Dependent and Independent Variables
Can you think of a relationship between two quantities in your life? How could you apply your knowledge of dependent and independent variables to the scenario?

Here,

We know that,

[tex] \large \bf \: Distance= \sqrt{(x_2 - x_ 1) {}^{2} + (y_ 2 -y_1) {}^{2} } [/tex]

So ,

Putting the value of x and y in the given formula

We get,

[tex]\large\sf{ \: = \sqrt{ \{ \: - 1 - ( - 9) \} {}^{2} + (8 - 8) {}^{2} } \: }[/tex]

[tex]\large\sf{ = \sqrt{(8) {}^{2} + (0) {}^{2} } }[/tex]

[tex]\large\sf{ \: = \sqrt{64 + 0} }[/tex]

[tex]\large\sf{ = \sqrt{64} }[/tex]

[tex]\large\bf \blue { = 8 \: unit}[/tex]

Thus,

[tex] {\underline {\underline{ \rule{200pt}{8pt}}}}[/tex]

We can conclude that cosθ = sin(90° - θ) for any right triangle with angle θ and its complementary angle (90° - θ).

To prove that cosθ = sin(90° - θ), we can use the properties of right triangles and their angles.
Consider a right triangle ABC, where angle A is θ, angle B is (90° - θ), and angle C is 90°.
According to the definition of trigonometric functions in a right triangle:
- cosθ = adjacent side (AB) / hypotenuse (AC)
- sin(90° - θ) = opposite side (AB) / hypotenuse (AC)
From the given definitions, we can see that both cosθ and sin(90° - θ) have the same ratio of side lengths, which is AB/AC.
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2081 is the estimated population of the deer as per the given table.

We assume that the characteristics of the sample of the deer population match the characteristics of the population as a whole. This will be the case if the sample is random: each deer is equally likely to be included in the sample.

So, if 85 of the 110 marked deer show up in the sample, we assume that the sample is 85/110 of the entire population.

For a sample size of 1608 deer, that means the population is estimated to be ...

 1608 = 85/110 × population

 population = 1608 × 110/85 = 2081

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The minimum number of times Johnny needs to use to the wheelbarrow until the delivery truck is filled up to 70%, whereby the shape of the truck is a rectangular prism is about 41 times

A rectangular prism is a three dimensional geometric figure with three pairs of parallel facing sides, and perpendicular adjacent faces.

Whereby the wheelbarrow and the truck are rectangular prisms, we get;

The dimensions of the wheel barrow are;

2 feet by 3 feet by 1.5 feet

The dimensions of the truck are;

11 feet by 8 feet by 6 feet

The amount of soil John puts in the truck = 70% of the volume of the truck

Therefore;

The amount of soil John puts in the truck = 11 feet × 8 feet × 6 feet × 70%

11 feet × 8 feet × 6 feet × 70% = 528 ft³ × 70% = 369.6 ft³

The number of times to use the wheelbarrow, n, is therefore;

n = 369.6 ft³ ÷ (2 ft × 3 ft × 1.5 ft) ≈ 41.0

The minimum number of times Johnny needs to use the wheelbarrow until the delivery truck is 70% filled is therefore 41  = 41 times

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The given equation is true when a and b are integers and x is not equal to 0.

The statement "xᵃ + xᵇ = x⁽ᵃ⁺ᵇ⁾ " is always true when a=b.

In this case, the exponents of x on both sides of the equation are equal, and the equation simplifies to xᵃ + xᵃ = x²ᵃ , which holds true.

When a=b, the exponents are the same, and the equation is satisfied.

However, the statement is not always true for all values of a and b. If a and b are not equal, then the equation does not hold true in general.

The statement is also not true when a=0 and b=0, as both sides of the equation become 1, but the equation becomes 1 + 1 = 1, which is not true.

Therefore, the statement "xᵃ + xᵇ = x⁽ᵃ⁺ᵇ⁾ " is always true when a=b, but not true in general for all values of a and b.

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Answer: The total cost will be equal to $2839.16. The expression to calculate the total cost is 119.92(m)+81=C(t).

Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.

Given that a person buys a phone for ​$81 and signs up for a​ single-line phone plan with 2000 monthly anytime minutes. The plan costs ​$119.92 per month.

The expression will be formed as below:-

119.92(m)+81=C(t)

119.92(23)+81=C(t)

2839.16=C(t)

Therefore, the total cost will be equal to $2839.16.

We find that Mary owes $13,564 in income tax. So the correct option is (a) $13,564. Option a

Mary's taxable income falls between $31,850 and $77,100. This means we apply the tax rate relevant to this bracket, which is $4,386.25 plus 25% of the amount over $31,850.

First, we need to subtract $31,850 from Mary's income to find the amount that we need to apply the 25% tax rate to:

$68562 - $31850 = $36712

Next, we take 25% of this result:

0.25 * $36712 = $9178

Then we add the base tax for this bracket ($4,386.25) to this result to get the total tax:

$4386.25 + $9178 = $13564.25

Rounding this to the nearest dollar, we find that Mary owes $13,564 in income tax. So the correct option is (a) $13,564.

Option  a

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Answer:

Step-by-step explanation:

To determine the time it takes for the same hose to fill the larger aquarium, we need to compare the volumes of the two aquariums and consider the rate at which the hose fills the smaller one.

The volume of a rectangular prism (such as an aquarium) is given by the formula: Volume = length × width × height.

Let's calculate the volume of the two aquariums:

For the first aquarium:

Volume = 9 inches × 13 inches × (height not given)

For the second (larger) aquarium:

Volume = 24 inches × 30 inches × 34 inches

Since the height of the first aquarium is not provided, we cannot directly compare the volumes. However, we can determine a proportion between the two based on the known measurements.

If the time it takes to fill the first aquarium is 2 minutes, we can establish a ratio:

(9 inches × 13 inches × height of the first aquarium) / (24 inches × 30 inches × 34 inches) = 2 minutes / x minutes

To find the value of x, the time it takes to fill the larger aquarium, we rearrange the equation:

x = (24 inches × 30 inches × 34 inches) / (9 inches × 13 inches × height of the first aquarium) * 2 minutes

Without knowing the height of the first aquarium, we cannot determine the exact time it takes to fill the larger aquarium.

Answer:

[tex]\sf g(x) = f(x\; \boxed{+ 3}\;) \;\boxed{- 6}[/tex]

Step-by-step explanation:

A translation is a transformation that moves every point of a figure the same distance and in the same direction. This means that the size, shape, and orientation of the figure are preserved, but its position is changed.

From inspection of the given diagram, we can see that the size, shape and orientation of the graph of function g is the same as that of the graph of function f, but its position has changed.  Therefore, the transformation is a translation.

We can use the vertex of both graphs to determine the translation.

Therefore, the graph of function f has been moved 3 units left and 6 units down to create the graph of function g.

When we move "a" units left, we add the value of "a" to the x-value of the function.

When we move "a" units down, we subtract the value of "a" from the function.

Therefore:

[tex]\sf g(x) = f(x\; \boxed{+ 3}\;) \;\boxed{- 6}[/tex]

Answer:

Perimeter is 20.57 yards.

Step-by-step explanation:

Perimeter is the distance around the outside of a shape. This shape is a semicircle, that is, half of a circle.

We can calculate the circumference of the circle (that's the distance around the whole circle) and then cut it in half. Then add on the flat, straight side.

Circumference is:

C = pi•d

C = pi•8

C = 25.1327

This is the distance around the whole circle. For the curved part of our shape, we only need half of that.

25.1327/2

= 12.56637

Lastly, we add on the straight side, which was given as 8 yds.

12.56637 + 8

= 20.56637

Round to two decimal places.

20.57

The perimeter is 20.57yds.

The number that belongs in the green box is 45.65

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

The triangle

If the number that belongs in the green box is x, then we have

6.78/sin(29) = 10/sin(x)

Take the inverse of both sides

sin(29)/6.78 = sin(x)/10

This gives

sin(x) = 10 * sin(29)/6.78

Evaluate

sin(x) = 0.7151

Take the arc sin of both sides

x = 45.65

Hence, the number that belongs in the green box is 45.65

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[tex]\huge\mathcal{\fcolorbox{Aqua}{azure}{\red{➳Answer}}}[/tex]

Base of the Parallelogram=14m

Height of the Parallelogram=8m

∴ Are of the Parallelogram= Base × height

= (14×8) sq. m

=112 sq.m

Now,

Diameter of the circle (unshaded portion)=8m

Radius of the circle= Diameter/2 =8/2 m= 4m

So, radius=4m

∴ Area of the circle = πr²

=(22/7 × 4 × 4)sq.m

=352/7 sq.m

=50.2857 sq.m

=50.286 sq.m

∴ Are of the Unshaded portion= Are of the Parallelogram - Area of the circle

=(112-50.286) sq.m

= 61.714 sq.m (Ans)

∴The area of the shaded portion is 61.714 sq.m

Hope it helps you

[tex]\red{\rule{200pt}{5pt}}[/tex]

[tex]\bold{Thank ~you~:)}[/tex]

___________________________________

Given:-

Base of the Parallelogram=14m

Height of the Parallelogram=8m

∴ Are of the Parallelogram= Base × height

= (14×8) sq. m

=112 sq.m

Now,

Diameter of the circle (unshaded portion)=8m

Radius of the circle= Diameter/2 =8/2 m= 4m

So, radius=4m

∴ Area of the circle = πr²

=(22/7 × 4 × 4)sq.m

=352/7 sq.m

=50.2857 sq.m

=50.286 sq.m

∴ Are of the Unshaded portion= Are of the Parallelogram - Area of the circle

=(112-50.286) sq.m

= 61.714 sq.m (Ans)

∴The area of the shaded portion is 61.714 sq.m

___________________________________

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Answer:

The is is 1

Step-by-step explanation:

The problem asks to convert a volume of 8000 [tex]cm^3[/tex] to [tex]m^3[/tex] which is 0.008 [tex]m^3[/tex].

Unit conversion is the process of converting a quantity expressed in one unit of measurement into another equivalent quantity expressed in a different unit of measurement.

To do this conversion, we need to use the fact that 1 m = 100 cm (or equivalently, 1 cm = 0.01 m) and that volume is a cubic measure.

So, first we need to convert the volume from cubic centimeters to cubic meters. To do this, we can divide the volume in [tex]cm^3[/tex] by the number of [tex]cm^3[/tex] in 1 [tex]m^3[/tex].

1 [tex]m^3[/tex] = (100 cm)^3 = 1,000,000 [tex]cm^3[/tex]

Therefore,

8000 [tex]cm^3[/tex] / 1,000,000 [tex]cm^3/m^3[/tex] = 0.008 [tex]m^3[/tex]

Thus, the volume of 8000 [tex]cm^3[/tex] is equal to 0.008 [tex]m^3[/tex].

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The dollar value of the stock that Raquel wants to purchase is $1,833.33.

If the margin requirement is 55%, then the amount of cash wanted is 45% of the total value of the inventory.

Consequently, we will set up the following equation like this:

0.45x = 825

Wherein:

x is the dollar value of the stock Raquel wants to purchase.

To solve for x, we are able to divide both aspects of the equation by way of 0.45:

x = 825 ÷ 0.45

x = $1,833.33

Therefore, the dollar value of the stock that Raquel wants to purchase is $1,833.33.

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The distance from the town in SC to Washington DC is,

⇒ 32.6 units

We have to given that;

Washington DC is the midpoint of Maine and South Carolina. If a town in SC is located at (-2,-20) and a town in Maine is located at (18,42),

Since, We know that;

The distance between two points (x₁ , y₁) and (x₂, y₂) is,

⇒ d = √ (x₂ - x₁)² + (y₂ - y₁)²

Hence, We get;

The coordinate for Washington is,

⇒ (18 - 2)/2, (- 20 + 42) / 2

⇒ (8, 11)

The distance from the town in SC to Washington DC is,

⇒ d = √ (8 - (- 2))² + (11 - (- 20))²

⇒ d = √10² + 31²

⇒ d = √100 + 961

⇒ d = √1061

⇒ d = 32.6 units

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The measure of angle U is given as follows:

m < U = 66º.

The Law of Cosines states that for any triangle with sides a, b, and c and angle C opposite to side c, the following equation is true to obtain the missing length:

c² = a² + b² - 2abcos(C)

Hence the length of side v is obtained as follows:

v² = 22² + 79² - 2 x 22 x 79 x cosine of 51 degrees

v² = 4537.48

[tex]v = \sqrt{4537.48}[/tex]

v = 67.36.

By the law of sines, we have that:

sin(51º)/67.36 = sin(U)/79

Hence the measure of angle U is obtained as follows:

sin(U) = 79 x sine of 51 degrees/67.36

sin(U) = 0.9114

m < U = arcsin(0.9114)

m < U = 66º.

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The product -2jk is negative if and only if either j or k is negative (but not both). Therefore, the possible conditions for j and k are:

j < 0 and k > 0

j > 0 and k < 0

For the expression -2jk to be negative, at least one of the factors j or k must be negative, since the product of two positive numbers is always positive, and the product of two negative numbers is positive as well. Therefore, the possible conditions for j and k are:

j is negative and k is positive.

j is positive and k is negative.

In other words, if j and k have opposite signs, their product will be negative. However, if both j and k are positive or both are negative, their product will be positive. It's important to note that there are infinite possibilities for j and k that satisfy these conditions, since j and k can take on any real value as long as one is negative and one is positive.

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Answer:

False

Step-by-step explanation:

Net worth is a combination of everything you own.

This includes the amount of money you have, or have ever made, plus your house, vehicle, belongings, etc.

The probability the point will be in the region labeled A is 2/15 if  point is selected at random inside the given figure.

Given that a point is selected at random from inside the given figure.

We are to find the probability that the point will be in the region labeled B.

From the figure, we note that the regions A, B and D are rectangles and the region C is a square.

The areas of all the regions are calculated as follows:

Area of region A is 5×(3+4) =35 sq in

Area of region B is 3×4  = 12 in

Area of region C is 4² = 16 sq in

Area of region D is 3×(4+5)= 27 sq. in

Therefore, the probability that the randomly chosen point will lie in the region B is given by P

P = 12/35+12+16+27

=12/90

=2/15

Hence,  the probability the point will be in the region labeled A is 2/15

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Answer:

[tex]w=\frac{82}{17}[/tex]

Step-by-step explanation:

Isolate the variable by dividing each side by factors that don't contain the variable.

Decimal form: 4.82352941

having a great day and thx for your inquiry :)

Answer: -4x^2-x+15

Step-by-step explanation:

Combine like terms: 2x^2 and -6x^2 combine to -4x^2.

Combine like terms: -4x and 3x combine to -x.

Combine like terms: 8 and 7 combine to 15.

The simplified expression is -4x^2-x+15.

The probability of someone being a right-handed chess player is 0.962, or approximately 0.96

Now, We have to given that;

The probability of being right-handed (R) is 0.9 and the probability of playing chess (C) given that someone is left-handed (L) is 0.6.

And, We are also told that dominant hand and playing chess are independent, so this means that the probability of playing chess is the same regardless of whether someone is left-handed or right-handed.

Hence, the probability of someone being a right-handed chess player, we can use the following formula:

P(R and C) = P(R) x P(C)

We know that P(R) = 0.9,

and since playing chess is independent of handedness,

P(C) = P(C|L) + P(C|R) = 0.6 + P(C|R).

We want to find P(C|R), the probability of playing chess given that someone is right-handed.

We can use the fact that the probabilities must add up to 1 to solve for P(C|R):

P(C|R) = 1 - P(not C|R)

We know that;

P(not C|R) = 1 - P(C|R),

so we can substitute to get:

P(C|R) = 1 - (1 - P(C|R)) = 2P(C|R) - 1

Now we can substitute this back into the original formula and solve for P(R and C):

P(R and C) = P(R) x P(C) P(R and C) = 0.9 x (0.6 + P(C|R))

P(R and C) = 0.54 + 0.9P(C|R)

Finally, we use the fact that the probabilities must add up to 1 to solve for P(C|R):

P(C|R) = 1 - P(not C|R) P(C|R)

= 1 - P(no C and R) / P(R) P(C|R)

= 1 - P(R and not C) / P(R) P(C|R)

= 1 - (0.1 x 0.4) / 0.9

P(C|R) = 0.5333

Now we can substitute this back into the equation for P(R and C) to get:

P(R and C) = 0.54 + 0.9(0.5333...)

P(R and C) = 0.962

Therefore, the probability of someone being a right-handed chess player is 0.962, or approximately 0.96

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The equation of the line in the graph passing through the points (-2,1) and (6,7) is  [tex]y = \frac{3}{4}x + \frac{5}{2}[/tex].

The slope-intercept form is expressed as;

y = mx + b

Where m is slope and b is the y-intercept.

From the graph, the line passes through point (-2,1) and (6,7).

First, we determine the slope:

[tex]m = \frac{y_2 - y_1}{x_2 - x_1} \\\\m = \frac{7 - 1}{6-(-2)} \\\\m = \frac{6}{6+ 2} \\\\m = \frac{6}{8} \\\\m = \frac{3}{4}[/tex]

Now, plug the slope m = 3/4 and point (-2,1) into the point-slope form:

( y - y₁ ) = m( x - x₁ )

[tex]y - 1 = \frac{3}{4}(x - (-2)) \\\\y - 1 = \frac{3}{4}(x + 2)\\\\y - 1 = \frac{3}{4}x + \frac{3}{2} \\\\y = \frac{3}{4}x + \frac{3}{2} + 1\\\\y = \frac{3}{4}x + \frac{5}{2}[/tex]

Therefore, the euation of the line is [tex]y = \frac{3}{4}x + \frac{5}{2}[/tex].

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The first plan has a higher rate of change compared to the second plan based on the given information.

To identify which function has the greater rate of change, we need to compare the rate at which the cost increases for each function. Given the table of values, let's analyze the rates of change for both functions.

First Plan:

The first plan charges a rate of $5 per day, regardless of the number of days. In this case, the rate of change is constant at $5 per day, meaning the cost increases by $5 for each additional day.

Second Plan:

The table of values provided doesn't directly represent the second plan. We need the values for the number of days and their corresponding costs for the second plan to determine its rate of change.

However, since we're comparing the rates of change between the two functions, we can calculate the average rate of change for the second plan using the given values. Let's find the average rate of change between the first and last points:

Change in cost = 102 - 24 = 78

Change in days = 20 - 2 = 18

The average rate of change = Change in cost / Change in days

= 78 / 18

≈ 4.3333

The average rate of change for the second plan is approximately 4.3333 dollars per day.

Comparing the rates of change:

The rate of change for the first plan is $5 per day, while the average rate of change for the second plan is approximately $4.3333 per day. Therefore, the first plan has a greater rate of change.

In conclusion, the first plan has a higher rate of change compared to the second plan based on the given information.

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If owners of a recreation area are filling a small pond with water.  The  total amount of water after 13 minutes is 1016 liters.

Let the total amount of water in the pond = W

Let the total number of minutes that water has been added = T

The equation relating W to T Is:

W = 600 + 32T

Where:

600=  initial amount of water in the pond

32T = additional water added over time

Now let find the  total amount of water in the pond after 13 minutes

Substitute T = 13 into the equation:

W = 600 + 32(13)

W = 600 + 416

W = 1016

Therefore the total amount of water is 1016 liters.

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Answer:

Step-by-step explanation:

Let’s assume the original price of the item is “x.” Then, using the first statement, the list price of the item is 80% of the original price, or 0.8x. If there is a discount applied, let’s say “d,” then the discounted price would be (1-d)(0.8x).

Using the second statement, if the price of the item has been reduced by 80%, then the discounted price would be 0.2x. This can be expressed as (1 - 0.8)(x).

So, we have the following two equations:

(1 - d)(0.8x) = 0.2x

0.64x - 0.8dx = 0.2x

Simplifying this equation, we get:

0.44x = 0.8dx

d = 0.55

This means that the discount applied in the first statement is 55%, not 80%.

The calculation for change in percentage (increase or decrease) involves finding the difference between two values and expressing it as a percentage of the original value. This is different from multiplying by percentages, which involves finding a percentage of the original value and subtracting or adding it to the original value.

The wording of percentage problems is important because it can affect the way the problem is interpreted and the calculation that is used to solve it. For example, the phrase “increased by 50%” could be interpreted as multiplying the original value by 1.5, while the phrase “increased to 50%” could be interpreted as finding 50% of the original value and adding it to the original value.

Examples:

A shirt is on sale for 30% off its original price of $50. The discounted price is (1-0.3)($50) = $35.

A company’s revenue increased from $100,000 to $120,000. The percentage increase in revenue is ((120,000 - 100,000) / 100,000) x 100% = 20%.