Laws of sine and cosine

1. The definition of a function is based on the concept of a mapping from one set to another, where each element in the domain has a unique image in the range. This is a fundamental concept in mathematics and is used to describe relationships that are deterministic in nature.

2. To determine if a set of ordered pairs is a function, one must ensure that no two ordered pairs have the same first component (x-value) and different second components (y-values). This is known as the vertical line test. If a vertical line can be drawn that intersects the graph of the relation at more than one point, then the relation is not a function.

3. The slope-intercept form is a standard form for linear equations that is particularly useful for graphing lines and for understanding the slope and y-intercept of the line. It is derived from the general form of a linear equation, y = mx + b, where m represents the slope and b represents the y-intercept.

4. In the given equation y = -2x + 6, the coefficient of x represents the slope, which is -2, indicating that for every 1 unit increase in x, y decreases by 2 units. The constant term, 6, represents the y-intercept, which is the value of y when x = 0.

5. A horizontal line has a slope of zero because there is no change in the y-value as the x-value changes. Thus, the slope, which is the ratio of the change in y to the change in x, is zero. The equation of a horizontal line is simply y = b, where b is the y-value of every point on the line.

A vertical line has an undefined slope because the change in x is zero (since x does not change), and division by zero is undefined in mathematics. The equation of a vertical line is x = a, where a is the constant x-value for every point on the line.

Answer:

E(-5, - 6) D(5,-6) C(0,0)

Step-by-step explanation:

The strategy here is run then rise/fall

For example: for E, you run back - 5, then fall - 6

Answer:

Basically when you use parantheses it protects that number from becoming negative...

-10^2 is -100 but with paranthesis is 100

Answer:

without the parenthesis it would just be 100 but with it, it would be -100 :)

The problem involves applying the concept of percentage. By using the given data, we find that 70 sixth graders went on the museum trip.

The subject of this problem is percentage and its application in a real-world situation. The problem stated that 70% of the sixth graders chose to go to the museum. As there are 100 sixth graders, to calculate the number of students who went on the museum trip, you would use the formula percentage/100 x total number.

So, 70/100 x 100 = 70 sixth graders. Hence, 70 sixth-graders from Sarah's school went on the field trip to the museum.

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Answer:

The number of students who have a test score in the interval 71-80 is 13.

Step-by-step explanation:

Given:

The frequency chart which show the cumulative numbers of students test scores.

Now, to find the number of students who score in the interval 71-80, we will look in the frequency chart. In the chart score range of 65-80 the cumulative number of students are 13. As the 71 to 80 comes in between the score range 65-80 and there are 13 students who scored in this range.

Therefore, the number of students who have a test score in the interval 71-80 is 13.

Answer:

$32

Step-by-step explanation:

The money she spent was $5 on the kitten and if you add $5 to the remaining $27 that is $32

Current amount that Penny have is $32.

Calculate the answer using a math operator is referred to as a mathematical operation.

Basic mathematical operations are addition, multiplication, subtraction and division.

Given that,

The amount that Penny have after  = $27.

Also, Penny pays $5 for her new kitten.

Since, amount $27 remains after paying $5.

So initial amount should more than $5.

The amount = current amount + amount paid for kitten

                              = 27 + 5

                              = 32

The required amount that Penny have = $32.

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Answer:

cdsnvj;vs

Step-by-step explanation:

Answer:

x=8, y=-7. (8, -7).

Step-by-step explanation:

2x-y=23

x-9=-1

-----------------

x=-1+9=8

x=8

-------------

2(8)-y=23

16-y=23

y=16-23=-7

x=8, y=-7. (8, -7).

Answer:

x = 27

Step-by-step explanation:

The given angles are vertical angles and congruent, thus

5(x - 4) = 4x + 7, that is

5x - 20 = 4x + 7 ( subtract 4x from both sides )

x - 20 = 7 ( add 20 to both sides )

x = 27

The distance from home to second base in a slow pitch softball diamond is approximately 87.68 feet. This calculation is achieved through the use of the Pythagorean theorem, which utilizes the square lengths of the diamond sides to calculate the diagonal path between the bases.

The distance from home to second base in a slow pitch softball diamond can be calculated using the Pythagorean theorem, as the path forms a right triangle. The side lengths of the diamond are 62 ft, so we can use these as the legs of our right triangle. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, it's a^2 + b^2 = c^2, where c is the distance we want to find.

So, the calculation will be √((62ft)^2 + (62ft)^2). Simplifying this gives √(3844ft^2 + 3844ft^2), which is √7688ft^2. Which gives us approximately 87.68 ft. So, the distance from home to second base is approximately 87.68 feet.

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To find the distance from home to second base on a slow pitch softball diamond, use the Pythagorean theorem to calculate the diagonal of the square. With each side being 62 feet, the distance comes out to approximately 87.7 feet.

How to Find the Distance from Home to Second Base

In a slow pitch softball diamond, the bases form a square with each side measuring 62 feet.

We must compute the diagonal for the square in order to get the distance between home plate and second base.

This can be done using the Pythagorean theorem.

Step-by-Step Solution

Label the sides of the square. Each side is 62 feet.

Recall the Pythagorean theorem formula: a² + b² = c², where 'a' and 'b' are the legs of a right triangle, and 'c' is the hypotenuse (which, in this case, is the diagonal).

Here, both 'a' and 'b' are 62 feet because the square’s sides are equal.

→ Substitute the values into the formula: 62² + 62² = c²

→ Calculate the squares: 3844 + 3844 = c²

→ Combine the values: 7688 = c²

→ Take the square root to find 'c': c = √7688

The diagonal (distance from home to second base) is approximately 87.7 feet.

Therefore, the distance from home to second base is about 87.7 feet.

Answer:

Option C. [tex]25\pi\ ft^{2}[/tex]

Step-by-step explanation:

The complete question is

The diameter of Jim’s circular flower bed is 10 feet. What is the area, in square feet, of Jim’s flower bed?

A) 10π

B) 20π

C) 25π

D) 100π

we know that

The area of a circle (Jim's circular flower bed) is equal to

[tex]A=\pi r^{2}[/tex]

where

r is the radius of the circle

we have

[tex]r=10/2=5\ ft[/tex] ----> the radius is half the diameter

substitute

[tex]A=\pi (5)^{2}[/tex]

[tex]A=25\pi\ ft^{2}[/tex]

Answer:

$256

Step-by-step explanation:

Each class got $256. I know this because I did 3,840 divided by 15 which got me the CORRECT answer of $256

Answer:

$256

Step-by-step explanation:

if there was $3840 donated and theres 15 classes that need to get an even amount so you would do 3840÷15 to see how many times 15 can go in to 3840 to find the even amount of money each class should recieve which is $256

Answer:

8

Step-by-step explanation:

The discriminant of ax² + bx + c = 0 is b² − 4ac.

Here, a = -1, b = 4, and c = -2.

b² − 4ac

(4)² − 4(-1)(-2)

16 − 8

8

Answer:

the radio of the Numbers of toucans to the Numbers of parrots at the zoo is 1:4

Step-by-step explanation:

For each toucan there are 4 parrots

15/60 = 1/4

Answer: 1:4

Step-by-step explanation: the answer would be 1:4.

Answer:

C)

Step-by-step explanation:

The third graph represents this system.

Answer:

Graph C

Step-by-step explanation:

Thinking process:

The are two ways to solve the system:

1. the substitution method:

for equation (1) y = 1/2x + 3

when x = 0, y = 3

when y = 0, x = -6

for equation (2) y = 3/2x - 1

when x= 0; y = -1

when y = 0; x = 2/3

this gives line C

2. another way to solve the simultaneous equations:

y = 1/2x + 3  ...1

y = 3/2x - 1    ...2

substuting (1) into  (2)

3/2x - 1 = 1/2x + 3

x = 4

substituting x = 4 into (1):

y = 1/2x + 3

y = 1/2(4)+ 3

  = 5

the point of intersection will be (4 , 5) which is line c

Either method works.

Answer:

Here are 3

Step-by-step explanation:

Are a U.S. citizen.

Meet your state's residency requirements.

Are 18 years old on or before Election Day.

The product of two square roots is equal to the square root of the product, so the product of √y3 and √y3 is y3.

The equation is asking you to multiply the square root of y cubed by itself.

Based on the properties of square roots, if you multiply the square root of a number by itself, you get the original number.

This is because multiplication is the inverse operation for square roots. Meaning, it undoes the square root.

So, in this scenario, you are multiplying [tex]\sqrt{y^{3} }[/tex] * [tex]\sqrt{y^{3} }[/tex]which results in y^3.

Thus, the product of [tex]\sqrt{y^{3} }[/tex] and  [tex]\sqrt{y^{3} }[/tex]is y^3, assuming y≥0.

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Answer:

5,070

Step-by-step explanation:

Answer:

5069.0

Step-by-step explanation:

The tenth value is the value directly after the decimal point. In this case, the number given is a whole number, and so to find the tenth place value, add a decimal point and a 0 to the back.

~

8,653,972 rounded to the ten-thousands is 8,650,000.

Hope this helps!

8,653,972 rounded to the ten thousand is

8,650,000

Answer:

[tex]f(x)=x-5[/tex]

Step-by-step explanation:

step 1

Find the slope m

The formula to calculate the slope between two points is equal to

[tex]m=\frac{y2-y1}{x2-x1}[/tex]

we have the ordered pairs

(10,5) and (2,-3)

substitute the values

[tex]m=\frac{-3-5}{2-10}[/tex]

[tex]m=\frac{-8}{-8}[/tex]

[tex]m=1[/tex]

step 2

Find the equation of the line in slope intercept form

[tex]f(x)=mx+b[/tex]

we have

[tex]m=1[/tex]

[tex]point\ (10,5)[/tex]

substitute and solve for b

[tex]5=(1)(10)+b[/tex]

[tex]5=10+b[/tex]

[tex]b=-5[/tex]

The linear function is equal to

[tex]f(x)=x-5[/tex]

The linear function with the values [tex]f(10)=5[/tex] and [tex]f(2)=-3[/tex] is [tex]f(x)=x-5[/tex].

Step-by-step explanation:

Given information:

The values of a liner function

[tex]f(10)=5[/tex]

and

[tex]f(2)=-3[/tex]

So, we can write the slope as:

[tex]m=\frac{y_2-y_1}{x_2-x_1} \\[/tex]

On, putting the values:

[tex]m=\frac{-3-5}{2-10} \\m=1[/tex]

Point (10,5)

Now ,substitute the values in the linear equation:

[tex]\text {y=mx+b}[/tex]

where, m is the slope.

[tex]5=1 \times 10 + \text b\\\text b=-5[/tex]

Now put the values to get the required linear function.

So, the linear function with the values  [tex]f(10)=5[/tex] and [tex]f(2)=-3[/tex] will be [tex]f(x)=x-5\\[/tex]

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Answer: m=34

Step-by-step explanation:

11+m=-23

add 23 on both sides

m=34

Answer:

[tex]\frac{ d^{2} y}{dx^{2} } = -10[/tex]

Step-by-step explanation:

Concept : We have to differentiate the given equation twice and then put the values of x and y at the given point.

The given point is (2,-5).

Given    xy - y = -5

Differentiating both sides,

[tex] x \times \frac{dy}{dx} + y - \frac{dy}{dx}[/tex] = 0

Substitute (x,y) as (2,-5)

[tex]2 \times \frac{dy}{dx} -5 - \frac{dy}{dx}[/tex] = 0

[tex]\frac{dy}{dx} = 5[/tex]

Differentiating again, we get

[tex]\frac{dy}{dx} + x \times \frac{ d^{2} y}{dx^{2} } + \frac{dy}{dx} - \frac{ d^{2} y}{dx^{2} } = 0[/tex]

Substitute values of x , y and \frac{dy}{dx} ,

[tex]5 + 2 \times \frac{ d^{2} y}{dx^{2} } + 5 - \frac{ d^{2} y}{dx^{2} } = 0[/tex]

[tex]\frac{ d^{2} y}{dx^{2} } = -10[/tex]

Answer:

Part A) The linear expression that represent the total field goal points scored in the first two quarters is (3x-4)

part B) The linear expression that represent the total points scored in the game is [tex]6x-1[/tex]

Step-by-step explanation:

Part A) we know that

To find out the total field goal points scored in the first two quarters, sum the field goal points scored in the first quarter plus the field goal points scored in the second quarter

we have that

1st quarter= 2x-6

2nd quarter = x+2

therefore

[tex](2x-6)+(x+2)[/tex]

Group terms

[tex](2x+x)+(-6+2)[/tex]

Combine like terms

[tex]3x-4[/tex]

therefore

The linear expression that represent the total field goal points scored in the first two quarters is (3x-4)

Part B) we know that

To find out the total points scored in the game, sum the field goal points scored in the first quarter, plus the field goal points scored in the second quarter plus the field goal points scored in the third quarter, plus the field goal points scored in the fourth  quarter, plus the total free throw points.

we have that

1st quarter= 2x-6

2nd quarter = x+2

3rd quarter=2x

4th quarter=x-6

Free throw points =9

therefore

[tex](2x-6)+(x+2)+2x+(x-6)+9[/tex]

Group terms

[tex](2x+x+2x+x)+(-6+2-6+9)[/tex]

Combine like terms

[tex]6x-1[/tex]

therefore

The linear expression that represent the total points scored in the game is [tex]6x-1[/tex]

Answer:

xy/2

Step-by-step explanation:

use m a t h w a y

Answer:

Cost of Tammy's lunch box before tax = $9.571

Step-by-step explanation:

Let the cost of Tammy's lunch box before tax be =$ [tex]x[/tex]

Sales tax charged = $0.727

Sales tax rate =7.596%

Sales tax charged in terms of  will be = 7.596% of the Original cost of lunch box[tex]=7.596\% \ of\ x =0.07596\ x[/tex]

So, we have,

[tex]0.07596\ x=0.727[/tex]

Dividing both sides by [tex]0.07596[/tex]

[tex]\frac{0.07596\ x}{0.07596}=\frac{0.727}{0.07596}[/tex]

∴ [tex]x=9.571[/tex]

∴ Cost of Tammy's lunch box before tax = $9.571

Answer:

The length of the sailboat is 80 feet.

Step-by-step explanation:

The function  

[tex]s(l) = 1.34 \sqrt{l}[/tex] ................ (1)  

models the maximum speed s in knots for a sailboat with length l in feet at the waterline.

So, using this model we have to find the length of a sailboat with a maximum speed of 12 knots.

Now, from equation (1) we get

[tex]12 = 1.34 \sqrt{l}[/tex]

⇒ [tex]\sqrt{l} = 8.955[/tex]

⇒ [tex]l = (8.955)^{2}  = 80[/tex] feet (Answer)

{The value is round to the nearest foot}

Each van carried 18 students and each bus carried 59 students.

Step-by-step explanation:

Let,

students carried by one van = x

students carried by one bus = y

According to given statement;

x+6y=372   Eqn 1

4x+12y=780   Eqn 2

Multiplying Eqn 1 by 4

[tex]4(x+6y=372)\\4x+24y=1488\ \ \ Eqn\ 3[/tex]

Subtracting Eqn 2 from Eqn 3

[tex](4x+24y)-(4x+12y)=1488-780\\4x+24y-4x-12y=708\\12y=708[/tex]

Dividing both sides by 12

[tex]\frac{12y}{12}=\frac{708}{12}\\y=59[/tex]

Putting y=59 in Eqn 1

[tex]x+6(59)=372\\x+354=372\\x=372-354\\x=18[/tex]

Each van carried 18 students and each bus carried 59 students.

Keywords: Linear equations, subtraction

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Each van carried 18 students and each bus carried 59 students.

Step-by-step explanation:

Let,

students carried by one van = x

students carried by one bus = y

According to given statement;

x+6y=372 Eqn 1

4x+12y=780 Eqn 2

Multiplying Eqn 1 by 4

Subtracting Eqn 2 from Eqn 3

Dividing both sides by 12

Putting y=59 in Eqn 1

Answer:

C) -5 -1/8

Step-by-step explanation:

You need to use the distributive property.

1) ( - 1/2 x 10 ) + ( - 1/2 x 1/4 )

2) 10 / -2 = -5, and -1/2 x 1/4 = -1/8

3) Make them into one equation. -5 - 1/8 is the same as - 1/2 ( 10 + 1/4 )

Final answer:

The equivalent expression to -1/2(10 + 1/4) is -5 - 1/8. Simplify the expression by distributing the -1/2 and combining the results.

Explanation:

The expression given is: -1/2(10 + 1/4)

To simplify this expression:

Distribute the -1/2: -1/2 * 10 = -5 and -1/2 * 1/4 = -1/8

Combine the results: -5 - 1/8

Therefore, the equivalent expression is: -5 - 1/8 (Option B)

Answer:

+1

Step-by-step explanation:

The y-intercept is +1.