The difference between five times a number and twelve times the number is –35.

Answer:

[tex]1+\sqrt{2}[/tex]

Step-by-step explanation:

We have been given a radical expression [tex]\sqrt{3+2\sqrt{2}}[/tex]. We are asked to simplify the given expression.

We will add [tex](\sqrt{2})^2-2[/tex] to our given expression as after adding and subtracting same quantity the value of our expression will be same.

[tex]\sqrt{3+2\sqrt{2}+(\sqrt{2})^2-2}[/tex]

[tex]\sqrt{3-2+2\sqrt{2}+(\sqrt{2})^2}[/tex]

[tex]\sqrt{1+2\sqrt{2}+(\sqrt{2})^2}[/tex]

Using perfect square formula [tex](a+b)^2=a^2+2ab+b^2[/tex] we can rewrite our expression as:

[tex]\sqrt{1^2+2\sqrt{2}\cdot 1+(\sqrt{2})^2}[/tex]

[tex]\sqrt{(1+\sqrt{2})^2}[/tex]

Applying radical rule [tex]\sqrt[n]{x^n} =x[/tex], we will get,

[tex](1+\sqrt{2})^{\frac{2}{2}}=1+\sqrt{2}[/tex]

Therefore, the simplified form of our given expression would be [tex]1+\sqrt{2}[/tex].

Answer:

1). Slope = (-2)

2). Midpoint = (2, -2)

3). Slope of the perpendicular bisector = (1/2)

4). Equation of perpendicular bisector will be x - 2y = 6

Step-by-step explanation:

A line segment has the endpoints at (4, -6) and (0, 2).

1). Then the slope of the given line segment will be = (y - y')/(x - x') = (2 + 6)/(0 - 4) = 8/(-4) = (-2)

2). Mid point of the line segment is given by [tex](\frac{x_{1}+x_{2}}{2}) , (\frac{y_{1}+y_{2}}{2})[/tex]

Therefore midpoints of the line segment will be [tex]\frac{4+0}{2},\frac{-6+2}{2}[/tex] = (2, -2)

3). Slope of the perpendicular bisector is represented by [tex]m_{1}.m_{2}=(-1)[/tex]

⇒ (-2)×m2 = (-1)

⇒ [tex]m_{2}=\frac{1}{2}[/tex]

4). Now we have to find the equation of perpendicular bisector passing through (2, -2) and slope (1/2).

Since standard equation of the line will be given as y = mx + c

[tex]y=\frac{1}{2}x+c[/tex] passes through (2, -2).

[tex](-2) = \frac{1}{2}(2) + c[/tex]

c = (-1) - 2 = -3

Finally the equation of perpendicular bisector will be

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

⇒ 2y = x - 6

⇒ 2y - x = -6

⇒ x - 2y = 6

Answer: the answers are in the explanation, they are also in order

Step-by-step explanation:

-2

(2,-2)

1/2

y=(1/2)x - 3

The chronological age of a person with a mental age of 40 and an IQ of 80 is calculated to be 50 years by using the direct and inverse proportionality of the IQ, Mental Age, and Chronological Age.

The subject of your question is related to a mathematical concept known as Direct and Inverse Proportion. In this scenario, the IQ is directly proportional to the Mental Age and inversely proportional to the Chronological Age. Therefore, the formula is given as: IQ = (Mental Age / Chronological Age) * 100.

From the first example that indicates a person with the mental age of 25 and a chronological age of 20 has an IQ of 125, we can determine the proportionality constant as follows: 125 = (25 / 20 ) * 100.

Now, to find out the chronological age of a person with a mental age of 40 and an IQ of 80, we use the same proportionality and solve the equation for chronological age: 80 = (40 / Chronological Age) * 100. Solving the equation, the Chronological Age equates to 50 years.

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To find the chronological age of a person with a mental age of 40 and an IQ of 80, we can use the fact that IQ varies directly with mental age and inversely with chronological age. The specific chronological age depends on the value of the constant of variation, k.

To find the chronological age of a person with a mental age of 40 and an IQ of 80, we can use the fact that IQ varies directly with mental age and inversely with chronological age. Let's denote the chronological age as x. We can set up the equation as follows:

IQ = (k * mental age) / chronological age

where k is the constant of variation.

Given that the mental age is 40 and the IQ is 80, we can plug in these values into the equation:

80 = (k * 40) / x

To solve for x, we can cross-multiply and then divide both sides by 80:

80x = k * 40

x = (k * 40) / 80

Since we don't have the value of k, we can't determine the specific value of x. However, we can determine the relationship between x and the mental age:

x = 40 / k

Therefore, the chronological age of a person with a mental age of 40 and an IQ of 80 depends on the value of the constant of variation, k.

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The expression x²- 4x - 21 represents the area of the basketball court in square meters.

The area can be defined as the space occupied by a flat shape or the surface of an object.

The area of a figure is the number of unit squares that cover the surface of a closed figure.

It is given that the area of a rectangle is (x − 7)(x + 3)

when It has a length of (x − 7) meters and a width of (x + 3) meters.

The simplified expression of the area is equal to

(x-7)(x-3)

This can also be written as

x *(x+3) - 7*(x+3)

x²+3x-7x-21

x²- 4x - 21

Therefore x²- 4x - 21 represents the area of the basketball court in square meters .

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Answer:

Option 1,3 and 4 are correct.

Step-by-step explanation:

Given is a quadratic equation

[tex]x^2-10x+25=0[/tex]

Marcus used quadratic formula to solve this

The correct use of formula is

[tex]\frac{-b+/\sqrt{b^2-4ac} }{2a}[/tex]

for the quadratic equation [tex]ax^2+bx+c=0[/tex]

Comparing we have a =1, b =-10: c=25

The student should have substituted -10 for b but he wrong did only 10

II option is wrong since 2 is substituted correctly

III option is right -4ac i.e. the product x 4 must have been subtracted

IV option is right since repeated root is there so only one solution

The solution of a linear system of an equations is (-1, 1).

The given linear system of equations are y = x+ 2 and 6x – 4y = –10.

A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.

The system of linear equations can be solved as follows

x-y=-2 --------(1)

6x – 4y = –10 --------(2)

Multiply equation (1) by 4

That is, 4x - 4y= -8 --------(3)

Subtract equation (3) from the equation (2)

So, 6x-4y-(4x-4y)=-10-(-8)

2x=-2

⇒ x=-1

Put x=-1 in x-y=-2

Now, y=1

So, solution is (-1, 1)

Therefore, the solution of a linear system of an equations is (-1, 1).

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Answer:

option C

Step-by-step explanation:

In United States national debt in 2010 was estimated to be nearly $13,000,000,000,000.

Now we have to convert this debt in scientific notation.

In scientific notation we have to convert this number in to 10ˣ.

13000000000000 = 1.3 × 10¹³

Therefore option C is the answer.

Answer:

The pressure of the gas 0°C is about 699.

Step-by-step explanation:

The given table of values is

Temperature      Pressure

     10                      723

     25                     763

     40                     800

     55                     845

     80                      900

The general form of a trend line is

[tex]y=a+bx[/tex]

where, b is slope and a is y-intercept.

[tex]b=\frac{\sum_{i=1}^nx_iy_i-n\overline{x}\overline{y}}{\sum_{i=1}^nx_i^2-n\overline{x}^2}[/tex]

[tex]a=\overline{y}-b\overline{x}[/tex]

Using graphing calculator, we get

[tex]a=699.006826[/tex]

[tex]b=2.552218[/tex]

The equation of line of best bit is

[tex]y = 2.552218x + 699.006826[/tex]

where, y is pressure at x°C.

We need to find the pressure of the gas at 0°C.

Substitute x=0 in the above equation.

[tex]y = 2.552218(0) + 699.006826[/tex]

[tex]y =699.006826[/tex]

[tex]y \approx 699[/tex]

Therefore the pressure of the gas 0°C is about 699.

Answer:

1335.90

Step-by-step explanation:

Answer:

Option 3rd is correct

4: 3

Step-by-step explanation:

Perimeter(P) of rectangle is given by:

[tex]P =2(l+w)[/tex]

where, l is the length and w is the width of the rectangle respectively.

Given that:

The dimensions of a smaller rectangle are 3 ft. by 9 ft.

Perimeter of smaller rectangle= 2(3+9) =2(12) = 24 ft.

It is also given that: The dimensions of a larger rectangle are 5 ft. by 11 ft.

Perimeter of Larger rectangle = 2(5+11) =2(16) = 32 ft.

We have to find  the ratio of the perimeter of the larger rectangle to the perimeter of the smaller rectangle.

[tex]\frac{\text{Perimeter of the larger rectangle}}{\text{Perimeter of the Smaller rectangle}}[/tex]

Substitute the given values we have;'

[tex]\frac{32}{24}=\frac{4}{3}[/tex]

Therefore,  the ratio of the perimeter of the larger rectangle to the perimeter of the smaller rectangle is, 4 : 3

The converse of the statement 'If two lines are parallel, they do not intersect' is 'If two lines do not intersect, they are parallel', which is false. A counterexample to this converse is skew lines that do not intersect but are not parallel. Thus, we can't write both statements as a biconditional

The subject of your question is regarding conditional statements and their converses in mathematics. A conditional statement is often expressed in the form: 'If p, then q', where p is a hypothesis and q is a conclusion. The converse of a conditional statement switches the hypothesis and the conclusion, i.e., 'If q, then p'.

In the context of your question, the conditional statement given is 'If two lines are parallel, they do not intersect'. The converse of this statement is 'If two lines do not intersect, they are parallel'. However, this converse is not true. A counterexample can be two lines that do not intersect but are not parallel because they are skew lines, which exist in three dimensions and do not lie in the same plane. Thus, the statements can't be written as a biconditional as one is not a valid conditional.

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