Trigonometric area formula: Area = 1/2 ab sin(c)

What is the area of triangle DEF? Round to the nearest tenth of a square unit.

10.3 square units
18.0 square units
20.0 square units
20.6 square units

The slope of a line perpendicular to the given line is -5/2.

The equation of the line given is 5y = 10 + 2x. To find the slope of this line, we need to rearrange the equation into the slope-intercept form, y = mx + b, where m represents the slope. Dividing both sides of the equation by 5, we get y = 2/5x + 2. Comparing this equation to the slope-intercept form, we can see that the slope is 2/5.

To find the slope of a line perpendicular to this line, we use the fact that the product of the slopes of two perpendicular lines is -1. Therefore, the slope of the perpendicular line is the negative reciprocal of the slope of the given line. The negative reciprocal of 2/5 is -5/2.

Therefore, the slope of a line that is perpendicular to the line with the equation 5y = 10 + 2x is -5/2.

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The angular speed of Earth is [tex]\fbox{\begin\\\ \bf 7.268\times 10^{-5}\ \text{radian per second}\\\end{minispace}}[/tex].

Further explanation:

It is given that the radius of Earth is [tex]6.37\times10^{6}\ \text{meters}[/tex] and it rotates once in every  [tex]24[/tex] hours.

The angular speed of earth is nearly constant and Earth takes one day to complete one rotation that means the time interval for earth to complete one revolution is [tex]1\ \text{day}[/tex].

The total angular displacement covered by the earth to complete one revolution is [tex]2\pi\text{radian}[/tex].

One day is equivalent to [tex]24\ \text{hours}[/tex], [tex]1\ \text{hour}[/tex] is equivalent to [tex]60\ \text{minutes}[/tex] and [tex]1\ \text{minute}[/tex] is equivalent to [tex]60\ \text{seconds}[/tex].

The total number of seconds in [tex]1\ \text{day}[/tex] is calculated as follows:

[tex]\fbox{\begin\\\ \begin{aligned}1\ \text{day}&=24\ \text{hours}\\&=24\times60\ \text{minutes}\\&=1440\ \text{minutes}\\&=1440\times60\ \text{seconds}\\&=86400\ \text{seconds}\end{aligned}\\\end{minispace}}[/tex]    

Therefore the time taken to complete one revolution is [tex]86400\ \text{seconds}[/tex].

The angular velocity is defined as the rate of change of angular displacement. So the formula for angular velocity is, as follows:

[tex]\fbox{\begin\\\ \omega=\dfrac{\theta}{t}\\\end{minispace}}[/tex]

In the above equation [tex]\omega[/tex] represents angular velocity, [tex]\theta[/tex] represents angular displacement and [tex]t[/tex] represents time.

To obtain the value of angular velocity of Earth substitute [tex]2\pi[/tex] for [tex]\theta[/tex] and [tex]86400[/tex]  for [tex]t[/tex] in the above equation.

[tex]\begin{aligned}\omega&=\dfrac{2\pi}{86400}\\&=\dfrac{2\times3.14}{86400}\\&=\dfrac{6.28}{86400}\\&=7.268\times10^{-5}\end{aligned}[/tex]

Therefore, the value of [tex]\omega[/tex] is [tex]7.268\times10^{-5}[/tex].

Thus, the angular speed of the Earth is [tex]\fbox{\begin\\\ \bf 7.268\times 10^{-5}\ \text{radian per second}\\\end{minispace}}[/tex].

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Answer details:

Grade: High school

Subject: Mathematics

Chapter: Angular motion

Keyword: Angular velocity, angular displacement, time, one revolution, radian/second, Earth, 86400 seconds, 2pi, speed, minutes, omega, theta.

The two numbers are 8 and 12.

What is Algebraic expression ?

Algebraic expressions are numbers expressed using letters or alphabets without specifying their values. Algebra taught us how to express unknown values using letters such as x, y, and z. There can also be a single value that is placed before and multiplied by a variable in an algebraic expression in addition to these letters, which are called variables.

Let us assume first number be = x

and, let the second number be = y

the,

One number is 4 less than a second number means :

x = y - 4........(1)

Twice the second number is 16 less than 5 times the first means :

2y = 5x - 16.......(2)

Substitute x equals from the first equation in the second equation.

2y = 5(y - 4) - 16

2y = 5y - 20 - 16

-3y = -36

y = 12

Now use the first equation for find x.

x = y - 4

x = 12 - 4

x = 8

Thus, The numbers are 8 and 12.

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Answer:

Rational.

Step-by-step explanation:

We are asked to determine the type of number 0.5.

We can see that our given number is a decimal.

We can also see that our given number is a terminating decimal, so we can represent it as a fraction.

[tex]0.5\times \frac{10}{10}=\frac{5}{10}=\frac{1}{2}[/tex]

Therefor, our given number is a rational number as we can represent it as a fraction.

To sketch a triangle with an acute angle θ and cot(θ) = 2, we can choose an angle in the first quadrant, such as 30 degrees, and draw a right triangle with sides 2 and 1. The hypotenuse can be found using the Pythagorean theorem.

To sketch a triangle with an acute angle θ and cot(θ) = 2, we can use the fact that cotangent is the reciprocal of the tangent function, so tan(θ) = 1/2. Since tangents are positive in the first and third quadrants, we can consider an angle in the first quadrant. Let's choose the angle as 30 degrees.

Now, we can draw a right triangle with one of the acute angles as 30 degrees. Since tan(θ) = opposite/adjacent, we can choose the side opposite the angle as 1 and the side adjacent as 2. Therefore, the triangle has sides 2, 1, and the hypotenuse can be found using the Pythagorean theorem.

The triangle has an acute angle θ and cot(θ) = 2. It has sides of length 2, and 1, and the hypotenuse can be found using the Pythagorean theorem.

Final answer:

To find the value of k so that the line containing the points (-5,k) and (2,10) is parallel to the line containing the points (5,5) and (1,-4), we need to find the slope of the second line and use it to find the value of k.

Explanation:

To find the value of k so that the line containing the points (-5, k) and (2, 10) is parallel to the line containing the points (5, 5) and (1, -4), we need to find the slope of the second line and use it to find the value of k. Firstly, let's find the slope of the second line:

The formula we apply is the slope

= (y2 - y1) / (x2 - x1).

slope = -9 / -4 = 9/4;

slope = (-4 - 5) / (1 - 5)

The lines now have the same slope since they are parallel.

So, the slope of the line passing through (-5, k) and (2, 10) should also be 9/4. We can find the slope using the formula:

slope = (y2 - y1) / (x2 - x1)

Substituting the coordinates (-5, k) and (2, 10):

9/4 = (10 - k) / (2 - (-5))

By simplifying the equation, we get:

9 = 7(10 - k)

Now, let's solve for k:

90 - 9k = 63

-9k = 63 - 90

-9k = -27

k = 3

The expected recorded score when the total number of points earned on the two parts is recorded is 20.4.

The expected recorded score when the maximum of the two scores is recorded is 25.7.

We have,

(a)

If the score recorded in the grade book is the total number of points earned on the two parts (x + y):

We will calculate the expected value (E) using the joint probability mass function (pmf) values given in the table.

E(x + y) = Σ[(x + y) * p(x, y)]

Using the table values:

E(x + y) = (0 * 0.02) + (5 * 0.06) + (10 * 0.02) + (15 * 0.10) + (20 * 0.04) + (25 * 0.13) + (30 * 0.20) + (35 * 0.10) + (40 * 0.01) + (45 * 0.15) + (50 * 0.16) + (55 * 0.01)

E(x + y) = 20.4

(b)

If the maximum of the two scores is recorded:

We need to find the maximum value between x and y for each combination of scores in the table and calculate the expected value using the joint pmf.

E(max(x, y)) = Σ[max(x, y) * p(x, y)]

Using the table values:

E(max(x, y)) = (0 * 0.02) + (5 * 0.06) + (10 * 0.15) + (15 * 0.20) + (20 * 0.20) + (25 * 0.15) + (30 * 0.15) + (35 * 0.15) + (40 * 0.16) + (45 * 0.16) + (50 * 0.16) + (55 * 0.01)

E(max(x, y)) = 25.7

Therefore,

The expected recorded score when the total number of points earned on the two parts is recorded is 20.4.

The expected recorded score when the maximum of the two scores is recorded is 25.7.

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The expected recorded score in the grade book (E(X+Y)) is calculated by multiplying each possible total score by its probability and summing these products. The expected maximum score is calculated similarly, but considers the probabilities of each possible maximum score. Application of these principles requires access to full probability mass function data.

The student's question relates to the calculation of expected values in the context of a two-part quiz, with scores registered as the total (x+y) or the maximum of the two scores. Given a joint probability mass function (pmf) for x and y, we can calculate expected scores in both cases.

Generally, the expected value E(X) of a discrete random variable X is calculated by multiplying each possible value of X by its probability, and then summing these products. This is represented by the formula E(X) = Σ xP(x). Assuming the same logic for the expected total score on the quiz, E(X+Y) would be calculated by summing the products of each possible total score and its probability.

To find the expected maximum score, we'd perform a similar calculation, but consider the probabilities of each possible maximum score instead.

Without explicit probabilities in the question, no exact solutions can be provided. However, the processes outlined here would allow the student to solve such problems with access to the full pmf data.

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62 +62 = 124

360 - 124 = 176

176/2 = 88

BED = 88 degrees