Find the distance between the points.
(-1,-7) and (3, - 7)

The final solution of the given integral ∫ ([tex]x^3[/tex] - 2) dx is ([tex]x^4[/tex])/4 - 2x + C.

To integrate ∫ (x^3 - 2) dx, we use the power rule of integration, which states that ∫[tex]x^n[/tex] dx = ([tex]x^{(n+1)}[/tex])/(n+1) + C, where C is the constant of integration.

In this case, we have [tex]x^3[/tex] as the inside function, so we apply the power rule by raising the exponent by 1 and dividing by the new exponent. This gives us ([tex]x^4[/tex])/4 + C as the antiderivative.

However, we also have the constant term -2. To integrate a constant term, we add it as a separate term to the antiderivative. So the final solution is ([tex]x^4[/tex])/4 - 2x + C.

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Any set of ordered pair cannot be called as a function , the ordered pairs that in which no two different ordered pairs have the same x coordinate is called as a Function .

What is a Function ?

The set of ordered pair (x,y) is called a function, in which no two different ordered pairs have the same x coordinate. and the equation that produces such a set of ordered pairs can be called as function.

For Example : Let us Consider ordered pairs , F = [tex][(1,5) , (3,6)][/tex] ;

we see that ,  the element 1 has the image as 5 , and the element 3 has the image as 6 .

So , we can conclude  that , the each value of x has exactly only one corresponding value of y .

Therefore , the ordered pair can be called as a function .

The given question  is incomplete , the complete question  is

Is any set of ordered pair is called a Function ?

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There are 80 dogs and 20 cats in the animal shelter.

The ratio of dogs to cats is 4:3. This means for every 4 dogs, there are 3 cats. We know that there are 40 more dogs than cats.

To calculate the number of each animal, we can set up the equation 40 = 4x - 3x, where x represents the number of cats.

After solving the equation, we get x = 20. This means that there are 20 cats in the animal shelter. To find the number of dogs, we can use the ratio. We know that for every 4 dogs, there are 3 cats. So, for 20 cats, there must be 4 × 20 = 80 dogs.

Therefore, there are 80 dogs and 20 cats in the animal shelter.

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Answer:

To find the area of a rectangle, multiply its height by its width. For a square you only need to find the length of one of the sides (as each side is the same length) and then multiply this by itself to find the area......

Step-by-step explanation:

Answer: 30,000

Step-by-step explanation:

150,000 x 0.2 = 30,000

Answer:

y = 1425 * (1 + 0.05)^x

Step-by-step explanation:

An exponential function of the form y = a * b^x can be used to model exponential growth or decay, where a is the initial value, b is the base (growth or decay factor), and x is the time period. In this case, the initial value is 1,425 and the rate of increase is 5%. We can write this as an exponential function as follows:

y = 1425 * (1 + 0.05)^x

where y is the final value after x time periods, and x is the number of time periods (in years, for example). To find the value of the function after x time periods, you can substitute the value of x into the equation.

Answer:

1326.78 minutes

Step-by-step explanation:

We know that the bathtub starts with 6012 gallons of water and that it drains at an average rate of 4.5 gallons per minute. To find how many minutes it will take to reach 24.5 gallons, we can use the formula:

time = (initial amount - final amount) / rate

Plugging in the given values, we get:

time = (6012 - 24.5) / 4.5

Solving for time:

time = 1326.78 minutes

It will take 1326.78 minutes for the bathtub to drain to 24.5 gallons of water.

The cost of one small box of grapefruit and one large box of grapefruit is

Simultaneous equations are two or more algebraic equations with the same unknown variables and the same value of the variables satisfies all such equations. This implies that the simultaneous equations have a common solution. Some of the examples of simultaneous equations are: 2x - 4y = 4, 5x + 8y = 3.

represent small box by x and large box by y

12x + 8y = 276 equation 1

6x + 9y = 228 equation 1

make the coefficients of x equal by multiplying equation 2 by 2

12x + 8y = 276 equation 3

12x + 18y = 456 equation 4

subtract equation 3 from 4

10y = 180

y = 180/10

y = 18

substitute 18 for y in equation 1

12x + 6×18 = 276

12x = 276-108

12x = 168

x = 168/12

x= 14

therefore the cost of one small boxes is $14 and the cost of one large box is $18

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The expression equivalent to - 8.2 - 5 is - 8.2 + (-5)

What is an expression?

A mathematical expression is a phrase that includes at least two numbers or variables, and at least one arithmetic operation. The mathematical operations used can be addition, subtraction, multiplication or division.

Note ,

         - × - = +

         - × + = -

         + × + = +

For the expression - 8.2 - 5

In the above expression, we are subtracting positive integer 5 from -8.2

  So we get the answer as

                 -8.2-5 = = -13.2

Now this is the solution we should get while solving the given options.

  a) 5 - ( -8.2)  = 5 + 8.2 = 13.2

  b) -8.2 + 5 = - 3.2

  c) -8.2 -5 [tex]\neq[/tex] 8.25 (as seen above)

   d) - 8.2 + ( -5) = -8.2 - 5 = -13.2

Comparing the solutions, we get that option(d) matches with the required answer.

Therefore option(d) is the correct expression that gives the same answer.

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The slope of the given equation y = 10x + 1 is 10.

If we draw the graph of the linear equation y = mx + c. The result is a line with m as the slope, and c as the y-intercept. This form of the linear equation y = mx + c is called the slope-intercept form, and the values of m and c are real numbers.

The slope, m, of the line represents the steepness of a line. This is also termed a gradient, sometimes. The y-intercept, c, of the line y = mx + c represents the y-coordinate of the point where the graph of the line intersects the y-axis.

Now, if we equate the standard slope-intercept form of the equation with the given equation, we get

y = mx + c

y = 10x + 1

m = 10 and c = 1

As we know the m is the slope which is equal to 10.

-- The given question is incorrect, the correct form of equation is

"What is the slope of y = 10x + 1?" --

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when x is 0.5 then y value is 3, x is 1.5 then y is 9 and x is 4.5 then y is 27.

A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0.

A proportion is an equation in which two ratios are set equal to each other

In the given table the value of x is 0.5 and value of y is 3

y/x=3/0.5=6

In the second row the value of x is 1.5 we need to find value of y.

3/0.5=y/1.5

4.5=0.5y

Divide both sides by 0.5

y=9

In third row the value of x is 4.5 and y is 27

Hence, when x is 0.5 then y is 3, x is 1.5 then y value is 9 and x is 4.5 then y is 27.

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Therefore , the solution to the given problem of perimeter model of the sports stadium has a 12 inch height. The right response is option C.

The perimeter of a shape is sometimes refers to as its entire length in geometry. The perimeter of something like a form can be calculated by summing the distances of all of its sides and edges. The lenght of all a form's sides can be added up to find its perimeter. A rectangle that is 24 yards long and 15 yards broad, for instance, will have a perimeter of 78 yards.

Here,

A man can cover 6 miles on foot in 2 hours.

A man will cover 3 miles on foot in one hour.

10 potatoes can be sliced by a machine in 5 minutes.

2 potatoes can be cut by a machine in a minute.

Those are

2 inches per thirty feet

By 30 multiply both sides.

1 foot Equals 2/30 inches.

Now,

1 foot Equals 2/30 inches.

180 times on both sides.

180 feet = 180 x 2/30 inch

180 feet equals 6 by 2 inches.

120 feet equals 12 inches.

Thus,

Considering that a foot equals two inches, a sports arena's height would be 180 feet.

The scale model of both the sports stadium has a 12 inch height.1 foot Equals 2/30 inches.

Now,

1 foot Equals 2/30 inches.

180 times on both sides.

180 feet = 180 x 2/30 inch

180 feet equals 6 by 2 inches.

120 feet equals 12 inches.

Thus,

Considering that a foot equals two inches, a sports arena's height would be 180 feet. The scale model of both the sports stadium has a 12 inch height.

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The value of h(-2)+g(4) by substitution is 6

The method of solving "by substitution" works by solving one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging this back into the other equation, "substituting" for the chosen variable and solving for the other. Then you back-solve for the first variable.

For instance if h(x)=3x+1 . This means h(-2) means we are going to substitute -2 for variable x

Therefore h(-2) = 3(-2)+1

= -6+1 = -5

Similarly g(x)=x²-5. This means g(4) means we are going to substitute 4 for x in the equation.

Therefore g(4) = 4²-5

= 16-5 = 11

Therefore h(-2)+g(4) = -5+11

= 6

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The formula of commutative property is α + β = β + α or α × β = β × α.

The commutative property is the property that deals with the arithmetic operations of addition and multiplication. So mathematically, if we change the order of the operand and it does not change the result of the arithmetic operation then that particular arithmetic operation is commutative.

If two numbers α and β are given, then the formula of the commutative property of numbers is given as,

α + β = β + α

α × β = β × α

α - β ≠ β - α

α ÷ β ≠ β ÷ α

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A two-step inequality is an inequality that requires more than one step to solve. An example of a two-step inequality is to solve the inequality 2x - 3 ≥ 5.

Example: Solve the inequality 2x - 3 ≥ 5

Step 1: Isolate the variable by adding 3 to both sides: 2x ≥ 8

Step 2: Divide both sides by 2: x ≥ 4

In this example, the first step is to isolate the variable x by adding 3 to both sides of the inequality. In the second step, we divide both sides by 2 to get the final solution of x ≥ 4.

It's worth noting that this inequality has a solution that is an interval, it is x ≥ 4, which means that all values of x that are greater or equal to 4 are solutions of the inequality.

It's important to keep in mind that inequalities may require multiple steps to solve and may not always have a single solution. It's important to check the solution by substituting back into the inequality to make sure it holds true.

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The jazz band has 15 seventh graders playing according to the ratio.

A ratio, which is a number, is a way to represent one quantity as a portion of another. A ratio can only be compared when the two numbers in it have the same unit. To compare two objects, ratios are utilized.

Given One fifth of the 75 seventh graders at Charles School participate in the jazz band.

Presented as a ratio

75 pupils, or 1/5 of them, participate in the jazz band.

75 pupils out of the total play in the jazz band.

15 kids in total participate in the jazz band.

15 seventh graders are therefore playing in the jazz band in accordance with the ratio.

Hence, The jazz band has 15 seventh graders playing according to the ratio.

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Answer:

  30%

Step-by-step explanation:

You want the percentage reduction each year that will result in a reduction of 51% over two years.

The growth factor is 1 added to the growth rate. If the desired growth rate over 2 years is -51%, then the growth factor for that period is ...

  1 -51% = 0.49

If the growth factor is the same for each of two years will be ...

  (1 +r)² = 0.49 . . . . . . . where r is the growth rate each year

Solving for r gives ...

  1 +r = √(0.49) = 0.7

  r = 0.7 -1 = -0.3 = -30%

The required growth rate each year is -30%.

The pollution must be reduced by 30% each year for 2 years.

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If Amy walked 25 seconds, she travelled 150 feet.

The definition of direct proportion states that "When the relationship between two quantities is such that if we increase one, the other will also increase, and if we decrease one the other quantity will also decrease, then the two quantities are said to be in a direct proportion". which means the ratio of two quantities is constant.

Given, In table

Relationship between distance and time of Amy travel is given

Feet   Second

 18           3

48           8

 c           25

From the above table ratio between distance and time = Distance/ Time

= 18/3 = 48/8 = c/25

= 6/1 = 6/1 = c/25

⇒ c/25 = 6/1

⇒ c = 6 × 25 = 150

So, In 25 seconds distance will be 150 feet.

Hence, in 25 seconds Amy will travel 150 feet.

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Answer:

If it's written like this: (2, 5) then it is a ordered pair

Step-by-step explanation:

Answer:

3 d shape is in rectangular

Therefore , the solution of the given problem of expression comes out to be the coefficient of given expression  XYZ  is 1.

By substituting each variable with a number and conducting mathematical functions, an algebraic formula must be checked. The solution variable in the previous example is identical to 6 since 79 + 8 = 87. To evaluate the expression, we can replace the variables values with ones we are familiar with.

Here,

Given :

XYZ

To find the coefficient of given expression ,

Thus,

From observing we can see XYZ

can be written as

=>1*XYZ

Where 1 is coefficient.

Therefore , the solution of the given problem of expression comes out to be the coefficient of given expression  XYZ  is 1.

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Answer:

P=Ix100/RT

therefore P is made the subject formula

The range is found by finding the vertex and noting that it is the minimum or maximum (depending on direction) of the function. You can find the vertex by using -b/2a for the x value and then plugging in to get y value.

What is a function?

A mathematical phrase, rule, or rule that establishes the link between explanatory variables or a dependent variable  In mathematics, curves exist everywhere, and they are crucial for constructing physical links in the sciences.

A Method can be defined by a module, an object, or another function. An internal feature of a class is called a method. Methods are listed inside a base class to make the connections between it class and thus the method clear. A method can be called or invoked using different syntax. We can convert a type into an object before being able to access its methods.

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The three steps of factoring include taking the common component out of the polynomial then simplifying the polynomial such that the most simple values are left and finally rearrangement.

If we are provided with the polynomial that is very complex in nature then first thing we have to do in order to solve the problem related to that polynomial is to factorise the polynomial.

The factory of that particular polynomial can be done by taking out the most common component out of the polynomial in a multiplication.

After this we have to simplify the polynomial such that only the simplest values are left in the polynomial and finally we have to rearrange the polynomial because after such operations on that polynomial the polynomial will not be easy to understand in the first look of it.

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The side opposite the 30-degree angle is called the shorter leg in a 30 − 60 − 90 triangle

A right triangle is any triangle that has a 90-degree angle.

A 30-60-90 triangle is a special type of right triangle that has a 30-degree angle and a 60-degree angle in addition to the right angle.

The side opposite the 30-degree angle is called the shorter leg.

The side opposite the 60-degree angle is called the longer leg.

The side opposite the right angle of 90 degrees is called the hypotenuse.

The hypotenuse is twice the shorter leg.

The longer leg is √3 times the shorter side.

if the length of the shorter leg is given, then these properties are used to find the longer side and the hypotenuse.

However, the 30-60-90 properties hold no matter which side is provided.

If the length of the hypotenuse is given, divide it by 2 for the shorter leg. Then multiply that result by √3 to get the length of the longer leg.

If the length of the longer leg is given, divide it by √3 to get the shorter leg. Then, multiply that result by two to get the length of the hypotenuse.

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The graph that shows the relationship is graph of a line passing through the points negative 2 comma 1 and 0 comma 0. Option D

We know that a straight line graph is the kind of graph that we can be able to represent by the use of the equation; y = mx + c. Now we could have a straight line graph that passes through points.

The points that the graph would have to pass through are also the points that we can see on the graph of the function whose table have been shown in the question.

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Answer:

(-2,1) (0,0) D

Trucks will be able to pass through tunnel B while tunnel A is destroyed.In light of this, the equations for tunnels A and B are x2 + y2 + 30x + 56 = 0 and x2 - 30x + 16y - 95 = 0, respectively.

A parabola's general equation is either y = a(x-h)2 + k or x = a(y-k)2 + h, where (h, k) stands for the vertex.

y2==4ax is the typical equation for a normal parabola.

Section A: x2+y2+30x+56=0

On both sides of an equation, add 152.

This is,

x²+30x+15²+y²=-56+15²

⇒ (x+15)²+y²=169⇒ (x+15)²+y²=13²

On both sides of the equation, divide 132.

⇒ (x+15)²/13² + y²/13² =1

Part B:x²-30x+16y-95=0

x2-30x+16y=95 now,

On both sides of an equation, add 152.

In other words, x2-30x+152+16y=95+152

⇒ (x-15)²=320-16y

⇒ (x-15)²=16(20-y) (20-y)

Part C:greatest height:A: 13 feet

B: 20 feet ,8 feet wide

A: x+15=8/2=4 y=12.37<13.5

B: x-15=8/2=4 y=19>13.5

In turn, tunnel A will sustain damage while the truck can travel via tunnel B.

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The steps in solving systems of equations using the substitution method include simplifying the given equation, solving one of the equations for any unknown value, substituting the step 2 solution in another equation, solving the new equation, and finally solving the equation to find the second variable.

The substitution method is the algebraic method to solve different linear equations. In this method of substitution, the value of one variable from either of the equation is substituted in the other equation.

In this technique the following steps are used to solve an equation by substitution method:

Step 1. Simplify the given equation by removing the parenthesis.

Step 2. Solve one of the equations for any unknown value.

Step 3. Substitute step 2 result in the other equation.

Step 4. Now solve the equation which gets from step 3 obtained using elementary arithmetic operations.

Step 5. Finally, solve the final equation for the second variable.

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Answer:16ft

√256=16ft

length=16ft