At a movie theater, 2 medium drinks and a large popcorn costs $13.85. At the same theater, a family bought 3 medium drinks and 2 large popcorns for a total of $23.90. Which system of equations could be used to find x, the cost of a medium drink and y, the cost of a large popcorn?
A) 2x + y = 13.85
3x + 2y = 23.9
B) 3x + 2y = 13.85
2x + y = 23.9

C) x + 2y = 13.85
3x + 2y = 23.9

D) 2x + y = 13.85
2x + 3y = 23.9

Answer:

a = 9 , b = 5 , c = 25

Step-by-step explanation:

using the Cosine rule in the triangle

a² = b² + c² - 2bc cosA

where a is the side opposite ∠ A and b, c the sides adjacent to ∠ A

here

a = y , b = x + 5 , c = 3x and A = Θ , then

y² = (x + 5)² + (3x)² - [ 2(x + 5) × 3x × cosΘ ]

y² = x² + 10x + 25 + 9x² - [ 6x(x + 5) × [tex]\frac{1}{6}[/tex] ]

y² = 10x² + 10x + 25 - x(x + 5)

y² = 10x² + 10x + 25 - x² - 5x

y² = 9x² + 5x + 25 ← in the form y² = ax² + bx + c

with a = 9 , b = 5 , c = 25

Answer:

Step-by-step explanation:

The math used shows how much cups it would take to make 4 batches, 3 cups. Therefor yes, he does have exactly enough to make 4 batches of muffins.

Answer:

No, you cannot simplify it.

Step-by-step explanation:

Because 2 and 3 are one of the lowest numbers that cannot be simplified.

The distance between the points H (-9, 1), and K (-1, 8) in the coordinate plane, obtained using the distance formula is; Distance: 10.63

The distance formula is a formula that is used to find the distance between two points on the coordinate plane.

The specified points are; H = (-9, 1), and K = (-1, 8)

The distance formula can be used to calculate the distance between the points as follows;

The distance formula is; d = √((x₂ - x₁)² + (y₂ - y₁)²)

Where; (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.

When, (x₁, y₁) = (-9, 1), and (x₂, y₂) = (-1, 8), we get;

d = √((-1 - (-9))² + (8 - 1)²) = √(8² + 7²) =

√(8² + 7²) = √(64 + 49) = √(113) ≈ 10.63 (rounded to the nearest hundredth)

The distance between the points is about 10.63 units

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The IQR is the best measure of variability for this data set, and its value is 4.

What is median?

Median is a measure of central tendency that represents the middle value in a dataset when the values are arranged in order of magnitude.

The correct answer is:

The IQR is the best measure of variability, and it equals 4.

The box plot shows the median (represented by the line in the box), the interquartile range (IQR) (represented by the length of the box), and the range (represented by the lines outside the box).

The IQR is the range between the first quartile (Q1) and the third quartile (Q3) of the data set, which contains the middle 50% of the data. In this case, the box extends from 17 to 21, indicating that Q1 is 17 and Q3 is 21. Therefore, the IQR is the difference between Q3 and Q1, which is 21-17=4.

The range is the difference between the maximum and minimum values in the data set, which is 27-10=17. However, the range can be affected by outliers and is not as robust as the IQR.

Therefore, the IQR is the best measure of variability for this data set, and its value is 4.

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Answer:

[tex] - \frac{2}{5} m( - \frac{4}{9} m - \frac{1}{7} ) = [/tex]

[tex] \frac{8}{45} {m}^{2} + \frac{2}{35} [/tex]

Answer:

6x² - 12x - 18

the factors are -18 and 6

6x² - 18x + 6x - 18

6x(x - 3) + 6 (x - 3)

(6x + 6) or (x - 3)

the factors are:

(6x + 6) or (x - 3)

After solving the given equation 1/4b - 2 = -1/2b + 4, the resultant value of b is 8/3 respectively.

An equation is a mathematical expression with two equal sides and an equal sign.

4 + 6 = 10 is an illustration of an equation.

On the left side of the equal sign, you can see 4 + 6, and on the right, you can see 10.

A formula exists in every equation.

Some equations do not have formulae.

Equations are designed to be solved for a variable.

We look over formulas.

So, we have the equation:

1/4b - 2 = -1/2b + 4

Now, solve the equation for b as follows:

1/4b - 2 = -1/2b + 4

1/4b = -1/2b + 6

1/4b + 2/1b = 6

1+8/4b = 6

9/4b = 6

b = 6 * 4/9

b = 2 * 4/3

b = 8/3

Therefore, after solving the given equation 1/4b - 2 = -1/2b + 4, the resultant value of b is 8/3 respectively.

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Since there are 52 cards in a standard deck, there are 52 possible outcomes for the first draw from the first deck. Similarly, there are 52 possible outcomes for the first draw from the second, third, and fourth decks. Therefore, the total number of outcomes for the first draw is 52 + 52 + 52 + 52 = 208.

For the second draw, there are only 51 possible outcomes for each deck, since one card has already been drawn. Therefore, the total number of outcomes for the second draw is 51 + 51 + 51 + 51 = 204.

For the third draw, there are only 50 possible outcomes for each deck, since two cards have already been drawn. Therefore, the total number of outcomes for the third draw is 50 + 50 + 50 + 50 = 200.

For the fourth draw, there are only 49 possible outcomes for each deck, since three cards have already been drawn. Therefore, the total number of outcomes for the fourth draw is 49 + 49 + 49 + 49 = 196.

The total number of outcomes for drawing one card each from four different standard decks of cards is the product of the outcomes for each draw: 208 x 204 x 200 x 196 = 338,401,792.

The least of the integers in arithmetic mean of n consecutive odd integers is 3.

The formula to calculate average or arithmetic mean of evenly distributed set is -

Average = sum of first and last term/2

Keep the values in formula to find the least integer. Since these are consecutively arranged, the first number will be the least number.

10 = first number + 17/2

First number + 17 = 10×2

Performing multiplication on Right Hand Side

First number + 17 = 20

First number = 20 - 17

Performing subtraction

First number = 3

Hence, the least integer is 3.

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When a study sample is representative of the target population, it allows for the results to be generalizable, meaning they can be applied to the broader population.

This is an important aspect of research, as it ensures the study's findings are relevant and useful beyond the specific sample studied.

When a study sample shows representativeness, it means that the sample accurately reflects the characteristics of the target population.

In this case, the study is most likely generalizable, with results that can be applied to the target population.
To achieve representativeness, researchers often use random sampling techniques, which help minimize the potential for sampling bias.

Sampling bias occurs when certain groups within the target population are over- or under-represented in the sample, which can lead to inaccurate conclusions.

In contrast, if a study used strict inclusion and exclusion criteria to limit the sample size or relied on specific inclusion criteria, it could result in a sample that is not representative of the target population.

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Answer: 163.6363...

Step-by-step explanation:

44% of c is 72, so then [tex]0.44*c=72\\[/tex].

Divide both sides by 0.44 to solve for c, and you get c=163.6363...

Tip: if you are ever trying to figure out what a number is a percentage of, like in this problem, just divide the number by the percent in decimal form.

Answer: The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.

For the original cone with radius r and height 12 cm, we have:

V = (1/3)πr^2h

V = (1/3)πr^2(12)

V = 4πr^2

For the second cone with 3 times the radius of the original cone and the same height of 12 cm, we have:

V' = (1/3)π(3r)^2h

V' = (1/3)π9r^2(12)

V' = 36πr^2

Therefore, the expression for the volume of the second cone is 36πr^2.

Step-by-step explanation:

Answer:

aₙ = 3n - 2

Step-by-step explanation:

The given sequence is an arithmetic sequence with the first term a₁ = 1 and the common difference d = 3.

The explicit formula for an arithmetic sequence is:

aₙ = a₁ + (n - 1)d

where aₙ represents the nth term in the sequence.

Substituting the values we have:

aₙ = 1 + (n - 1)3

Simplifying the equation we get:

aₙ = 3n - 2

Therefore, the explicit formula for the given sequence 1, 4, 7, 10, ... is aₙ = 3n - 2.

The three numbers are:

(-1 + √(97))/2, (-1 ± √(97))/2, (-1 ± 7√(97))/2

What is an arithmatic progression?

An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the preceding term. The fixed constant is called the common difference, denoted by d.

Let the three numbers in the GP be a/r, a, and ar, where a is the middle term.

Then we have:

a/r + a + ar = 26 (sum of GP)

(a/r + 5) + (a + 9) + (ar + 5) = 3(a + 4) (sum of AP)

Simplifying the second equation, we get:

a(r - 1) = 3

Substituting this into the first equation, we get:

a/r + a + ar = 26

1/r + 1 + r = 26/a

r^2 + r - 26/a = 0

Solving for r using the quadratic formula, we get:

r = (-1 ± √(1 + 104/a))/2

Since r > 0 (because the terms are in GP), we take the positive root:

r = (-1 + √(1 + 104/a))/2

Substituting this into the equation a(r - 1) = 3, we get:

a(-1 + √(1 + 104/a))/2 - a = 3

-a + a √(1 + 104/a) - 2 = 0

a √(1 + 104/a) = -a + 2

a² (1 + 104/a) = a² - 4a + 4

104a = 4a² - 4a + 4

a² - a + 1 = 26

(a - 1/2)² + 3/4 = 26

(a - 1/2)² = 97/4

a - 1/2 = ±√(97)/2

a = (1 ± √(97))/2

Since the terms are in GP, we can use the value of a to find the other two terms:

a/r = (1 ± √(97))/(2(-1 + √(1 + 104/a))/2) = (-1 ± √(97))/2

ar = (1 ± √(97))/(2(-1 - √(1 + 104/a))/2) = (-1 ± 7√(97))/2

Therefore, the three numbers are:

(-1 + √(97))/2, (-1 ± √(97))/2, (-1 ± 7√(97))/2

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The resulting triangle would have twice the area and all sides would be twice as long as the original triangle, but the angles would remain the same.

A triangle is a geometric shape that consists of three straight line segments or sides that are connected at three points called vertices. Triangles are two-dimensional and are the simplest polygon that can exist in Euclidean space. Triangles can be classified according to the size and shape of their sides and angles. Triangles are used in many different areas of mathematics, science, and engineering. They are also common in everyday life, for example in construction, architecture, and design.

Here,

To dilate a triangle with a center point C and a scale factor of 2, you would first draw a ray from C through each vertex of the triangle. Then, you would measure the distance from C to each vertex of the original triangle and multiply it by the scale factor (2 in this case) to find the distance from C to each corresponding vertex of the dilated triangle. Finally, you would draw lines connecting the dilated vertices to form the dilated triangle.

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Therefore , the solution of the given problem of trigonometry comes out to be -1 is the simplified formula.

Some claim that the fusion of various fields contributed to the development of astrophysics. With the aid of precise mathematical methods, many metric issues can be resolved or the output of a computation can be determined. The scientific investigation of all six fundamental geometric calculations is known as trigonometry. They are known by a number of names and abbreviations, such as sine, variance, angle, and others. (csc).

Here,

Let's first use trigonometric identities to separately simplify each term:

=> cos(x - 90°) equals sin(x)

=>  cos(x - 180°) = -cos(x)

=> tan(x) = cos(x) / sin(x)(x)

=>  x(cos(-)) = x(x)

=> (180° + x) sin = -sin(x)

When we replace the equation with these simplifications, we get:

=> cos(x) * sin(180° plus x) * cos(x) * (-cos(x)) + (sin(x) / cos(x))

=>  sin(180° plus x) * sin(-cos2(x))

=>  -cos²(x) - sin(x)*sin(180° + x) (using the equation sin(x)*sin(x)) = -sin^2(x))

Using the identity sin2(x) +cos2(x) = 1, we get = -1.

Consequently, -1 is the simplified formula.

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Answer:

O=0

T=-5

c = 0

R = -2

6 = 8

2 = 8

11=-3

Step-by-step explanation: Sorry for the late answer It took me a bit to solve all the problems but I hope this helps you!!

Your chance of being chosen to attend the festival is approximately: Probability = 1 / 23 ≈ 0.0435 or 4.35% when rounded to four decimal places.

To determine your chance of being chosen to attend the festival, you need to find the probability of your name being

drawn from the basket.

There are 22 other people in class, and each person, including you, will have their name on a piece of paper.

Total number of names in the basket = 22 (other people) + 1 (you) = 23

Since each person has an equal chance of being chosen, the probability of your name being drawn is:

Probability = (Number of favorable outcomes) / (Total number of outcomes)

Probability = 1 (your name) / 23 (total names)

So, your chance of being chosen to attend the festival is approximately:

Probability = 1 / 23 ≈ 0.0435 or 4.35% when rounded to four decimal places.

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A. The Utilization rate is 0.5714 or 57.14%, B. The average number of students in line is 0.9 students and C. The average time of waiting in line was 4.57 minutes.

The utilization rate refers to the proportion of time Judy Gumshoes spends processing student requests.

To calculate this rate, divide the average service time by the average time between arrivals. In this case, the average service time is 8 minutes, and students arrive every 14 minutes on average.

a) Utilization rate: 8 minutes / 14 minutes = 0.5714 or 57.14%

b) To determine the average number of students in line,

we can use the formula L = λW, where L represents the average number of students in the system, λ is the arrival rate (1 student per 14 minutes), and W is the average time a student spends in the system (waiting and being served). Since we know the utilization rate,

we can calculate the average waiting time (Wq) using the formula,

Wq = (U^2) / (1 - U), where U is the utilization rate.

In this case, Wq = (0.5714^2) / (1 - 0.5714) = 0.3265 or 32.65% of 14 minutes = 4.57 minutes.

W = Wq + service time

= 4.57 minutes + 8 minutes

= 12.57 minutes.

Finally, L = λW = (1 student / 14 minutes) * 12.57 minutes

= 0.9 students.

c) The average time of waiting in line is the same as the average waiting time in the system (Wq), which we calculated as 4.57 minutes.

In summary, the utilization rate of the service counter is 57.14%, the average number of students in line is 0.9, and the average waiting time in line is 4.57 minutes.

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Answer: 200 teachers attended the conference

Step-by-step explanation:

        To answer this question, we will find 5% of 4,000. A percent divided by 100 becomes a decimal, and "of" means multiplication in mathematics.

        5% / 100 = 0.05

        4,000 * 0.05 = 200 teachers attended the conference

The answer is 200

If 5% of teachers at a university attended a conference and 4000 teachers are enrolled at the university, about 200 teachers attended the conference. What is the problem asking for? The problem is asking us to calculate how many teachers attended a conference given that 5% of the total number of teachers at a university attended the conference and that the university has 4000 teachers .What is the formula for percentage? The formula for percentage is given by:(Part/Whole)*100Let T be the total number of teachers at the university, and let x be the number of teachers that attended the conference. We know that 5% of teachers at a university attended a conference, therefore:(5/100)*T = x,. Given that there are 4000 teachers in the university, we can substitute this value into the equation: (5/100)*4000 = x. Simplifying the equation, we get: x = 0.05 * 4000x = 200.

Therefore, about 200 teachers attended the conference.

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The equation is not a proportion, and the value of z in the equation is 17

Given that

2Z = Z + 17

To solve for z in the proportion:

2z = z + 17

We can start by simplifying the equation by subtracting z from both sides:

2z - z = 17

Simplifying further, we get:

z = 17

Therefore, the solution to the proportion is z = 17.

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The required volume of the cube is increasing at a rate of 15,000 cubic centimeters per second when the length of each edge is 50 centimeters and the edges are increasing at a rate of 2 centimeters per second.

Let us consider V be the volume of the cube,

And s be the length of one edge of the cube.

Volume of a cube is equal to,

V = s^3

To find how fast the volume is changing with respect to time,

Use the chain rule of differentiation,

dV/dt = dV/ds × ds/dt

where dV/dt is the rate of change of volume with respect to time,

dV/ds is the rate of change of volume with respect to the length of one edge of the cube,

And ds/dt is the rate of change of the length of one edge of the cube with respect to time.

ds/dt = 2 cm/s,

find dV/dt when s = 50 cm.

First, find dV/ds,

dV/ds = 3s^2

Then, we can plug in s = 50 cm,

dV/ds = 3(50)^2

          = 7500 cm^2

Finally, we can plug in the values we have to find dV/dt,

dV/dt = (dV/ds) × (ds/dt)

         = 7500 cm^2/s × 2 cm/s

         = 15,000 cm^3/s

Therefore, the volume of the cube is increasing at a rate of 15,000 cubic centimeters per second .

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Answer:

PQ = 165.60m

Step-by-step explanation:

Since triangles are similar

PQ/70 + 210 = 124.2/210

PQ = 124.2 (70 + 210)/210 = 165.60m

the length of NR is approximately 22.8, rounded to the nearest tenth.

To find the length of NR, we need to first identify any relationships between the chords and the arcs they intercept in the circle. From the diagram, we can see that chord PQ intersects arc ONR and chord NO intersects arc PQN.

There are two main properties we can use to solve this problem: the chord-chord power theorem and the angle-arc theorem. The chord-chord power theorem states that if two chords intersect in a circle, the product of the lengths of their segments is equal. That is:

(PQ)(QR) = (NO)(NR)

We know that PQ = 48 and NO = 21, so we can plug those values into the equation and solve for NR:

(48)(QR) = (21)(NR)

QR = (21NR)/48

To find QR, we can use the angle-arc theorem. This theorem states that the measure of an angle formed by two intersecting chords is equal to half the sum of the measures of the arcs they intercept. That is:

∠QNR = (arc PN + arc OQ)/2

We know that arc PN is equal to arc PQ (since they are intercepting the same arc), and we know that arc OQ is equal to the entire circumference of the circle minus arc PQ. The circumference of a circle is given by 2πr, where r is the radius. We don't know the radius of the circle, but we can find it using the Pythagorean theorem. We know that NO and PQ are both chords, so they must intersect at the center of the circle. Therefore, the line segment connecting the center of the circle to the midpoint of chord NO is a perpendicular bisector of NO. This gives us a right triangle with legs of 10.5 and 24 (half of 21 and half of 48). Using the Pythagorean theorem, we can find that the radius of the circle is approximately 26.2.

Now we can plug in our values for arc PN and arc OQ:

∠QNR = (arc PQ + (2πr - arc PQ))/2

∠QNR = πr

We know that QR is the side opposite the angle ∠QNR in right triangle QNR. Therefore, we can use the sine function to find QR:

sin(∠QNR) = QR/r

sin(πr) = QR/26.2

QR = 26.2sin(πr)

Now we can substitute this value for QR into our equation from the chord-chord power theorem:

(48)(26.2sin(πr)) = (21)(NR)

NR ≈ 22.8

Therefore, the length of NR is approximately 22.8, rounded to the nearest tenth.

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Answer:

(6-3)

Step-by-step explanation:

Answer: SHawn

Step-by-step explanation:

Answer:

Graph H

Step-by-step explanation:

intercept (0,2)

From the given percentage values, 326.25 points come from amount owed and length of credit history.

What is percentage?

Percentage is a method of representing a value as a fraction of 100. This is frequently used to make comparisons, demonstrate ratios, or determine a fraction of a whole.

The borrower's credit score is 725, and we need to find out how many points come from amount owed and length of credit history.

From the information given, we know that these two categories make up a total of 45% of the score calculation (30% for amount owed and 15% for length of credit history).

We can set up a proportion:

(30% + 15%) / 100% = x / 725

Simplifying the left side:

45% / 100% = 0.45

0.45 = x / 725

Multiplying both sides by 725:

x = 0.45 * 725

x = 326.25

Therefore, 326.25 points come from amount owed and length of credit history.

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Therefore, the general form of the circle is:

[tex]x^2 + y^2 - 10x-10y+34=0[/tex].

A circle is a geometric shape that is defined as the set of all points in a plane that are at a fixed distance (called the radius) from a given point (called the center). The distance from the center to any point on the circle is always the same. Circles are often represented using the symbol "O" or "⚪" and are important in mathematics, geometry, and physics. They have many properties, such as circumference (the distance around the circle), area (the space inside the circle), and diameter (the distance across the circle through the center). Circles are used in various fields, including architecture, engineering, and art.

To convert the standard form of a circle to the general form, we expand and simplify the equation as follows:

[tex](x - 5)^2 + (y - 5)^2 = 16[/tex]

[tex]x^2 - 10x + 25 + y^2 - 10y + 25 = 16[/tex] (using the identity [tex](a - b)^2 = a^2 - 2ab + b^2)[/tex]

[tex]x^2 + y^2 - 10x-10y+34=0[/tex]

Therefore, the general form of the circle is:

[tex]x^2 + y^2 - 10x-10y+34=0[/tex]

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