An 80-ounce bottle of apple juice contains 32 ounces of water.

Part A: equate the expressions 6x - 8 = 3x - 5

collect the like terms 6x -3x = -5 + 8

Divide by the coefficient x = 1

Part B: The values are (1, -3)

To determine the value of the variables,  we need to consider the equations.

From the information given, we have the equations given as;

y = 3x − 5

y = 6x − 8

Now, equate the expressions, we get;

6x - 8 = 3x - 5

collect the like terms, we have;

6x - 3x = -5 + 8

add or subtract the like terms

3x = 3

Divide by the coefficient

x = 1

Substitute the value

y= 6(1) - 9

expand the bracket

y = 6 - 9 = -3

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Derek is saving money for the down payment on a used car. So far, he has saved $200. His parents have said they help him put aside money, they will pay him 6% interest each month on the money Derek has saved. It will take Derek approximately 28 months to save at least $1,000 for the down payment on the used car if the only additions to his savings account are his parents' interest payments.

Month Process

0 Derek saves $200 (initial savings)

1 Derek receives 6% interest on $200 (interest = $200 * 0.06 = $12), total savings = $200 + $12 = $212

2 Derek receives 6% interest on $212 (interest = $212 * 0.06 = $12.72), total savings = $212 + $12.72 = $224.72

3 Derek receives 6% interest on $224.72 (interest = $224.72 * 0.06 = $13.48), total savings = $224.72 + $13.48 = $238.20

4 Derek receives 6% interest on $238.20 (interest = $238.20 * 0.06 = $14.29), total savings = $238.20 + $14.29 = $252.49

5 Derek receives 6% interest on $252.49 (interest = $252.49 * 0.06 = $15.15), total savings = $252.49 + $15.15 = $267.64

6 Derek receives 6% interest on $267.64 (interest = $267.64 * 0.06 = $16.06), total savings = $267.64 + $16.06 = $283.70

It will take Derek approximately 28 months to save at least $1,000 for the down payment on the used car if the only additions to his savings account are his parents' interest payments.

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The notation which is the most appropriate notation when calculating conditional probabilities is option B.

The appropriate notation for calculating conditional probabilities is often denoted using the vertical bar symbol "|". This symbol represents "given" or "conditional on."

In a mathematical expression, the conditional probability of event A given event B is written as P(A | B), where "P" stands for probability. This notation indicates that the probability of event A occurring is being calculated assuming that event B has already occurred.

Thus option B is correct.

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Answer: B

Step-by-step explanation:

The coefficients 9 & 6, divided by 3, equal 3 & 2.

For the exponents, if you divide the entire equation by xy, you need to subtract the corresponding exponents of x and y. Subtract the denominator's exponent from the numerator's exponent.

So, for the first part x^8/x^1=x^7

y^2/y^1=y

For the second part: x^3/x=x^2

y^4/y=y^3

So, if you put the coefficients and both parts of the equation together, you get 3x^7y - 2x^2y^3 (answer letter B).

Hope this helped!

Answer:

Let’s solve this problem step by step. Since a + b = 28 and a = 2b + 7, we can substitute the value of a in the first equation to get 2b + 7 + b = 28. Solving for b, we get 3b + 7 = 28, which means 3b = 21 and b = 7. So, the second number is 7.

The two other positive angles of rotation that take A to B are 19π/6 and 31π/6.

The unit circle is rotationally symmetric, meaning that any angle of rotation on the unit circle will take a point to another point on the unit circle with the same distance from the origin. This means that if we rotate point A by 7π/6 radians, we can also rotate it by 19π/6 or 31π/6 radians and end up at the same point B.

The point A has polar coordinates (1, 0). This means that it is located at a distance of 1 unit from the origin and has an angle of 0 radians with the positive x-axis.

The point B has polar coordinates (√3/2, 1/2). This means that it is located at a distance of √3/2 units from the origin and has an angle of 7π/6 radians with the positive x-axis.

If we rotate point A by 19π/6 radians, we get a point with polar coordinates (√3/2, -1/2). This point is located at the same distance from the origin as point B and has the same angle with the positive x-axis.

Therefore, the two other positive angles of rotation that take A to B are 19π/6 and 31π/6.

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The total invoice is $788.80, we find this by find the cost of each item.

To find the invoice total, we need to calculate the total cost of each item.

For the tablecloth:

Quantity: 16

Unit Price: $35.00

Total Cost = Quantity × Unit Price = 16 × $35.00 = $560.00

For the rings:

Quantity: 8

Unit Price: $6.50

Total Cost = Quantity × Unit Price = 8 × $6.50 = $52.00

For the cups (cases):

Quantity: 4 cases

Unit Price: $25.30

Total Cost = Quantity × Unit Price = 4 × $25.30 = $101.20

For the bowls:

Quantity: 12

Unit Price: $6.30

Total Cost = Quantity × Unit Price = 12 × $6.30 = $75.60

Now, let's calculate the invoice total by adding up the total costs of each item:

Invoice Total = $560.00 + $52.00 + $101.20 + $75.60

= $788.80

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The volume of the bar magnet with dimensions 0.5 in. (length), 2.25 in. (width), and 0.25 in. (thickness) is 0.28125 cubic inches.

The expression for the volume of a bar magnet can be calculated by multiplying its length, width, and thickness.

In this case, the dimensions provided are:

Length = 0.5 in.

Width = 2.25 in.

Thickness = 0.25 in.

To find the volume, we multiply these dimensions together:

Volume = Length × Width × Thickness

= 0.5 in. × 2.25 in. × 0.25 in.

Simplifying this expression, we get:

Volume = 0.28125 in³

Therefore, the volume of the bar magnet with dimensions 0.5 in. (length), 2.25 in. (width), and 0.25 in. (thickness) is 0.28125 cubic inches.

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The maximum revenue, to the nearest whole dollar, is $180,625 (option c).

To find the maximum revenue, we need to determine the vertex of the quadratic function represented by the revenue equation.

The revenue equation is given as:

R(p) = -4p² + 1700p

The vertex of a quadratic function in the form of f(x) = ax² + bx + c can be found using the formula:

x = -b / (2a)

For the given revenue equation, a = -4 and b = 1700. Plugging these values into the formula, we have:

p = -1700 / (2 × -4)

p = -1700 / -8

p = 212.5

The maximum revenue occurs at p = 212.5.

To find the maximum revenue, we substitute this value back into the revenue equation:

R(p) = -4(212.5)² + 1700(212.5)

R(p) = -4(45256.25) + 361250

R(p) = -181025 + 361250

R(p) = 180625

Therefore, the maximum revenue, to the nearest whole dollar, is $180,625 (option c).

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The value of an automobile company's stock fell 5 points over the last month.  The integer that represents the change in the stock's value is -5.

In the context of stock market performance, positive values typically indicate an increase in the stock's value, while negative values indicate a decrease. In this case, since the stock's value fell by 5 points, we assign a negative sign to represent the change.

By convention, negative numbers are used to represent decreases or losses, while positive numbers represent increases or gains. When the stock's value decreases, it is common to use negative integers to signify the change.

In this scenario, a decrease of 5 points is denoted by the negative sign (-) followed by the numerical value 5. Therefore, the integer -5 represents the change in the stock's value.

Using this notation helps provide clarity and consistency in understanding the direction and magnitude of changes in stock prices. It allows investors, analysts, and market participants to communicate and interpret market movements accurately and effectively.

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This one's a special case of a right angled triangle with sides (3, 4, and 5 units)

[tex]\qquad\displaystyle \tt \dashrightarrow \: 5 \sin \bigg( \frac{ \pi}{3} x \bigg) = 3[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \sin \bigg( \frac{ \pi}{3} x \bigg) = \frac{3}{5} [/tex]

Now, check the triangle, sin 37° = 3/5

therefore,

[tex]\qquad\displaystyle \tt \dashrightarrow \: \sin \bigg( \frac{ \pi}{3} x \bigg) = \sin(37 \degree) [/tex]

[ convert degrees on right side to radians ]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \sin \bigg( \frac{ \pi}{3} x \bigg) = \sin \bigg(37 \degree \times \frac{ \pi}{180 \degree} \bigg ) [/tex]

There are three more possible values as :

[tex]\qquad\displaystyle \tt \dashrightarrow \: \sin( \theta) = \sin(\pi - \theta) [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: sin( \theta) = \sin \bigg( { 2\pi}{} + \theta \bigg) [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: sin( \theta) = \sin \bigg( \frac{ 3\pi}{} - \theta\bigg) [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \frac{ \pi}{3} x = 37 \times \frac{ \pi}{180} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: x = 37 \times \frac{ \cancel \pi}{180} \times \frac{3}{ \cancel \pi} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: x = \frac{37}{60} [/tex]

or in decimals :

[tex]\qquad\displaystyle \tt \dashrightarrow \: x = 0.616666... = 0.6167[/tex]

[ 6 repeats at third place after decimal, till four decimal places it would be 0.6167 after rounding off ]

similarly,

[tex]\qquad\displaystyle \tt \dashrightarrow \: \pi - \frac{ \pi}{3} x = 37 \times \frac{ \pi}{180} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \pi \bigg(1 - \frac{x}{3} \bigg ) = 37 \times \frac{ \pi}{180} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 1 - \frac{x}{3} = \frac{37}{180} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: - \frac{x}{3} = 0.205 - 1[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \frac{x}{3} = 0.795[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: x = 3 \times 0.795[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: x = 2.385[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 2\pi + \frac{ \pi}{3} x = 37 \times \frac{ \pi}{180} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \pi \bigg(2 + \frac{x}{3} \bigg ) = 37 \times \frac{ \pi}{180} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 2 + \frac{x}{3} = \frac{37}{180} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \frac{x}{3} = 0.205 - 2[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \frac{x}{3} = - 1.795[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: x = 3 \times -1 .795[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: x = -5.385[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 3 \pi - \frac{ \pi}{3} x = 37 \times \frac{ \pi}{180} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \pi \bigg(3 - \frac{x}{3} \bigg ) = 37 \times \frac{ \pi}{180} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 3 - \frac{x}{3} = \frac{37}{180} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: - \frac{x}{3} = 0.205 - 3[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \frac{x}{3} = 2.795[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: x = 3 \times 2.795[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: x = 8.385[/tex]

" x can have infinite number of values here with the same result, here are the four values as you requested "

I hope it was helpful ~

Hello!:

24/12 - 8

= 2 - 8

= -6

The amplitudes of functions are a) 8 and b) 6.

Given are the functions we need to determine the amplitude of function,

a) y = 8 Sin (x/2) + 3

b) y = 6 Cos x + 2

So,

To determine the amplitude of a trigonometric function, you can follow these steps:

For a sine function of the form y = A×sin(Bx + C) + D:

The amplitude is equal to the absolute value of the coefficient A.

For a cosine function of the form y = A×cos(Bx + C) + D:

The amplitude is equal to the absolute value of the coefficient A.

Let's apply these steps to the given functions:

a) y = 8×sin(x/2) + 3

The coefficient of sin in this function is 8, so the amplitude is |8| = 8.

Therefore, the amplitude of function a) is 8.

b) y = 6×cos(x) + 2

The coefficient of cos in this function is 6, so the amplitude is |6| = 6.

Therefore, the amplitude of function b) is 6.

Hence the amplitudes of functions are a) 8 and b) 6.

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Answer:

Mr Kalada's present age is 45 years old

Step-by-step explanation:

Answer:

x = 6
JN = 10

Step-by-step explanation:

if the values of SN and SP are the same, so are the corresponding sides next to the right angles.
which means:
if SP = SN, then JK = LM.
Use the reverse version of the function for calculating LM (3x + 2).
-2 * 1/3
Now apply that function to 20.
(20-2)/3
= 18/3
= 6.
x = 6
Now JK is a chord.
and Radius SQ intersects JK at a perpendicular right angle (at point N).
So JN would be half the length of JK.
20/2 = 10.
JN = 10

By algebra properties, the simplified form of the trigonometric equation cot² x / (csc x - 1) is  csc x + 1.

In this problem we find the case of a trigonometric equation, whose simplified form must be found. The simplification can be done both by algebra properties and trigonometric formulas. Now we proceed to determine the simplificated form:

cot² x / (csc x - 1)

(csc² x - 1) / (csc x - 1)

[(csc x + 1) · (csc x - 1)] / (csc x - 1)

csc x + 1

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The area of the triangle whose dimensions are given in a vertices form would be = 9

To calculate the area of the triangle, the formula that should be used is the coordinate geometry formula: area = the absolute value of Ax(By - Cy) + Bx(Cy - Ay) + Cx(Ay - By) divided by 2

Where; W=A, X= B, Y = C

Ax= 8

By= -2

Cy= -4

BX= 5

Ay = -8

CX= 3

That is:

= 8(-2--4)+5(-4--8)+3(-8--2)/2

= 16+20-18/2

= 18/2

= 9

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Area of shape 1:

Area of triangle = 1/2 × b × h

Area of triangle = 1/2 × 6 × 8

Area of triangle = 24 cm²

Area of shape 2:

Area of Semicircle = πr²/2

Radius of Semicircle = Diameter / 2

Radius of Semicircle = 3 cm

Then,

Area of semicircle = π(3)²

Area of semicircle = 9π cm²

Total Area = Area of semicircle + Area of triangle

Total Area  = (24 + 9π) cm²

Total Area = 52.27 cm² .

Hence Total area of the figure combining triangle and semicircle is 52.27 cm².

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Answer:

There are 6 countries with more than 60 confirmed cases

Step-by-step explanation:

- Draw your stem and leaf plot

-There is a stem without a leaf (There are no values in this class)

-Count the values in the stem and leaf plot and ensure they are 15. Each value represents the number of confirmed case in a county.

There are six numbers greater than 60 which are 62,63,63,67,65 and 97

Hence, there are 6 countries which have more than 60 confirmed cases

Answer:

measure of angle 7 + measure of angle 8 = 180°.

Step-by-step explanation:

5(2x – 3) = x – 15 + 9x

10x - 15 = x - 15 + 9x

10x - 9x - x = -15 + 15

0 = 0

Answer = A) Infinity many solutions

The new height above sea level would be Option (C) 3980m.

New sea-level = Previously given sea level + 20m

Given,

Height above sea previously = 4000m

Increase in sea level = 20m

Now, since there has been a rise in sea level due to global warming the distance between the previous 4000m mark and the sea level would decrease. This shall now be less than 4000m. The distance/height would reduce by 20m. It can also be observed that the distances below the sea level would increase due to a rise in the sea level mark.

∴ New height above sea level = 4000m - 20m

                                                 = 3980m.

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Answer:

B). No two lines are parallel

Step-by-step explanation:

In spherical geometry, there are no parallel lines. This is because any two lines on a sphere will eventually intersect.

The required median is 110- 120 and the upper quartile of the scores is 110-120.

Given that, the data from the histogram is

CI                 |             f

80-90          |            5

90-100,        |            3

100-110,        |            4

110-120,        |            4

120-130,       |             3

130-140,       |             3

140-150,       |             2

150-160        |             2

To find the median and upper quartile of the given scores, make the cumulative frequency table.

The cumulative frequency table is given below.

CI                 |             f               |             cf            

80-90          |            5               |             5

90-100,        |            3               |             8

100-110,        |            4               |            12

110-120,        |            4               |             16

120-130,       |             3               |             19

130-140,       |             3               |             22

140-150,       |             2               |             24

150-160        |             2               |             26

N = 26.

The median is the N/2 th score when N is even.

Median  = 13th score = 110-120 scores

To find the upper quartile, the product of (3/4 and N )th score gives the upper quartile.

Upper quartile = (3/4 x 26)th score.

Upper quartile = 19.5th  score.

The upper quartile score range is 110 - 120.

Therefore,the required median is 110- 120 and the upper quartile of the scores is 110-120.

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The volume of the largest piece is 10, 597. 5 cm³

The largest part of the cone takes the shape of a cylinder.

Now, the formula for calculating the volume of a cylinder is expressed as;

V = πr²h

The parameters of the formula are enumerated as;

Now, substitute the values, we get;

Diameter = 2 radius

Radius = 30/2

Radius = 15cm

Height = 15cm

Now, substitute the values, we get;

Volume = 3.14 × 15² ×15

Find the square value and substitute, we have;

Volume = 10, 597. 5 cm³

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The total cost of producing the additional units is $767.50.

In order to determine the total cost of producing 10 additional units assuming 2 units are currently being produced, we would have to integrate the marginal cost function for this handbag manufacturer with respect to x, and over the interval [10, 2].

Based on the information provided above, the marginal cost function for this handbag manufacturer is given by this function;

C'(x) = 0.046875x² − x + 100

₂∫¹⁰C'(x)dx = ₂∫¹⁰(0.046875x² − x + 100)dx

₂∫¹⁰C'(x)dx = ₂∫¹⁰(0.046875x²)dx − ₂∫¹⁰(x)dx + ₂∫¹⁰(100)dx

₂∫¹⁰C'(x)dx = 0.046875x³/3|¹⁰₂ - x²/2|¹⁰₂ + 100x|¹⁰₂

₂∫¹⁰C'(x)dx = 0.015625(10³ - 2³) - 1/2(10² - 2²) + 100(10 - 2)

₂∫¹⁰C'(x)dx = 0.015625(1000 - 8) - 0.5(100 - 4) + 100(8)

₂∫¹⁰C'(x)dx = 0.015625(992) - 0.5(96) + 800

₂∫¹⁰C'(x)dx = 15.5 - 48 + 800

₂∫¹⁰C'(x)dx = $767.50

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The increase in certain individuals leads to:

An increase in certain individuals within a bacterial population can lead to:

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Using the trigonometric ratios:

(1). sinA = 5/13, cosA = 12/13 and tanA = 5/13. And by Pythagoras rule:

(2). x = √116

(3). x = 90

The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.

The basic trigonometric ratios includes;

sine, cosine and tangent.

(1). sinA = 5/13 {opposite/hypotenuse}

cosA = 12/13 {adjacent/hypotenuse}

tanA = 5/13 {opposite/adjacent}

By Pythagoras rule

(2). x = √(4² + 10²)

x = √(16 + 100)

x = √116

(3). 106² = x² + 56²

x² = 106² - 56²

x = √(11236 - 3136)

x = √8100

x = 90

Therefore, using trigonometric ratios:

(1). sinA = 5/13, cosA = 12/13 and tanA = 5/13. And by Pythagoras rule;

(2). x = √116

(3). x = 90

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