formula: commission = (sales * rate)

The commission rate for sales above $4,000 is 6% and the commission earned is $670.

So, we can set up the equation as : 670 = (sales * 0.06)

We can solve this equation to find the value of sales by dividing both sides of the equation by 0.06

sales = 670 / 0.06

sales = 11,166.67

So the answer for part B is 11,166.67 dollars.

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If one is the sole factor that connects a fraction's numerator and denominator, then that fraction is said to be in its simplest form.The mixed fraction is 33*3/4

As an illustration, the fraction 89 has only one common component between the digits 8 and 9, which is 1.Since fractions always work or can be calculated when they are in their simplest form, we simplify them.When two numbers are represented together, they are called mixed numbers.

Based on the given conditions, formulate:45 / 1/1/3

Convert the expression into a fraction: 45/1/1+1/1

Expand the fraction to get the least common demominator: 45/ 3/1*3+1/3

Calculate the product or quotient:45/3/3 +1/3

Write the numerators over common denominator:45/3+1/3

Calculate the sum or difference:45/4/3

Divide a fraction by multiplying its reciprocal:45* 3/4

Write as a single fraction:45* 3/4

Calculate the product or quotient:135/4

Convert to mixed fraction: 33*3/4

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The  vertices of the rotation of 180⁰  are P'(2, 2), Q'(-1, 2), R'(-2, 4) and S'(3, 4)

The angle of rotation is a measurement of the amount, of namely angle, that a figure is rotated about a fixed point, often the center of a circle

When we rotate a figure of 180 degrees about the origin either in the clockwise or counterclockwise direction, each point of the given figure has to be changed from (x, y) to (-x, -y) and graph the rotated figure.

Before Rotation                                 After Rotation

(x, y)                                                          (-x, -y)                            

Let P(-2, -2), Q(1, -2), R(2, -4) and S(-3, -4) be the vertices of a four sided closed figure. If this figure is rotated 180° about the origin, find the vertices of the rotated figure and graph.

Here, the given is rotated 180° about the origin. So, the rule that we have to apply here is

(x, y) ----> (-x, -y)

Based on the rule, we have to find the vertices of the rotated figure.

Also, (x, y) ----> (-x, -y)

P(-2, -2) ----> P'(2, 2)

Q(1, -2) ----> Q'(-1, 2)

R(2, -4) ----> R'(-2, 4)

S(-3, -4) ----> S'(3, 4)

There, the vertices of the rotated figure are

P'(2, 2), Q'(-1, 2), R'(-2, 4) and S'(3, 4)

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Complete question:

Let P(-2, -2), Q(1, -2), R(2, -4) and S(-3, -4) be the vertices of a four sided closed figure. If this figure is rotated 180° about the origin, find the vertices of the rotated figure and graph.

Answer:

3 slices

Step-by-step explanation:

37.5% = 0.375

We take

8 times 0.375 = 3 slices

So, 3 slices of pie remain after dinner.

The expression would be "4 + (7 - m)" .

This expression takes the value of m, subtracts it from 7, and then adds the result to 4.

Answer:

y = -8[tex]x^{2}[/tex] + 32x + 40

Step-by-step explanation:

y= -8(x-5)(x+1)

y = (-8x + 40) (x + 1)

y = -8[tex]x^{2}[/tex] -8x + 40x + 40

y = -8[tex]x^{2}[/tex] + 32x + 40

A polynomial of degree 2 is called a quadratic equation. It is a type of equation that can be solved by using the quadratic formula, which looks like this:

where x is the unknown variable and y is the solution to the equation. The quadratic equation can be written in several different ways, but all of them involve solving for y using equations similar to the one above.

Polynomials of degree 2 are common in everyday life, and are used to solve problems such as solving for the slope or height of a line, determining the points at which two lines intersect, or finding the distance between two points.

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Answer:

$182.90

Step-by-step explanation:

Find the cost per kg. To do so, divide the total cost, with the amount of kg that is bought:

(Total Cost/Total weight) = (cost/unit):

[tex]\frac{106.20}{18} = 5.9\\\\[/tex]

$5.9 per kg of apple.

~

Next, solve for the amount that 31 kg of apples would cost. Multiply the unit cost with the amount of kg of apples purchased:

(Total weight of apples x Cost per kg for apples) = Total cost:

[tex]31 * $5.9 = Total Cost\\Total Cost = 182.9[/tex]

$182.90 is your answer.

~

Answer:

  2x -1

Step-by-step explanation:

You want to show that (3x+4) is a factor of f(x) = 6x^2+5x-4, and you want the other factor.

When the product, f(x), is divided by one of its factors, the remainder will be zero and the quotient will be the other factor.

The attached long division shows the remainder (bottom line) is zero, and the quotient (top line) is (2x -1), the other factor.

The other factor is (2x -1).

Answer:

[tex] \sf \: 34 \: isosceles \: triangle \: - - - - - - - - - - - - - - - - - - - - - - - - [/tex]

Step-by-step explanation:

[tex]b = 144 - 2a < 2a [/tex]

So, that

[tex]a > 36[/tex]

and 2a < 144 so that

[tex]a < 72[/tex]

since a = b implies a = 48

{37,38,39.......71}/{48}

There are 71-37 = 34 isosceles triangle

The equation for the altitude AR of triangle ABC will be 4y-3x=0 i.e. B.

What is a triangle?

Triangles are polygons in geometry that have three sides and three vertices. This figure is two-dimensional and has three straight sides. A triangle is a three-sided polygon. A triangle's three angles added together equals 180°. A single plane contains the triangle. The triangle is classified into six forms based on its sides and angles.

A triangle is categorized into three categories based on its sides, namely:

Scalene Triangle - Each side has a distinct length.

Isosceles Triangle - A triangle with two sides of equal length and one side of a different length.

Equilateral Triangle - A triangle has three sides that are of the same length.

A triangle is categorized into three categories based on its angles, namely:

Acute Angle Triangle - A triangle with all of its angles smaller than 90°.

Obtuse Angle Triangle - A triangle with one of its angles larger than 90°.

Triangle with a Right Angle - A triangle with one of its angles equal to 90°.

Now,

The general form of the linear line is:

y = mx + c where m is the slope and c is the y-intercept

Step 1: getting the slope:

Since AR is the altitude of ABC, therefore, AR is perpendicular on BC which means that slope of AR = -1/slope of BC

Therefore, we will need to get slope of BC:

We have point B = (0,4) and point C = (3,0)

slope of BC = (y2-y1) / (x2-x1) = (0-4) / (3-0) = -4/3

This means that:

slope of AR = m = 3/4

Step 2: getting the y-intercept:

Since point A belongs to the altitude, therefore, point A satisfies the equation of the line. So, to get the y-intercept, we will use point A to substitute in the equation of the line and solve for c as follows:

y = mx + c

y = (3/4) x + c

0 = (3/4)(0) + c

c = 0

Hence,

           y = (3/4) x

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The gradient is −3 and the y-intercept is 6.

The gradient of a function is defined to be a vector field. Generally, the gradient of a function can be found by applying the vector operator to the scalar function. (∇f (x, y)).

This kind of vector field is known as the gradient vector field. Now, let us learn the gradient of a function in the two dimensions and three dimensions.

The gradient should take a scalar function (i.e., f(x, y) and produces the vector function (∇ f).

The vector ∇f(x, y) should lie in the plane.

The gradient of a function can be found by applying the vector operator to the scalar function. I.e., ∇f (x, y).

y+3x=6

Rearrange the equation to the slope-intercept form,

y=mx +b,

where ,

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

Subtract 3x from both sides.

y=−3x+6

The gradient is −3 and the y-intercept is 6.

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The total entry code can create by the dataset given is 17576000.

Assume the entry character that is n. As per requirement given are the first three entry are 3 letter then then n1 = 26 , n2 = 26, n3 = 26. Then, the three after that is a single digit number, so n4 = 10, n5 = 10, n6 = 10. Since we have find the value of each entry character then we can determine total entry code as follow:

Total entry code = n1 x n2 x n3 x n4 x n5 x n6

Total entry code = 26 x 26 x 26 x 10 x 10 x 10 = 26³ x 10³

Total entry code = 17576 x 1000

Total entry code = 17576000

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The algebraic expression for the a number that has been multiplied by three is: x + 3*x. In this statement, x stands for the number, while 3*x, or 3x, stands for the number times three.

An algebraic expression is a combination of numbers, variables, and mathematical operations, such as addition, subtraction, multiplication, and division. Algebraic expressions can be evaluated for specific values of the variables in order to find a numerical result.

A mathematical statement with variables, constants, coefficients, or algebraic operations is known as an algebraic expression. A good example of an algebraic expression is 5x2+6xyc. Algebraic expression do not utilize equal signs, in contrast to algebraic equations.

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Right angled triangle PQR, whose vertices are located at P(-6, -2) and Q(6,6). It is further given that PQR = 90° . y = 0.666666666667 x + 2 is the equation of the straight line .

y = 0.6666666667x + 2 Points ( -6 , -2 ) , ( 6 , 6 ) ( 6 , 6 ) Formula               (y = mx + b) y = 0.666666666667 x + 2 Slope in metres : 0.66666666667 , 2. Y-intercept 0. 6666666666 x Plus any number of parallel lines 1.5 x + any number for parallel lines.

Any line without curves is said to be straight. Consequently, a straight line is one that is uncurved and stretches beyond infinity on both sides. a diagram of a graph in which each edge is a section of a straight line. Additional terms to consider are grid drawing, orthogonal drawing, planar straight-line graph, and Bresenham's algorithm. A straight line has the equation y=mx+c, where c is the height at which the line crosses the y-axis, also known as the y-intercept, and m is the gradient.

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Total thirty-nine months will be take by Johnson family to pay off the balance, $10,000.00.

The monthly payment is the amount paid per month to pay off the principle in the time period of the loan. If your rate is r%, divide r/100 by 12 to calculate your monthly interest rate. Formula for monthly payment is, M = Pr(1+r)ⁿ/{(1+r)ⁿ - 1 }

where, P --> principle

r--> rate of interest in months

n --> number of periods (in months )

In this problem, Johnson family bought the furniture and appliances for a new house that is principle, P = $10,000

Rate of interest, R = 15%

In terms of months, R = 0.15/12

= 0.0125 per month

Monthly payment, M = $325.00

Plugging all known values in above formula we get, $325.00 = {$10,000 × 0.0125(1 + 0.0125)ⁿ}/{(1+ 0.0124)ⁿ - 1 }

=> 325 = 125(1.0125)ⁿ/{(1.0125)ⁿ - 1}

=> 325 {(1.0125)ⁿ - 1} = 125(1.0125)ⁿ

=> 325(1.0125)ⁿ - 325 = 125(1.0125)ⁿ

=> 325(1.0125)ⁿ - 125(1.0125)ⁿ = 325

=> 200 (1.0125)ⁿ = 325

=> (1.0125)ⁿ = 325/200 = 13/8 = 1.6250

Taking natural logarithm both sides

=> ln(1.0125)ⁿ = ln (1.6250)

=> n ln(1.0125) = ln (1.6250) = 0.4855

=> n×0.0124 = 0.4855

=> n = 0.4855/0.0124 = 39.153 ~ 39

Hence, required months are 39.

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Answer:

PART A : points to graph (7,0)(1,-1)(2,-5)

PART B: (x, y) => (-x+3, y-2)

Explanation step by step :

Reflected across y axis means y-coordinate of every point remains same and x-cordinate become opposite

(x, y) become (-x, y)

Here,

(-4, 2) become (4, 2)

(2, 1) become (-2, 1)

(1,-3) become (-1,-3)

Similarly a point P(x, y) translated by vector <3, -2> means the point P become P'(x+3, y-2)

(4, 2) become (7,0)

(-2, 1) become (1, -1)

(-1,-3) become (2,-5)

So the composition of function for the given sequance of transfermation is (x, y) = (-x+3, y-2)

Note : graph not accurate, just for explanation.