Q3: Use the image to determine the direction and angle of rotation.

graph of triangle ABC in quadrant 4 and a second polygon A prime B prime C prime in quadrant 3

90° clockwise rotation
90° counterclockwise rotation
180° clockwise rotation
360° counterclockwise rotation

By using the properties of logarithms, The logarithm "log₅(98)" can be written as log₅(2) + log₅(7) + log₅(7).

We use the "product' and "quotient" logarithmic identities to write log₅(98) in terms of log₅(2) and log₅(7):

⇒ logₐ(b × c) = logₐ(b) + logₐ(c)      ....(product rule)

⇒ logₐ(b/c) = logₐ(b) - logₐ(c)           ...(quotient rule)

First, we write the number "98" as the product of 2 and 7, which is :

⇒ log₅(98) = log₅(2 × 7 × 7),

Using the product rule, we can split this into three logarithms;

⇒ log₅(2 × 7 × 7) = log₅(2) + log₅(7) + log₅(7),

So, we have,

⇒ log₅(98) = log₅(2) + log₅(7) + log₅(7),

Therefore, log₅(98) is equal to log₅(2) + log₅(7) + log₅(7).

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The given question is incomplete, the complete question is

Use the properties of logarithms to write the logarithm in terms of log₅(2) and log₅(7).

The logarithm is log₅(98).

Answer:

The distance from home to second base is approximately 84.85 feet in a slow pitch softball diamond that is 60 feet on each side.

Step-by-step explanation:

In a softball diamond that is 60 feet on each side, the distance from home to second base is approximately 84.85 feet. This is based on the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. In this case, the distance from home to second base forms the hypotenuse of a right triangle with legs that are each 60 / sqrt(2) feet long. Using the Pythagorean theorem, we can calculate that the distance from home to second base is approximately 84.85 feet.

The proof of cot²(x) - tan²(x) = csc²(x) - sec²(x) is the help of trigonometry

identity and Pythagorean identity .

A trigonometric identity is an equation that is true for all values of the variables in the equation, where the variables are trigonometric functions such as sine, cosine, tangent, cotangent, secant, or cosecant.

The Pythagorean identity is a fundamental trigonometric identity that relates the three basic trigonometric functions: sine, cosine, and tangent, in terms of the unit circle or right triangle.

The identity you provided, cot²(x) - tan²(x) = csc²(x) - sec²(x), is a trigonometric identity that can be derived from the Pythagorean identity and the reciprocal identities. Here's one way to prove it:

Start with the Pythagorean identity:

sin²(x) + cos²(x) = 1

Divide both sides by sin²(x):

1 + cos²(x)/sin²(x) = 1/sin²(x)

Recall that cot(x) = cos(x)/sin(x) and tan(x) = sin(x)/cos(x):

1 + cot²(x) = csc²(x)

Subtracting sec²(x) from both sides:

cot²(x) - tan²(x) = csc²(x) - sec²(x)

Therefore, cot²(x) - tan²(x) = csc²(x) - sec²(x) is a valid trigonometric identity.

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Answer:

B

Step-by-step explanation:

the area (A) of a triangle is calculated as

A = [tex]\frac{1}{2}[/tex] bh ( b is the base and h the height )

here b = 7 and h = 4 , then

A = [tex]\frac{1}{2}[/tex] × 7 × 4 = [tex]\frac{1}{2}[/tex] × 28 = 14 cm²

The function with y-intercept as 1 is h(x) = 0.5(2)ˣ + 0.5.

The y-intercept is the point where the graph intersects the y-axis. In other words, the y-intercept of a function is the value of y when x = 0.

Therefore, let's find the function with y-intercept as 1.

Hence,

h(x) = 0.5(2)ˣ + 0.5

Let's make x = 0 to check if the value of y = 1

Therefore,

h(x) = 0.5(2)ˣ + 0.5

h(0) = 0.5(2)⁰ + 0.5

h(0) = 0.5(1) + 0.5

h(x) = 0.5 + 0.5

h(x) = 1

Therefore, the function with y-intercept as 1 is h(x) = 0.5(2)ˣ + 0.5

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92 grams of recycled plastic were used to make Chanasia's bucket

Chanasia buys a plastic bucket

The bucket weighs 460 grams

The label on the bucket says it was made with 20% recycled plastic

The number of grams of recycled plastic that was used to make the bucket can be calculated as follows

= 460/100

= 4.6

= 4.6 × 20

= 92

Hence 92 grams of recycled plastic was used to make the bucket

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Answer:

B. (2, 2)

Step-by-step explanation:

(x, y)

3x = 8 - y

a) 3(3) = 8 - 1

9 = 8 - 1

9 = 7  False

b) 3(2) = 8 - 2

6 = 6  True

c) 3(8) = 8 - 0

24 = 8 False

With additive inverse and by grouping like terms the simplified form of the expression is 6a²-13ab-b².

The given expression is 3a²-5ab+b²-(-3a²+2b²+8ab).

Here,

3a²-5ab+b²+3a²-2b²-8ab

Group like terms, that is

(3a²+3a²)+(-5ab-8ab)+(b²-2b²)

= 6a²-13ab-b²

Therefore, with additive inverse and by grouping like terms the simplified form of the expression is 6a²-13ab-b².

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A- Tanya's gross income for the first year was $7,940, and b- her net income was $5,840 after subtracting her total expenses.

a. To calculate Tanya's gross income for the first year, we need to multiply the rent per month by the number of months the house was rented:

Gross income from rent = $620/month x 7 months = $4,340

Tanya also paid $175/month in average expenses for the whole year, so her total expenses were:

Total expenses = $175/month x 12 months = $2,100

Adding the mortgage interest paid, Tanya's gross income for the first year was:

Gross income = $4,340 + $3,600 = $7,940

b. To calculate Tanya's net income or net loss, we need to subtract her total expenses from her gross income:

Net income = Gross income - Total expenses

Net income = $7,940 - $2,100

Net income = $5,840

Therefore, Tanya's net income for the first year was $5,840.

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If the radius of smaller-saw is 1.25 in, and radius of larger-saw is 2 inches, then larger-saw is 4.71 inches greater the smaller-saw.

The "Circumference" of a circle is defined as the distance around the boundary of a circle, it is also called perimeter of circle;

The diameter of the smaller saw can be calculated as:

⇒ diameter = 2 × radius = 2 × 1.25 inches = 2.5 inches;

The diameter of the larger saw is 1.5 inches greater than the diameter of the smaller saw, which means,

⇒ diameter of larger saw = diameter of smaller saw + 1.5 inches,

⇒ diameter of larger saw = 2.5 inches + 1.5 inches = 4 inches;

So, Circumference of circle is = 2 × π × radius;

The Circumference of "smaller-saw" = 2 × 3.14 × 1.25 inches,

⇒ circumference of smaller saw = 7.85 inches;

circumference of "larger-saw" = 2 × 3.14 × 2 inches

⇒ circumference of larger saw = 12.56 inches;

The difference in circumference between the two saws is:

⇒ circumference of larger saw - circumference of smaller saw

⇒ 12.56 inches - 7.85 inches = 4.71 inches;

Therefore, the circumference of the larger saw is 4.71 inches greater than the circumference of the smaller saw.

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The given question is incomplete, the complete question is

A carpenter has two different-sized circular saws. The smaller saw has the radius as 1.25 inch. The larger saw has a diameter that is 1.5 inches greater than the diameter of the smaller one.

How much greater is the circumference of the larger saw than the smaller saw? Use 3.14 for π

The values of b and c include the following:

c = 8

b = 4√3

In order to determine the length of b, we would apply the law of tangent because the given side lengths represent the adjacent side and opposite side of a right-angled triangle.

Tan(θ) = Opp/Adj

Where:

Therefore, we have the following tangent trigonometric function:

Tan(θ) = Opp/Adj

Tan(30°) = 4/b

√3/3 = 4/b.

b = 4√3

From Pythagorean Theorem, the length of c is given by;

c² = (4√3)² + 4²

c² = 48 + 16

c = √64

c = 8

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

The equation for f(x) can be written as f(x)= 15(3/5)ˣ

An exponential function is a mathematical function of the form f(x) = aˣ, where 'a' is a positive constant and 'x' is any real number. The graph of an exponential function is characterized by a rapid increase or decrease as 'x' increases or decreases, respectively. Exponential functions have many applications in fields such as finance, biology, and physics.

From this table we know this thing that when x=0, y=15

when x=1, y=9

a×b⁰=15⇒ a=15 as b⁰=1

a×b¹=9⇒ ab=9

so b=9/a

b= 9/15⇒3/5

After putting the values we get the result f(x)=15×(3/5)ˣ

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The correct answer is C. The circumference of a circle varies directly as the radius.

It is calculated by multiplying the diameter of a circle by pi, or by measuring the length of a curved line that encloses the shape.

The relationship between the circumference of a circle and its radius is a linear equation, where the circumference is equal to two times pi times the radius, or C = 2πr.

The circumference of a circle varies directly as the radius, meaning that if the radius of a circle is doubled, the circumference will also double.

This linear relationship can be used to approximate the value of pi, which is a constant ratio between the circumference and the diameter of any circle.

Therefore, the statement that justifies why the measurements of the circumference and radii of circles with different areas can be used to approximate the value of pi is that the circumference of a circle varies directly as the radius.

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Answer:

x = 68

Step-by-step explanation:

X + 22 = 90 degrees

90 - 22 = 68 degrees

Answer:

C.

Step-by-step explanation:

because x value cannot be repeated in an ordered pair

Thus, the length of chord BD of the given circle is found to be 8.5 units.

A chord is indeed a straight line that connects two points on a circle's circumference. Because it connects to points on the circle's perimeter, the diameter is thought to be the chord with the longest length.

The lengthiest chord in a circle is called the diameter, and the chord's perpendicular distance to the circle's centre is zero.

Given that: From diagram.

Now, in right triangle, MAD, using the Pythagorean theorem:

(AD)² = (AM)² + (MD)²

(9.5)² = (4)² + (MD)²

(MD)² = 9.5² -  4²

(MD)² = 90.25  - 16

(MD)² = 74.25

MD = 8.5

Thus, the length of chord BD of the given circle is found to be 8.5 units.

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Answer:

-2a⁴ b⁶ c³

Step-by-step explanation:

2a² b c³ (-a² b⁵)

determine the signs for multiplication or division

-2a²bc³a²b⁵

multiply the monomials

-2a⁴b⁶c³

(then done)

The coordinates of the vertices of rectangle are (-3,3),(6,3),(6,-3), (-3,-2).  the rectangle is 9 units long and 5 units wide.

In mathematics and computer science, a graph is a collection of vertices (also called nodes) and edges that connect pairs of vertices. Graphs are used to represent various types of relationships between objects, such as social networks, road networks, electrical circuits, and many others.

Vertices in a graph are typically represented by circles or points, while edges are represented by lines or arcs that connect pairs of vertices. A graph can be directed, where each edge has a direction, or undirected, where edges have no direction.

The coordinates of the vertices of rectangle are (-3,3),(6,3),(6,-3), (-3,-2).

To find the length and width of the rectangle, we need to determine the distance between the points on the x and y axes.

The length of the rectangle is the distance between points (-3,3) and (6,3) on the x-axis:

Length = 6 - (-3) = 9 units

The width of the rectangle is the distance between points (-3,3) and (-3,-2) on the y-axis:

Width = 3 - (-2) = 5 units

Therefore, the rectangle is 9 units long and 5 units wide.

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Answer: Box plot:    |       o

19 |          

18 |          

17 |          

16 |          

15 |       o  

14 |     o      

13 |          

12 |   o     o

11 | o  o  o  

10 |          

9 | o  o      

8 |   o      

7 |  o        

  |___________

Line plot: 19|

18|

17|

16|

15|     o

14|   o

13|

12| o   o

11|o o o

10|

9|o o

8|  o

7|o

 |___________

   1 2 3 4 5 6 7 8 9 10  (data index)

Step-by-step explanation:

It should be noted that to determine the amplitude of a sinusoidal function from a graph:

Identify the highest point (peak) and the lowest point (valley) of the graph of the sinusoidal function.

Find the vertical distance between the peak and the midline of the graph (the line halfway between the highest and lowest points).

The amplitude of the sinusoidal function is half of this vertical distance.

Identify the highest point (peak) and the lowest point (valley) of the graph of the sinusoidal function.

Find the vertical coordinate of the midline of the graph (the line halfway between the highest and lowest points).

Write the equation of the midline as y = the vertical coordinate of the midline.

To determine the maximum value of a sinusoidal function from a graph:

Identify the highest point (peak) of the graph of the sinusoidal function.

The maximum value of the sinusoidal function is the vertical coordinate of this peak.

To determine the minimum value of a sinusoidal function from a graph:

Identify the lowest point (valley) of the graph of the sinusoidal function.

The minimum value of the sinusoidal function is the vertical coordinate of this valley.

To determine the period of a sinusoidal function from a graph:

Identify two consecutive peaks or valleys of the graph of the sinusoidal function.

Find the horizontal distance between these two points.

The period of the sinusoidal function is twice this horizontal distance.

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To show that the angle bisector of an equilateral triangle is perpendicular to the base​, we can assume an equilateral triangle with sides ABC. Next, we can get an angle bisector in the middle represented by BD which becomes perpendicular to the base BC.  

After getting the angle bisector, we would see that the line segment BD, divides angle BAC into two equal angles, each measuring 30 degrees. Now, we can draw a perpendicular line from point D to the base BC.

We could also draw a line E, which is the perpendicular line that intersects the base BC. We want to show that BD is perpendicular to BC, which means we need to show that angle BDE is a right angle.

Since the sum of angles BDA and ADE is 90 degrees (they form a right angle), and the sum of angles BAD, ABD, and BDA is 180 degrees, it follows that angle BDE must be a right angle.

Therefore, we have shown that the angle bisector of an equilateral triangle is perpendicular to the base.

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The result of adding -3 (row 1) to row 2 is determined as (0  10) |-14.

The result of the row multiplication in the matrix is calculated by applying the following method;

row 1 in the given matrix = [1  -4] | 8

To multiply row1 by -3, we will multiply each entity by -3 as shown below;

= -3(1  -4) | 8

= (-3  12) | -24

To add the result to 3;

(-3  12) | -24  +  (3   -2)|10

= (0  10) |-14

Thus, the result of the row multiplication is determined by multiplying each entry in row 1, by - 3.

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Answer:

Since there are only yellow and green balls in the tub, there is no possibility of picking a red ball. Therefore, the probability of picking a red ball is zero.

(i) The probability of picking a yellow ball can be calculated by dividing the number of yellow balls by the total number of balls. In this case, there are two yellow balls and three green balls, so the total number of balls is 2 + 3 = 5. The probability of picking a yellow ball is 2/5 or 0.4, which is 40%.

(ii) Similarly, the probability of picking a green ball can be calculated by dividing the number of green balls by the total number of balls. In this case, there are three green balls and five total balls. Therefore, the probability of picking a green ball is 3/5 or 0.6, which is 60%.

(iii) As mentioned earlier, there are no red balls in the tub. Hence, the probability of picking a red ball is 0.

The proportion of homes in Oregon heated by natural gas is different than the reported national average of 45%.

Null hypothesis: The proportion of homes in Oregon heated by natural gas is the same as the reported national average of 45%.

Alternative hypothesis: The proportion of homes in Oregon heated by natural gas is different than the reported national average of 45%.

We can use a two-tailed z-test to test this hypothesis.

The test statistic is calculated as:

z = (p - P) / √[P(1-P)/n]

where p is the sample proportion, P is the hypothesized population proportion (0.45), and n is the sample size.

Using the values given in the problem, we have:

p = 175/314 = 0.557

P = 0.45

n = 314

Plugging these values into the formula, we get:

z = (0.557 - 0.45) / √[0.45(1-0.45)/314] = 3.039

Using a standard normal distribution table, we find that the probability of getting a z-value of 3.039 or higher (in absolute value) is less than 0.005.

Since the significance level is 0.05, which is greater than the p-value, we can reject the null hypothesis and conclude that there is evidence to support the alternative hypothesis.

Therefore, the proportion of homes in Oregon heated by natural gas is different than the reported national average of 45%.

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Answer:

We can use the cosine double angle formula to find cos(2π/3):

cos(2θ) = cos^2(θ) - sin^2(θ)

Letting θ = π/3, we get:

cos(2π/3) = cos^2(π/3) - sin^2(π/3)

Using the values of cosine and sine of π/3 (which are known), we have:

cos(2π/3) = (1/2)^2 - (√3/2)^2

Simplifying, we get:

cos(2π/3) = 1/4 - 3/4

cos(2π/3) = -1/2

Therefore, cos(2π/3) is equal to -1/2.

Answer:

cos(2π/3) = cos(π/3 + π/3)

= cos²(π/3) - sin²(π/3)

= (1/2)² - (√3/2)²

= 1/4 - 3/4

= -2/4

= -1/2

Step-by-step explanation:

here are the steps:

Therefore, cos(2π/3) = -1/2

Answer:

B

[tex]6^{-7}[/tex] is equivalent to 1/279936 which is basically [tex]\frac{1}{6^{7}}[/tex]

The length of LG in the plane is

To find the distance between two points in a 2D plane, we can use the distance formula:

distance = √((x2 - x1)^2 + (y2 - y1)^2)

Using this formula, we can find the distance between L(4,4) and G(9,9):

distance = √((9 - 4)^2 + (9 - 4)^2)

= √(5^2 + 5^2)

= √50

= 5√2

Therefore, the distance between L(4,4) and G(9,9) is 5√2, which is approximately 7.07 units.

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The probability that the science teacher trips over the cords for the first time during the 4th period of the day is 0.0279.

A discrete probability distribution has countable alternative values for the random variable and a stated probability for each one. Since the potential values are 0, 1, or 2, and the probability of each conceivable value can be determined, the probability distribution of the number of heads acquired after two coin flips is an example of a discrete probability distribution.

In contrast, a continuous probability distribution has a continuous range of possible values for the random variable and a probability of zero for any given value.

Given that, the probability of tripping is given as: 0.35.

Now, the probability of tripping on the 4th period is given by:

[tex]P(X=k) = (1-p)^{(k-1)} * p[/tex]

Now, for k = 4 and p = 0.35 we have:

[tex]P(X=4) = (1-0.35)^{(4-1)} * 0.35[/tex]

P(X=4) = 0.0279

Hence, the probability that the science teacher trips over the cords for the first time during the 4th period of the day is 0.0279.

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