In the Reflecting Ball Game, a ball can be launched from points 1, 2, 3, 4, 5 or 6 in the direction shown. When a ball hits a side of rectangle $ABCD$, it bounces at a $90^{\circ}$ angle back into the playing field. The path of the ball ends when it hits corner point $A$, $B$, $C$, or $D$. The path for starting point 5 is shown in the diagram. Each of the $15$ non-overlapping squares of the playing field has an area of $2$ square centimeters. What is the length of the longest possible path for a ball launched from a starting point?

Answer:

Step-by-step explanation:

a

By evaluating the linear equation we can see that the complete table is:

x|  -4   -2  0  2  4

y| 17    13  9   5  1

Here we have a linear function:

y = -2x + 9

And we want to complete the given table, to do so, we just need to evaluate the linear function in the given values of x.

if x = -2

y = -2*-2 + 9 = 4 + 9 = 13

if x = 0

y = -2*0 + 9 = 9

if x = 2

y = -2*2 + 9 = 5

if x = 4

y = -2*4 +9 = 1

Then the complete table is:

x|  -4   -2  0  2  4

y| 17    13  9   5  1

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The statement "a quadrilateral is a square if and only if it is a rectangle" made by Ruby is a true biconditional statement in mathematical terms.

A quadrilateral is a four-sided shape that can have different properties based on its sides and angles.

Ruby says that a quadrilateral is a square if and only if it is a rectangle. This statement can be written as:

(Q is a square) if and only if (Q is a rectangle)

To understand if this statement is a true biconditional statement, we need to examine the definition of a square and a rectangle and how they relate to each other.

A square is a type of quadrilateral where all four sides are equal in length and all four angles are equal (90 degrees). In other words, a square is a rectangle that has four equal sides.

A rectangle is a type of quadrilateral where all four angles are equal to 90 degrees. This means that the opposite sides are parallel and equal in length.

Based on the definitions, we can conclude that the statement "a quadrilateral is a square if and only if it is a rectangle" is a true biconditional statement.

This means that if a quadrilateral is a square, it is also a rectangle and vice versa. If a quadrilateral is not a square, it cannot be a rectangle.

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One needs to connect the points where these arcs intersect the circle to form the desired shape. Finally, the angles need to be labeled to complete the unit circle.

A unit circle is a circle with a radius of 1. It is a tool used in trigonometry to measure angles and lengths of arcs. To calculate the length of an arc, one needs to know the angle it subtends at the center of the circle. The formula for finding this length is L = rθ, where L is the length of the arc, r is the radius, and θ is the angle subtended at the center. To fill in a unit circle, one needs to first calculate the lengths of the arcs subtended by the angles in the circle. Then, one needs to connect the points where these arcs intersect the circle to form the desired shape. Finally, the angles need to be labeled to complete the unit circle.

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The main step to add rational expressions with unlike denominators is to take least common multiple of denominator and make them like before applying the required operation.

Therefore, the addition of rational expressions with unlike denominators is simplified by taking least common multiple.

The above question is incomplete, the complete question is:

Write steps for adding rational expressions with unlike denominators with example ?

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Answer:

Step-by-step explanation:

The circumference of the circle is 30 cm.

Any unit for measuring length can be used to calculate the circumference, including feet, miles, metres, millimetres, etc.

By substituting 3 for, we can use the following formula to determine the circle's circumference:

C = 2πr

where r is the radius and C is the circumference.

We can observe from the provided diagram that the circle's radius is 5 cm. Thus, by changing r = 5 cm and = 3 in the formula, we obtain:

C = 2(3)(5) = 30

The circle's circumference is 30 cm as a result.

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:Answer: $90000

i = prt (interest = principle x rate x time)

i = 60000 x 0.025 x 20

i = $30000 interest

b = p + i (balance = principle + interest)

b = 60000 + 30000

b = $90000 balance

The requried measure of length JK on the coordinate plane is 10 units.

The distance between two points on the coordinate plane can be evaluated by the distance formula as,
D = √((x₂ - x₁)^2 + (y₂ - y₁)²)

Here,
The coordinates of the two points J(-7, -7) and K(-7, 3) suggest that they lie on a vertical line, where the x-coordinate is constant (-7) and only the y-coordinate changes. The distance between these points is equal to the difference in their y-coordinates.
Therefore, the length of JK is,
JK = |y-coordinate of K - y-coordinate of J|
    = |3 - (-7)| = |3 + 7| = 10

So, the length of JK is 10 units.

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Answer:

y+75°=180°[co interior angle]

or,y=180-75

•°•y=105

again,

y=z(alternate angle)

•°•z=105°

now

z+x+68°=180(sum of angle of triangle is 180)

or,105+68+x=180

or,x+173=180

or,x=180-173

•°•x=7

Answer: y=6x^2 opens downward. y=11x^2 opens upward

Step-by-step explanation:

-6 is a negative so it opens downward. 11 is a positive number so it will open upward.

Answer: The type of function that best models the data would depend on the actual change in the number of students and the specific data points for each year. However, if the change in the number of students was the same for each time period, a linear function would be a good candidate for modeling the data. This function would have the form y = mx + b, where m represents the rate of change (the change in the number of students over a certain time period) and b is the y-intercept (the number of students at the starting year).

Step-by-step explanation:

45% in decimal form is 0.45.

Percentages are a way of expressing a fraction or proportion of a whole, with the whole being represented by 100%. To convert a percentage to a decimal, we simply divide by 100.

In this case, we have 45%. Dividing 45 by 100 gives us 0.45, which is the decimal equivalent of 45%.

Decimal numbers are a base-10 system that uses the digits 0 through 9 to represent all possible values. They are commonly used in mathematics, finance, and science for their precision and ease of calculation.

Converting percentages to decimals is an important skill to have when dealing with calculations involving percentages, such as taxes, discounts, and interest rates.

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The equation of the circle graphed is given as follows:

(x + 3)² + (y - 4)² = 41.

The equation of a circle of center [tex](x_0, y_0)[/tex] and radius r is given by:

[tex](x - x_0)^2 + (y - y_0)^2 = r^2[/tex]

The center in this problem has the coordinates given as follows:

(-3,4).

Hence:

(x + 3)² + (y - 4)² = r².

x = -1 and y = 9 is a point on the circumference of the circle, hence the radius of the circle is obtained as follows:

r² = (-1 + 3)² + (9 - 4)²

r² = 41.

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P(A∩B) = 0 (the probability of both events occurring is zero).

P(A|B) = 0.

No. The student's argument that mutually exclusive events are independent is incorrect.

If events A and B are mutually exclusive, then by definition they cannot occur at the same time.

Thus, P(A∩B) = 0 (the probability of both events occurring is zero).

Since events A and B are mutually exclusive, they cannot be independent. In fact, if two events are mutually exclusive, it means that the occurrence of one event precludes the occurrence of the other event, so they are completely dependent on each other.

To calculate P(A|B), we can use the formula:

P(A|B) = P(A∩B) / P(B)

Since P(A∩B) = 0 and P(B) = 0.70, we have:

P(A|B) = 0 / 0.70 = 0

This means that if we know event B has occurred (i.e., P(B) > 0), then the probability of event A occurring (i.e., P(A|B)) is zero. Therefore, events A and B are not independent.

The student's argument that mutually exclusive events are independent is incorrect. Mutually exclusive events are by definition dependent events.

In general, the concept of mutually exclusive events and the concept of independent events are not the same. Mutually exclusive events are dependent events, while independent events are events that do not depend on each other.

In this problem, the fact that events A and B are mutually exclusive means that they are dependent events, and this is reflected in the probability calculation for P(A|B), which is zero. Therefore, we can conclude that mutually exclusive events are not independent events.

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Answer: x^6+6x^4+8x^2+2

Step-by-step explanation:

Since g comes after f then you will take g's equation and plug it into the f equation so it would turn out to be if you plugged it in

(x^2+2)^3-4(x^2+2)+2 which would equal x^6+6x^4+8x^2+2

a) Height of section A is 15.21 m, b) the length of the wire structure from the point O to the top of section A is 51 units and the length of the wire structure from the point O to the top of section B is 41.56 units

Height is the measurement of an item in the vertical direction, whereas distance is the measurement of an object in the horizontal direction from a certain location.

Given that an arrangement of wires and tower, making it as two right triangles, Let us consider the foot of the tower be C (refer to image attached)

Using the concept of height and distance, we will find the asked measurements,

a) What is the height of the section A?

CA = tan 45° × CO [tan 45° = 1]

CA = 36 m

Similarly,

CB = tan 30° × CO [tan 30° = 0.577]

CB = 20.78

Height of section A = CA - CB

= 15.21 m

b) What is the length of the wire structure from the point O to the top of section A?

OA = AC / sin 45°

OA = 36 / 0.707

OA = 51

Therefore, the length of the wire structure from the point O to the top of section A is 51 units

c) What is the length of the wire structure from the point O to the top of section B?

OB = BC / sin 30°

OB = 41.56

Therefore, the length of the wire structure from the point O to the top of section B is 41.56 units

Hence, the answers are a) Height of section A is 15.21 m, b) the length of the wire structure from the point O to the top of section A is 51 units and the length of the wire structure from the point O to the top of section B is 41.56 units

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The complete question is :-

Radio towers are used for transmitting a range of communication services including radio and television. The tower will either act as an antenna itself or support one or more antennas on its structure, including microwave dishes. They are among the tallest human-made structures. There are 2 main types: guyed and self-supporting structures.

On a similar concept, a radio station tower was built in two sections A and 6. Tower is supported by wires from a point O. Distance between the base of the tower and point O is 36 m. From point O, the angle of elevation of the top of section B is and the angle of elevation of the top of section A is

On the basis of the above information, answer any three of the following questions

*(i) What is the height of the section A?*

*(ii)What is the length of the wire structure from the point O to the top of section A?*

*(iii) What is the length of the wire structure from the point O to the top of section B ?*​

The  perimeter of the painting 7 feet.

Perimeter is the distance around the edge of a shape. To find the perimeter by adding up the side lengths of various shapes.

Given:

A rectangle with side lengths y and z has perimeter 2y+2z.

Elizabeth bought a rectangular painting whose length = 2 ft and width = 1.5 ft.

So, the  perimeter of the painting

= 2l+ 2w

= 2(2)+ 2(1.5)

= 4 + 3

= 7 feet

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Answer: -3

Step-by-step explanation:

Using the trapezoid midsegment theorem, the length of the midsegment of trapezoid PQRS is 11 units,

The Trapezoid Midsegment Theorem states that the length of the midsegment of a trapezoid is equal to the average of the lengths of the bases (the parallel sides). Mathematically, this can be expressed as:

length of midsegment = (length of base₁ + length of base₂) / 2

In the case of trapezoid PQRS, the bases are QR and PS.

The length of QR is 8 units and the length of PS is 14 units.

Therefore, the length of the midsegment can be calculated as:

length of midsegment = (8 + 14) / 2

length of midsegment = 22 / 2

= 11 units

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Answer: Robert has 35 cans of paint - 30 cans of paint that he has already used = 35 - 30 = 5 cans of paint left.

Since each section of trim takes 1/2 can of paint, Robert can paint 2 sections of trim with 1 can of paint.

Therefore, Robert can paint 5 cans of paint * 2 sections of trim per can = 10 sections of trim.

The inequality can be written as 10 <= sections of trim <= 10.

In words, the number of sections of trim that Robert can paint is less than or equal to 10.

On a number line, this solution can be represented as follows:

[9, 10, 11, ...]

 ^

 |

sections of trim

The solution makes sense because Robert has only 5 cans of paint left, so it is not possible for him to paint more than 10 sections of trim.

Step-by-step explanation:

Note that using relationship 1 where

A = 50°; and

Y = 110°

Angle b = 70°; and

Angle c = 60°

This means that the triangle is a scalene triangle.

Since a = 50° - Given; and

y = 110° - Given

to determine the m∠b, you must use the concept of sum of angles on a straight line. Sum of angles on a straight line must always equal 180°

Thus,

b = 180 - y

b = 180 - 110

b = 70°

To determine the m∠c, we used the sum of angles in a Δ theorem. This states that all angles in Δ must equal 180°

Since

a = 50° - Given; and

b = 70° - as derived above,

c = 180 - (a + b)

c = 180 - (50 + 70)

c = 180 - 120

c = 60°

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The total area for the parallelogram-shaped tiles Felix needs to buy exists 80cm² at the price of $40.

In calculating the area of a parallelogram, the base exists multiplied by the height, in an exact way as computing the area of a rectangle.

here, we have,

Given that, the base of one parallelogram-shaped tile exists 4cm with a height of 2 cm.

Area of one parallelogram shaped tile = 4cm × 2cm

Area of one parallelogram-shaped tile = 8cm²

Total area for the 5black and 5white tiles = 10 × 8cm²

Total area for the 5black and 5white tiles = 80cm²

Therefore, Felix must buy a total area of 80cm² of 10 parallelogram-shaped tiles.

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Based on the graph of this exponential function, the domain and range are as follows;

Domain = {-∞, ∞}.

Range = {0, ∞}.

In Mathematics, a domain is the set of all real numbers for which a particular function is defined.

Additionally, the vertical extent of any graph of a function represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.

By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:

Domain = {-∞, ∞}.

Range = {0, ∞} or {y|y > 0}.

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To convert grams to moles per gram, you need to find the molar mass of the substance in grams per mole. Then, divide the number of moles by the given mass to get moles per gram.

To convert grams to moles per gram, you first need to determine the molar mass of the substance you are interested in. The molar mass is the mass of one mole of the substance, expressed in grams. Once you know the molar mass, you can use it to convert grams to moles, and then divide by the total mass to get moles per gram.

Here are the steps to follow:

Determine the molar mass of the substance. This is the sum of the atomic masses of all the atoms in the molecule, expressed in grams per mole. You can find the atomic masses on the periodic table.

Convert the given mass from grams to moles, using the formula:

moles = mass / molar mass

where mass is the given mass in grams, and molar mass is the calculated molar mass in grams per mole.

Divide the number of moles by the given mass, to get the moles per gram. This gives you the number of moles of the substance per gram of the sample.

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The simplified answer in improper fraction is -3 ¹/₅.

To simplify the equation we will solve the equation and obtain the value of x as shown below.

7 ( x + 1 ) = 1 - 2( 5 - x )

open the bracket by multiplying by 7 and 2

7x  +  7  = 1 - 10 + 2x

collect similar terms as shown below;

7x - 2x = 1 - 10 - 7

5x = -16

x = -16 / 5

x = -3 ¹/₅

Thus, the simplified answer in improper fraction is obtained by finding the value of x.

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Answer:

Like the other solver, I believe they are correct because I arrived at the same solution:

3/4w <= 225

Multiplying both sides by 4/3 to isolate w:

w >= (225 * 4)/3

w >= 300

Solution is w >= 300. In order to meet the inequality, the value of w must be equal to or larger than 300.

The equation for A is A=4b(t+D) from t = A/4b-D.

Two or more expressions with an Equal sign is called as Equation.

The given equation is t = A/4b-D

t equal to A by four times of b minus D

The variables in the equation are A, b and D.

The formula is given for the variable t.

We need to change the formula to find the value of A.

We need to solve for A.

Add D on both sides

t+D=A/4b-D+D

The positive and negative D on right side of the equation will be cancelled and left with A/4b.

t+D/1=A/4b

Now apply cross multiplication

A=4b(t+D)

Hence, the equation for A is A=4b(t+D) from the t = A/4b-D.

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The ordered pair that show a combination of 12 coins from the graph is (10, 2)

from the question, we have the following parameters that can be used in our computation:

The graph

The purple line represents the combination of 12 coine

Using the above as a guide, we have the following:

The equation is

x + y = 12

From the list of options, we have

(10, 2): 10 + 2 = 12

Hence, the ordered pair is (10, 2)

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Answer:

2/5 bottle of juice

2/15L of juice

Step-by-step explanation:

We originally have 2 bottle of juice and each bottle contains 1/3 liter. From this, we can find that there are 2/3L of juice in total. Since five friends equally drink the juice, each of them drank 2/5 bottle of juice and 2/15L of juice.

I hope this makes sense to you. I am generally good with following up with additional questions. Feel free to leave a comment if you have any questions or you can also message me as well!