Renting a roller rink for your birthday costs $150 plus $6 ​ per person. The total charge for your party was $ 282. Translate this information into an equation using
x ​ to represent the number of guests at your party

The value of the trigonometric identities are;

1. cos R = 15/17

cos S = 8/17

2. cos R = 12/13

cos S = 5/13

Trigonometric Identities are simply seen as the equalities involving trigonometry functions.

It also holds true for all the values of variables given in the equation.

There are various distinct trigonometric identities involving the side length as well as the angle of a triangle. They are;

We have cosine represented as;

cos θ = adjacent/hypotenuse

For the first triangle, we have;

cos R = 30/34 = 15/17

cos S = 16/34 = 8/17

For the second triangle;

cos R = 24/26 = 12/13

cos S = 10/26 = 5/13

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The constant of proportionality between the plant's height and time in weeks is 2 inches/week. So,correct answer is (D) 2.

The constant of proportionality is a factor that relates two variables that are directly proportional, indicating the ratio of change between them.

We can use the given information to determine the constant of proportionality between the plant's height and time in weeks.

The plant's height increased by 6 - 0 = 6 inches in the first three weeks (from week 0 to week 3). Therefore, the rate of growth during this period was 6 inches / 3 weeks = 2 inches/week.

Similarly, the plant's height increased by 10 - 6 = 4 inches during the next two weeks (from week 3 to week 5), so the rate of growth during this period was 4 inches / 2 weeks = 2 inches/week.

Finally, the plant's height increased by 14 - 10 = 4 inches during the last two weeks (from week 5 to week 7), so the rate of growth during this period was also 4 inches / 2 weeks = 2 inches/week.

Since the plant grows at a constant rate, we can assume that the rate of growth was 2 inches/week throughout the entire period from week 0 to week 7.

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Answer:

The potential energy (PE) of an object is given by:

PE = mgh

where m is the mass of the object, g is the acceleration due to gravity (9.8 m/s² near the Earth's surface), and h is the height of the object above a reference level.

In this case, the mass of the clam is 0.11 kg, the height above the beach is 11.5 m, and the acceleration due to gravity is 9.8 m/s². Therefore, the potential energy of the clam is:

PE = (0.11 kg) × (9.8 m/s²) × (11.5 m) ≈ 12.7 J

So the potential energy of the clam is approximately 12.7 Joules.

As a result, the answer to the following question, As a result, the length triangle of side JS is 18.

A triangle is a polygon because it contains four or more parts. It features a simple rectangular shape. A triangle ABC is a rectangle with the edges A, B, and C. When the sides are not collinear, Euclidean geometry produces a single plane and cube. If a triangle contains three components and three angles, it is a polygon. The corners are the points where the three edges of a triangle meet. The sides of a triangle sum up to 180 degrees.  

We must apply the Pythagorean theorem to answer question 19. We know that JN is the hypotenuse of a right triangle with legs of 6 and 8 lengths. As a result, we may apply the formula:

[tex]JN^2 = 6^2 + 8^2\\JN^2 = 36 + 64\\JN^2 = 100\\JN = square root (100)\\JN = 10\\[/tex]

As a result, the length of side JN is 10.

JS/NS = JM/JN

When we substitute the provided values, we get:

12/10 = JS/NS

When we simplify the left side, we get:

6/5 = JS/NS

When we multiply both sides by NS, we get:

JS = (6/5)NS

We also know that NS is 15, therefore we may substitute that number for:

[tex]JS = (6/5) * 15\sJS = 18[/tex]

As a result, the length of side JS is 18.

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Answer:

-51 would be the answer for this equation.

Step-by-step explanation:

Kiran is correct in saying that a solution to the equation x + 4 = 20 must also be a solution to the equation 5(x + 4) = 100. This is because the second equation is simply the first equation multiplied by 5. To see this, we can distribute the 5 on the left side of the second equation to get 5x + 20 = 100. We can then subtract 20 from both sides to get 5x = 80, and finally divide both sides by 5 to get x = 16.

Since x = 16 satisfies the first equation, it must also satisfy the second equation. This is because if we substitute x = 16 into the first equation, we get 16 + 4 = 20, which is true. If we substitute x = 16 into the second equation, we get 5(16 + 4) = 100, which is also true. Therefore, any solution to the first equation will also be a solution to the second equation when the second equation is just the first equation multiplied by a constant factor.

Answer:

Kiran is correct. To see why, let's first simplify the second equation, 5(x + 4) = 100, by multiplying both sides by 1/5:

5(x + 4) = 100

⇒ (1/5) * 5(x + 4) = (1/5) * 100

⇒ x + 4 = 20

Now we can see that the second equation simplifies to the first equation, x + 4 = 20. This means that any solution that satisfies the first equation (x + 4 = 20) will also satisfy the second equation (5(x + 4) = 100).

In other words, if we find a value of x that makes x + 4 = 20 true, then substituting that value of x into 5(x + 4) = 100 will also make it true. Therefore, any solution to the equation x + 4 = 20 will also be a solution to the equation 5(x + 4) = 100.

Step-by-step explanation:

a. The surface area of the outer square pyramid is 11.25 square centimeters. b. The side length of the inner square pyramid is 2.25 centimeters.

a. To find the surface area of the outer square pyramid, we need to calculate the area of each of its faces and add them together. The outer square pyramid has four triangular faces and a square base.

Area of a triangular face = (1/2) x base x height

Area of a triangular face = (1/2) x 1.5 cm x 3 cm = 2.25 cm²

The area of the square base can be found using the formula for the area of a square:

Area of square base = side length²

Area of square base = 1.5 cm x 1.5 cm = 2.25 cm²

Therefore, the total surface area of the outer square pyramid is:

Surface area = 4 x area of triangular face + area of square base

Surface area = 4 x 2.25 cm² + 2.25 cm²

Surface area = 11.25 cm²

Therefore, the surface area of the outer square pyramid is 11.25 square centimeters.

b. The dimensions of the inner square pyramid have a ratio of 2:3 to the dimensions of the outer square pyramid. Let's call the side length of the inner square pyramid "x". Since the ratio of the dimensions is 2:3, we know that the side length of the outer square pyramid is (3/2)x.

The volume of a square pyramid can be calculated using the formula:

Volume = (1/3) x base area x height

Since the two pyramids are similar, the ratio of their volumes is the cube of the ratio of their corresponding side lengths:

Volume of inner pyramid / Volume of outer pyramid = (x / (3/2)x)³ = (2/3)³

We also know that the volume of the outer pyramid is:

Volume of outer pyramid = (1/3) x base area x height

The height of the two pyramids is the same, since they are stacked on top of each other, so we can write:

Volume of inner pyramid / Volume of outer pyramid = (1/3) x base area of inner pyramid / (1/3) x base area of outer pyramid

Simplifying this expression, we get:

(x / (3/2)x)³ = (1/3) x² / (1/3) (3/2x)²

Solving for x, we get:

x = (3/2)²

x = 2.25

Therefore, the side length of the inner square pyramid is 2.25 centimeters.

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The complete question is :

a. For the outer square pyramid, the side length of the base is 1.5 centimeters and the height of one of the triangular faces is 3 centimeters. What is the surface area of the outer square pyramid?

b. The dimensions of the inner square pyramid have a ratio of 2:3 to the dimensions of the outer square pyramid. What are the dimensions of the inner square pyramid?

1) The solution is (2, 4).

2) The solutions are (-1, -5) and (2, 4).

3) The solutions are (3, -5) and (-2, 5).

1) Y = x^2 and y = 4x - 4

Substitute y from the second equation into the first equation to get:

x^2 = 4x - 4

Simplifying and rearranging:

x^2 - 4x + 4 = 0

(x - 2)^2 = 0

x = 2

Substitute x = 2 into the second equation to get:

y = 4(2) - 4 = 4

Therefore, the solution is (2, 4).

2) Y = -x^2 + 2x - 4 and y = 3x - 2

Substitute y from the second equation into the first equation to get:

-x^2 + 2x - 4 = 3x - 2

Simplifying and rearranging:

-x^2 - x - 2 = 0

(x + 1)(x - 2) = 0

x = -1 or x = 2

Substitute x = -1 into the second equation to get:

y = 3(-1) - 2 = -5

Substitute x = 2 into the second equation to get:

y = 3(2) - 2 = 4

Therefore, the solutions are (-1, -5) and (2, 4).

3) Y = x^2 - 3x - 5 and y = -2x + 1

Substitute y from the second equation into the first equation to get:

x^2 - 3x - 5 = -2x + 1

Simplifying and rearranging:

x^2 - x - 6 = 0

(x - 3)(x + 2) = 0

x = 3 or x = -2

Substitute x = 3 into the second equation to get:

y = -2(3) + 1 = -5

Substitute x = -2 into the second equation to get:

y = -2(-2) + 1 = 5

Therefore, the solutions are (3, -5) and (-2, 5).

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For the given options, the only element that not already appears in one of the pairs is c = 13, so that is the correct option.

The basic definition of a one-to-one function is the mapping of two sets. If each element in the range of a function g matches precisely one in the domain of g, then the function g is one-to-one. 1-1 is another way to represent one-to-one. A function or formula, such as f() connects the values of two variables in such a way that the values of the first variable's elements determine the values of the second variable in exactly the same ways.

We have:

{(1, 2), (2, 3), (3, 5), (4, 7), (5, 11), (6, c)}

We can see the number of the 2nd position is prime and in sequence so, we can easily determine that 2, 3, 5, 7, 11, and next will be 13.

For the given options, the only element that not already appears in one of the pairs is c = 13, so that is the correct option.

When every element in a function's range and domain match exactly one another, the function is said to be one-to-one. 1-1 is also used to indicate one-to-one. A function or formula, such as f() links the elements and values of one variable to those of another in such a way that the elements of the first variable exactly predict the elements of the second variable.

The complete question is-

For what value of c is the function one-to-one?

{ (1, 2), (2, 3), (3, 5), (4, 7), (5, 11), (6, c) }

a)2

b) 5

c)11

d)13

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According to the given information, the terminal side of an angle of 7 radians is in the second quadrant.

What is the terminal angle?

The terminal angle is the angle formed by the terminal side of an angle in standard position (i.e., with its initial side along the positive x-axis) and the nearest x-axis. It is typically measured in a counterclockwise direction from the positive x-axis.

An angle of 7 radians is greater than 2π radians (which is approximately 6.28 radians), so it corresponds to more than one full revolution around the unit circle.

To find the terminal side of an angle of 7 radians, we can subtract 2π radians (or 360 degrees) from 7 radians until the result is between 0 and 2π radians. We have:

[tex]$$7 \text{ radians} - 2\pi \text{ radians} \approx 1.716 \text{ radians}$$[/tex]

Since 1.716 radians is less than π radians (which is approximately 3.14 radians), the terminal side of an angle of 7 radians is in the second quadrant.

The terminal side of an angle of 7 radians is in the second quadrant.

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Answer:

x = 18

Step-by-step explanation:

[tex]\frac{6}{7}=\frac{x}{21} \\\\6*21=7*x\\\\126=7x\\\\x=18[/tex]

a) 6 adult tickets and 6 child tickets cost  is £102. b) 4 adult tickets and 7 child tickets cost  is  £77 c) 3 child tickets cost is £21

The elimination method involves manipulating these equations to eliminate one of the variables. This is done by multiplying one or both of the equations by a constant so that the coefficients of one of the variables are equal in both equations, but with opposite signs. The equations are then added or subtracted, depending on the signs, to eliminate one of the variables.

Let x be the cost of one adult ticket and y be the cost of one child ticket.

From the given information, we can write two equations:

2x + 2y = 38 (equation 1)

2x + 5y = 59 (equation 2)

To solve for x and y, we can use elimination method.

Multiplying equation 1 by 5 and equation 2 by 2, we get:

10x + 10y = 190

4x + 10y = 118

Subtracting equation 2 from equation 1, we get:

6x = 72

x = 12

Substituting x = 12 into equation 1, we get:

2(12) + 2y = 38

2y = 14

y = 7

Therefore, one adult ticket costs £12 and one child ticket costs £7.

a) 6 adult tickets and 6 child tickets:

Cost = 6(£12) + 6(£7) = £102

b) 4 adult tickets and 7 child tickets:

Cost = 4(£12) + 7(£7) = £77

c) 3 child tickets:

Cost = 3(£7) = £21

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Jethro must get a mark of 92 out of 100 for the last test to have a mean mark of at least 77 for all 6 tests.

To work out the least mark that Jethro needs to get for the last test, we can use the Mean formula: Mean = (Sum of all scores) / (Number of scores).

We know that the Mean of the 5 tests that Jethro has already sat is 74, so we can calculate the sum of the 5 test scores by multiplying the mean by the number of tests (74 x 5 = 370). We also know that Jethro wants his mean mark for all 6 tests to be at least 77, so the sum of all 6 test scores must be higher than (77 x 6 = 462).

To calculate the least mark Jethro needs to get for the last test, we can combine these two facts. We know the sum of the 5 test scores is 370, so the sum of all 6 test scores must be 462 or higher. This means the least mark Jethro needs to get for the last test must be (462 – 370) = 92. This means that Jethro must get a mark of at least 92 out of 100 for the last test to have a mean mark of at least 77 for all 6 tests.

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Answer:

  (D)

Step-by-step explanation:

You want the graph that shows speed decreasing to zero.

"Slowing down" means speed is decreasing. On a graph of speed that is indicated by a negative slope.

When an object is stopped, its speed is zero. On a graph of speed, points on the horizontal axis indicate the speed is zero.

The attached graph shows the speed of a car that is slowing to a stop.

Answer:

Step-by-step explanation:

If the value of the car has decreased by 44%, it still retains 56% of its value. Thus, the solution for the depreciation is 63,765(.56) = $35,708.40

Stephen's hourly pay will be $10.42 in two years, calculated by plugging the principal amount of $9.45, the rate of interest of 5%, and the number of times interest is compounded per year of 1 into the compound interest formula.

Stephen's current hourly pay is $9.45. In order to calculate his hourly pay in two years time, we need to use the compound interest formula. The formula is A = P (1 + r/n)^nt, where A is the amount after n years, P is the principal or initial amount, r is the rate of interest, and n is the number of times interest is compounded per year. In this case, the principal is $9.45, the rate of interest is 5%, and the number of times interest is compounded per year is 1. Plugging these into the formula yields A = 9.45(1 + 0.05/1)^2*1, which equals $10.42. That means Stephen's hourly pay will be $10.42 in two years.

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Answer:

A=[tex]\pi[/tex]r(r+h2+r2)=[tex]\pi[/tex]·7·(7+122+72)≈459.44884

rounded off to 500 square millimetres

Step-by-step explanation:

The total surface area of a cone is the combination of the curved surface as well as the base area of a cone. The formula to calculate the total surface area of the cone is:

TSA of cone = [tex]\pi[/tex]r^2 + [tex]\pi[/tex]r l = r (l + r) square units.

Answer:YES

Step-by-step explanation:

you should write 1 instead of every x and 5 instead of every y.

6*1+2*5=6+10=16

1+3*5=1+15=16

Answer ! :)

If Jazmin spent  $38.75 on all

Then it would 15miles ( 15m ) to get home.

As your adding the extra $5 which starts the process.

Basically a equation could be:

(5 + 2.25 = 7.25) + ( 2.25 x 14 )

Extra info if needed more explanation: ( 2.25 x 14 = 31.5 ) Which 31.5 + 7.25 = 38.75

Info on counting:

7.25, 9.50, 11.75, 14, 16.25, 18.50, 20.75, 23, 25.25, 27.50, 29.75, 32, 34.25, 36.50, 38.75

Based on the above, the 7 melons together will weigh about 21,000 grams.

We know that 2 melons of the same size together weigh 6 kg, therefore each melon weighs:

6kg / 2 = 3kg

To know how many grams the 7 melons weigh, we first need to know how many grams a kilogram weighs:

1kg = 1000g

So each melon weighs 3 kg * 1000 g/kg = 3000 g.

Therefore, 7 melons equal to the ones Alberto bought will weigh:

7 * 3000g = 21,000g

Therefore, the 7 melons together will weigh 21,000 grams.

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See text below

Alberto bought 2 melons of the same size and together they weigh 6 kg. How many grams will 7 melons equal to the ones Alberto bought weigh?

The length of the missing side of the given triangle ABC which is similar to triangle ILN would be = 7cm

Given that triangle ILN and ABC are similar this shows that their sides are equal in measurement too.

Therefore, LI = 25cm = AB = c

AC = 24 cm = LN = b

BC = X = IN = a

Using the Pythagorean formula;

C² = a² + b²

25² = a² + 24²

625 = a² + 576

625 -576 = a²

a² = 49

a = √49

a = 7cm

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Answer:

Step-by-step explanation:

2a = -b - d + c

     = c - b - d

x = c/2 - b/2 - d/2

The vector a in terms of other vectors for the given vectors in the image is  c/2 - b/2 - d/2.

A vector is a quantity that determines both an object, its magnitude, and its direction.

The resultant vector is obtained when the tail of one vector is attached to the head of another vector such the resultant vector is formed from the sum of the two vectors.

Let a be the total vector of the given figure.

The vectors are written as:

2a = -b - d + c

a = c - b - d

a = c/2 - b/2 - d/2

Hence, the vector a in terms of other vectors is  c/2 - b/2 - d/2.

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Answer:

Step-by-step explanation:

Impossible. Using simple logic skills, the only possible outcomes from rolling a dice are 1,2,3,4,5,6. Hence, rolling a 9 is an impossible outcome.

The total expense for producing 2 widgets is $18,006.00.

To find the total expense for producing 2 widgets, we need to find the total revenue from selling 2 widgets and then subtract the total expense of producing them.

The total revenue from selling 2 widgets is simply the price per widget times the quantity demanded at that price. Since the price is not specified in the problem, we can use the demand function to find it.

We know that the quantity demanded when 2 widgets are produced is:

q = 2

So we can solve the demand function for p:

q = -112p + 4,500

2 = -112p + 4,500

-112p = -4,498

p ≈ 40.16

Therefore, the price per widget is approximately $40.16.

The total revenue from selling 2 widgets at this price is:

revenue = price × quantity demanded

revenue = $40.16 × 2

revenue = $80.32

Now we need to find the total expense of producing 2 widgets. The problem gives us an expression for the total expense:

E = 3.00q + 18,000

Substituting q = 2, we get:

E = 3.00(2) + 18,000

E = $18,006.00

Therefore, the total expense for producing 2 widgets is $18,006.00.

Finally, we can calculate the profit as the difference between the revenue and the expense:

profit = revenue - expense

profit = $80.32 - $18,006.00

profit ≈ -$17,925.68

Since the profit is negative, Willie's Widgets would not make any profit from producing 2 widgets at this price. They would lose money.

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Therefore, the two possible values for AC are approximately 13.266 cm and 16.734 cm.

An assortment of similarly spaced out points in a plane make up a circle. The center is where the point is located, but the radius is the distance from the center. Two times the radius equals the diameter.

We can see that triangle ADC is a right triangle since it's inscribed in a semi-circle (the smaller circle). So we can use the Pythagorean theorem to find AC:

AC² = AD² - CD²

Since AD is the radius of the larger circle (15 cm) and CD is the radius of the smaller circle (7 cm), we have:

AC² = 15² - 7²

AC² = 176

AC = √(176)

AC ≈ 13.266 cm

So one possible value for AC is approximately 13.266 cm.

Now let's consider the other intersection point, D. We can see that triangle BDC is also a right triangle since it's inscribed in a semi-circle (the smaller circle). So we can use the Pythagorean theorem again to find BD:

BD² = BC² + CD²

Since BC is the radius of the larger circle (15 cm) and CD is the radius of the smaller circle (7 cm), we have:

BD² = 15² - 7²

BD² = 176

BD = √(176)

BD ≈ 13.266 cm

Since BD is a diameter of the larger circle, we have:

AC + BD = 2 * 15 = 30

So the other possible value for AC is:

AC = 30 - BD

AC ≈ 16.734 cm

Therefore, the two possible values for AC are approximately 13.266 cm and 16.734 cm.

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Step-by-step explanation:

let the smallest angle = x

tan x = 5/8

x = arctan 5/8

x = 32°

Answer: The smallest angle of the triangle is 32°

Step-by-step explanation:

Given: 

one side of the triangle= 8 cm

The other side of the triangle = 5 cm

Angle between AB and BC = 90°

⇒ ∠ABC = 90°

ΔABC is a right angled triangle

Use trigonometric function: For X

tanx= AB/CB

tanx= 8/5

x=tan-1 (8/5)

x= 58°

Use trigonometric function: For Y

tany= BC/AB

tany=5/8

y=tan-1 (5/8)

y = 32°

Making 32° be the smallest angle of the triangle

The given sequence 22, 12, 2, -8 is arithmetic, and the common difference is -10.

An arithmetic sequence is a sequence where the difference between any pair of consecutive terms is a constant knowed as the common difference, and if k is that common difference, we can write the recursive formula as:

a(n) = a(n - 1) + k

Here we have the sequence:

22, 12, 2, -8

Taking the differences between consecutive terms we get:

12 - 22 = -10

2 - 12 = -10

-8 - 2 = -10

The differences are all equal, then this is an arithmetic sequence, and the common difference is -10.

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