How do you prove theorem 8.9 Class 9?

The function shown on the graph is f(x) = - 1/2 cos x.

What is a function?

A function in mathematics from a set X to a set Y allocates exactly one element of Y to each element of X. The sets X and Y are collectively referred to as the function's domain and codomain, respectively. Initially, functions represented the idealized relationship between two changing quantities.

Here,

We have given a graph and we have to determine which function is shown on the given graph.

We concluded that

The amplitude of the first graph is (1/2) which is the height from the midline to the peak.

The value of the function at x = 0 is -(1/2), where sin(0) = 0, therefore, the function is a cosine function

The period of the graph is 2·π, which is the period of the parent cosine function.

Hence, the function shown on the graph is f(x) = - 1/2 cos x.

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The missing sides or angles that make the triangle ABC congruent to PDF by SAS are AC and PF, ∠A and ∠P and ∠B and ∠D

If all three corresponding sides and all three corresponding angles are equal for two triangles, they are said to be congruent. The corresponding sides and angles of two triangles must be equal for them to be considered congruent.

The triangles are congruent by AAS if two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle (angle angle side). Hence, the missing side or the angle that makes the triangle ABC  congruent to triangle PDF are as follows:AC ≅ PF∠A ≅ ∠P∠B ≅ ∠D

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If f(x) = 5 - 3x and g(x)= 4x + 1, then the value of x such that f(x) = g(x) is, x 4/7

Function is a relation between a set of inputs and a set of outputs which are permissible. In a function, for particular values of x we will get only a single image in y. It is denoted by f(x).

Vertical line test:-

Whenever we want to check whether a given expression is a function or not we can apply a vertical line test, in this test we check for a single image of x , we are getting a single image or more.

If we get more images then it will not be a function.

For example, let us take, y² = 4ax

y = ±√4ax

For single value of x we get two values of y

Hence, it will not be a function.

Given that,

Two functions, f(x) = 5 - 3x and g(x)= 4x + 1,

The value of x, for which  f(x) = g(x)

Now, equating both the functions

⇒ 5 - 3x = 4x + 1

⇒ 4  = 7x

⇒ x = 4/7

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The linear equation that can be used to find the width of the rectangle is 34 = 6w, and w = 5.67cm

A linear equation is an algebraic equation where each term has an exponent of 1 and when this equation is graphed, it always results in a straight line.

The length of the rectangle is twice the width. Therefore, l = 2w

The perimeter of a rectangle is expressed as : P = 2(l+w)

l = 2w

P= 2(2w + w)

P = 6w

34 = 6w

therefore the equation that can be used to find the width of the rectangle is 34 = 6w

and the value of w = 34/6 = 5.67cm

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Ves = 169.56 cm³ is  the amount of empty space within a cylinder containing three solid spheres, where each sphere has a radius of 3 cm.

Let call Vc   volume of cylinder

If the cylinder has to contain three spheres of radius 3 cm the height of the

cylinder is  3 * 2 * 3 = 18 cm   ( we have three diameters )

The radius of the base of the cylinder is 3 cm

And Vc = π*r²*h    ⇒ Vc = (3,14) * 9 * 18   ⇒  Vc = 508.68 cm³

On the other hand we have Vs (volume of one sphere)

Vs  = 4/3 * π * r³     ⇒   Vs  = 4/3 * 3,14 * (3)³     ⇒  Vs = 113,04 cm³

as we have 3 spheres the total volume is   3 * 113.04

Vts = (total volume of spheres)  =  339,12 cm³

The empty space withing the cylinder is the difference between Vc and Vts

Ves ( volume of empty space ) =  Vc - Vts   =   508.68  -  339,12

Ves = 169.56 cm³

Ves = 169.56 cm³ is  the amount of empty space within a cylinder containing three solid spheres, where each sphere has a radius of 3 cm.

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A point on the real line with the conventional topology is a frequent example of this.

In that topology, a point a ∈ R is not an open point; rather, it is the intersection of open intervals.

{a} = ∩ (a - [tex]\frac{1}{n}[/tex] , a + [tex]\frac{1}{n}[/tex] ) { for n=1 to ∞ }

What is Open Set?

The fundamental building blocks of topology are open sets. The open sets have a natural description in the well-known context of a metric space, which can be conceptualized as a generalization of an open interval on the real number line. In the same manner that an interval's endpoints are not contained within the interval, an open set is logically defined as a set that does not contain its border.

Therefore our suitable example will be a point in the real line.

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The expression −11 + |−5| − |−15| containing modulus sign is solved to get -21

Without taking direction into account, absolute value describes how far away from zero a certain number is on the number line.

A number can never have a negative absolute value.

The equation is written in the form below

= −11 + |−5| − |−15|

removing the absolute value sign

= −11 + 5 − 15

performing the mathematical operations (addition and subtraction)

= -11 - 10

= -21

The absolute value equation was solved to get -10

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keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the equation above

[tex]x-2y=17\implies -2y=-x+17\implies y=\cfrac{-x+17}{-2} \\\\\\ y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{1}{2}} x-\cfrac{17}{2}\impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}[/tex]

so we're really looking for the equation of a line whose slope is 1/2 and that it passes through (-10 , 4)

[tex](\stackrel{x_1}{-10}~,~\stackrel{y_1}{4})\hspace{10em} \stackrel{slope}{m} ~=~ \cfrac{1}{2} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{4}=\stackrel{m}{ \cfrac{1}{2}}(x-\stackrel{x_1}{(-10)}) \implies y -4= \cfrac{1}{2} (x +10) \\\\\\ y-4=\cfrac{1}{2}x+5\implies {\Large \begin{array}{llll} y=\cfrac{1}{2}x+9 \end{array}}[/tex]

Answer:

[tex]x^2-12x+y^2-4y+39=0[/tex]

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{5 cm}\underline{Equation of a circle}\\\\$(x-a)^2+(y-b)^2=r^2$\\\\where:\\ \phantom{ww}$\bullet$ $(a, b)$ is the center. \\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]

Given:

Substitute the given values into the equation of a circle formula:

[tex]\implies (x-6)^2+(y-2)^2=1^2[/tex]

Simplify by expanding:

[tex]\implies (x-6)(x-6)+(y-2)(y-2)=1[/tex]

[tex]\implies x^2-12x+36+y^2-4y+4=1[/tex]

[tex]\implies x^2-12x+y^2-4y+39=0[/tex]

On solving the provide question, we can say that the correct number of significant figures, the area should be expressed as 5 cm and 10 dm and 2500 mm and 750 cm

The number of significant digits in a value—often a measurement—contributes to the accuracy of the value. From the first non-zero digit, begin counting the important digits. To the right of the last non-zero decimal place, all zeros are acceptable. For instance, the number 0.0079800 has five significant digits. When they come from the measurement, any zeros to the right of the final non-zero digit are important. equivalently rounds an integer to 3 significant digits and 3 decimal places. Start counting at the first non-zero digit and go up to three. After that, remove the final digit. Add zeros to the last two numbers to the right of the decimal point.

he sides of a rectangular piece of plywood as =  3.7 cm and 4.85 cm

the correct number of significant figures, the area should be expressed as

1. 5 cm

2. 10 dm

3. 2500 mm

4. 750 cm

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Depending on the comparable model, Energy Star appliances will save you 10% to 50% of the energy used;

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Using an energy efficient bulb will result in a significant cost saving over the long term.

Does utilising less energy cost less money?

A normal incandescent bulb uses 60W of power and lasts around 1000 hours, while a LED bulb uses around 8-12W of power and can last up to 25,000 hours.

So if you use the light 5 hours a day, 365 days a year, a 60W incandescent bulb will consume 11.25 kWh per year and cost you approximately £1.35 per year to run. In contrast, a 8W LED bulb will consume approximately 0.6 kWh per year, costing you approximately £0.072 per year to run.

Therefore, using an energy efficient bulb will result in a significant cost saving over the long term.

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Answer: 25°F is approximately $375.

Step-by-step explanation:

The scatter plot shows a positive correlation between the average monthly temperature (x) and the family's monthly heating cost (y). The line of best fit appears to be a linear equation, and it can be approximated by the equation y = mx + b, where m is the slope and b is the y-intercept.

(a) One possible equation of the line of best fit for the data is y = 15x + 300. This equation is only an approximation as it is hard to determine the exact line of best fit without more data points.

(b) Using the equation y = 15x + 300, the predicted monthly heating cost for a month with an average temperature of 25°F can be found by plugging in x = 25 into the equation.

y = 15(25) + 300 = 375

Therefore, the predicted monthly heating cost for a month with an average temperature of 25°F is approximately $375.

Answer:

1 < x < 13

Step-by-step explanation:

The longest leg of a triangle must be less than the sum of the shorter legs.

c < a + b

If the third piece is the longest leg, then:

x < 6 + 7

x < 13

If the 7-ft piece is the longest leg, then:

7 < 6 + x

x > 1

Therefore, the third leg is between 1 ft and 13 ft.

Answer:84652646.4

Step-by-step explanation:

The positive intervals are the y values over (0,∞ )

The negative intervals are the y values over (0,9]

What is Function?

A relation between a collection of inputs and outputs is known as a function. In plain English, a function is an association between inputs in which each input is connected to precisely one output. A domain, codomain, or range exists for every function.

The periods where a function is above the x-axis are considered its positive regions. Where the y-values are positive is where it is (not zero). Periods where a function is below the x-axis are referred to as its negative regions. This is the case when the y-values are negative (not zero).

The graph extends to infinity from the origin above x-axis

So, the positive intervals are the y values over (0,∞ )

The graph extends to y=9 from the origin below x-axis

So, the negative intervals are the y values over (0,9]

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A stable solution is a solution to a problem that does not change when the parameters of the problem are slightly varied. It is a point at which there is no net change, or a point of equilibrium.

A stable solution is a solution to a problem that does not change when the parameters of the problem are slightly varied. The parameters of a problem can refer to the data, assumptions, constraints, or other factors that are used to define the problem. When the parameters are changed, a stable solution will remain intact, indicating that it is the best solution to the problem.

For example, if a mathematics problem has a solution of 3, this solution is considered to be stable, as changing any of the parameters of the problem will not alter the solution of 3.

Stable solutions are important as they provide a point of equilibrium, or a point at which there is no net change. This can help to ensure that the solution is the most effective one to the problem.

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From the beginning of the second trimester a pregnant woman needs approximately an additional 300 calories a day.

Second Trimester:

Pregnancy helps you think about pregnancy because the changes you and your baby are going through fall into three broad categories: early, mid, and late. Metaphase, represents the 13th to 26th week. For many women, one of the best things about this trimester is when the nausea starts to subside.

I have had personal experiences with this, as to the fact I have 3 siblings and I have helped raise them. 300 calories are just the right amount for the mother and the growing baby. If she ate more than that, she would gain much more baby weight.

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The probability she will have at least one friend at the restaurant is 1/3.

Probability is the chance of occurrence of a certain event out of the total no. of events that can occur in a given context.

From the given information the total number of friends (F) is 3 and the total number of restaurants N(R) is 6.

Therefore,

The likelihood that she will be accompanied by at least one friend to the restaurant is,

P(F) = N(F)/N(R).

P(F) = 3/6.

P(F) = 1/3.

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The missing sides or angles that makes the triangle ABC congruent to PDF by SAS are AB and PD, BC and DF and ∠B and ∠D

Two triangles are said to be congruent if all three corresponding sides are equal and all the three corresponding angles are equal .

In other words, two triangles are congruent when the corresponding sides and angles are equal.

Hence, let's find the missing side of angle that makes the triangle ABC congruent to triangle PDF by SAS.

Hence,

If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent by SAS.

Therefore, the missing sides are as follows:

AB ≅ PD

BC ≅ DF

∠B ≅ ∠D

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Question 14

[tex]\frac{\sin x}{26}=\frac{\sin 49^{\circ}}{25}\\\\\sin x=\frac{26\sin 49^{\circ}}{25}\\\\x \approx 51.7^{\circ}, 128.3^{\circ}[/tex]

Both of these values satisfy the angle sum of a triangle.

Question 16

[tex]\frac{\sin x}{8}=\frac{\sin 105^{\circ}}{11}\\\\\sin x=\frac{8 \sin 105^{\circ}}{11}\\\\x \approx 44.6^{\circ}, 135.4^{\circ}[/tex]

However, the second case exceeds the angle sum of a triangle, so [tex]x \approx 44.6^{\circ}[/tex].

Answer:

x = 5

y = 8

Step-by-step explanation:

2x + y = 18

y = 4x - 12

2x + 4x - 12 = 18

6x - 12 = 18

6x = 30

x = 5

2x + y = 18

2*5 + y = 18

10 + y = 18

y = 8

Width = 8 inch

Length = 10 inch

This can be solved using rectangular area formula.

The area a rectangle takes up inside its four sides or limits is known as the area of the rectangle. A rectangle's sides determine how much space it has. The product of the rectangle's length and width is essentially the area formula.

By multiplying the rectangle's length by its breadth, we may get its area.

length of the picture = l inch

width of the picture = w inch

given that, length of a picture is 2 more than the width.

so, l = w + 2

There is a frame of uniform width around the picture that is 3 inches wide, now length becomes: l = w + 2 + 3

l = w + 5

and the width becomes: w = w + 3

the area of everything is 224 inch²

l × w = 224 inch²

or, (w + 5)  × (w + 3) = 224 inch²

or, w² + 8w + 15 = 224

or, (w + 19) (w - 8) = 0

or, w = 8, -19

Width never be negative so w = 8 inch

Now, length = w + 2

l = 8 + 2

l = 10 inch

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The dimensions of the picture are 10 × 8 in if the area of everything is 224 in².

The area a rectangle takes up inside its four sides or limits is known as the area of the rectangle. A rectangle's sides determine how much space it has. The product of the rectangle's length and width is essentially the area formula.

By multiplying the rectangle's length by its breadth, we may get its area.

Length of the picture = l inch

Width of the picture = w inch

Given that, the length of a picture is 2 more than the width.

So, l = w + 2

There is a frame of uniform width around the picture that is 3 inches wide, now length becomes: l = w + 2 + 3

l = w + 5

The width becomes: w = w + 3

The area of everything is 224 inch².

l × w = 224 inch²

(w + 5)  × (w + 3) = 224 inch²

w² + 8w + 15 = 224

(w + 19) (w - 8) = 0

w = 8, -19

Width never be negative so w = 8 inch

Now, length = w + 2

l = 8 + 2

l = 10 inch

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Answer:

We need 166.5 ml of soy sauce when we have 111 ml of vinegar.

We need 74 ml of vinegar when we have 111 ml of soy sauce.

Step-by-step explanation:

I think that you meant 150 and not 150150150 and 100 and not 100100100

[tex]\frac{soy sauce}{vinegar}[/tex] = [tex]\frac{soy sauce}{vinegar}[/tex]

[tex]\frac{150}{100}[/tex] is the same as [tex]\frac{3}{2}[/tex]  (simplified by dividing the top and the bottom by 50)

[tex]\frac{3}{2}[/tex] = [tex]\frac{x}{111}[/tex]  Cross multiply and solve for x

2x = 3(111)

2x = 333  Divide both sides by 2

[tex]\frac{2x}{2}[/tex] = [tex]\frac{333}{2}[/tex]

x = 166.5

We need 166.5 ml of soy sauce when we have 111 ml of vinegar.

[tex]\frac{3}{2}[/tex] = [tex]\frac{111}{x}[/tex]  Cross multiply and solve for x

3x = 2(111)

3x = 222  Divide both sides by 3

[tex]\frac{3x}{3}[/tex] = [tex]\frac{222}{3}[/tex]

x = 74

We need 74 ml of vinegar when we have 111 ml of soy sauce.

The probability that the students will have a pop quiz on a randomly selected day 0.77

Mrs. Barnes’s class determined the probability that she will check homework on a randomly chosen day is 0.42.

They also determined the probability that she will give a pop quiz when she checks homework is 0.6, and the probability that she will give a pop quiz when she does not check homework is 0.9.

Then the probability that Mrs. Barnes does not check homework if the students take a pop quiz will be

P= 0.42*0.60 + 0.58*0.90

P = 0.252 + 0.522

P = 0.774

P ≈ 0.77

The probability that the students will have a pop quiz on a randomly selected day 0.77

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A) 38% of 124 is 46.32 (as a decimal)

B) 138% of 124 is 171.12 (as a decimal)

What is the decimal form?

The decimal form of a number is a way of expressing that number as a decimal (base 10) number. A decimal number is made up of a whole number part and a fractional part, separated by a decimal point.

A) To work out 38% of 124, we can use the following formula:

(percentage/100) * value = result

So, in this case:

(38/100) * 124 = 0.38 * 124 = 46.32

So, 38% of 124 is 46.32 (as a decimal)

B) To work out 138% of 124, we can use the same formula:

(138/100) * 124 = 1.38 * 124 = 171.12

So, 138% of 124 is 171.12 (as a decimal)

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Answer:

Mar Apr May

Receipts $1,155 $1,250 $

Expenses $1,100 $990 $900

Net Cash Flow $55 $ $45

Cumulative Balance $ $315 $360

The probability of a right-handed student being forced to use a left-handed arm tablet is 0.085 or 8.5% using either the binomial or normal model.

A binomial model is appropriate to use for this problem since there are two possible outcomes (left or right-handed) and the probability of each outcome is independent of the other. The probability of a right-handed student being forced to use a left-handed arm tablet is calculated by subtracting the number of left-handed arm tablets (30) from the total number of students in the class (188) and dividing that by the total number of arm tablets (200). This yields a probability of 0.085 or 8.5%.

Using the normal model, the probability of a right-handed student being forced to use a left-handed arm tablet can also be calculated. This model is appropriate since there are a large number of students (188) and the probability of a student being right or left-handed is close to the population average (13%). The probability can be calculated by subtracting the number of left-handed arm tablets (30) from the total number of students in the class (188) and dividing that by the total number of arm tablets (200). This yields a probability of 0.085 or 8.5%.

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After simplification, we can express the given expression as "Negative twelve and one-half."

Option (B) is correct.

What is a simplification in mathematics?

In mathematics, simplification refers to the process of making an expression or equation simpler or easier to understand or work with. This can involve combining like terms, removing unnecessary terms or factors, or rearranging the order of the terms.

To simplify the expression: Negative two and one-third minus negative ten and one-sixth, we will convert the mixed numbers to fractions and then perform the subtraction.

First, we convert the mixed numbers to fractions:

Negative two and one-third = -2 + (1/3) = -2 + 1/3

Negative ten and one-sixth = -10 + (1/6) = -10 + 1/6

Then we perform the subtraction:

(-2 + 1/3) - (-10 + 1/6) = -2 -10 + 1/3 - 1/6 = -12 + 2/6 + 1/6 = -12 + 3/6 = -12 + 1/2

So the simplified expression is Negative twelve and one-half.

Hence, after simplification, we can express the given expression as "Negative twelve and one-half."

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Answer: hexagon

Step-by-step explanation: it has six sides !!

Answer:

[tex]\textsf{1)} \quad y=x+\dfrac{17}{4}[/tex]

[tex]\textsf{2)} \quad y=2x+4[/tex]

Step-by-step explanation:

Differentiation is an algebraic process that finds the gradient (slope) of a curve.

At a point, the gradient of a curve is the same as the gradient of the tangent line to the curve at that point.

Given function:

[tex]y=4+2x-x^2[/tex]

Differentiate the given function:

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=2-2x[/tex]

If the tangent to the parabola forms an angle of 45° with the negative direction of the x-axis, then the gradient of the tangent is 1.

Set the differentiated function to 1 to find the x-value of the point where the gradient of the tangent is 1:

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=1[/tex]

[tex]\implies 2-2x=1[/tex]

[tex]\implies -2x=-1[/tex]

[tex]\implies x=\dfrac{1}{2}[/tex]

Substitute x = 1/2 into the given function to find the y-value of the point:

[tex]\implies y=4+2\left(\dfrac{1}{2}\right)-\left(\dfrac{1}{2}\right)^2=\dfrac{19}{4}[/tex]

Therefore, the point at which the tangent of the parabola forms an angle of 45° with the negative direction of the x-axis is:

Substitute the found point and slope into the point-slope formula to create an equation of the tangent:

[tex]\implies y-y_1=m(x-x_1)[/tex]

[tex]\implies y-\dfrac{19}{4}=1\left(x-\dfrac{1}{2}\right)[/tex]

[tex]\implies y=x+\dfrac{17}{4}[/tex]

--------------------------------------------------------------------------------------

Given function:

[tex]y=4+2x-x^2[/tex]

Differentiate the given function:

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=2-2x[/tex]

If the tangent to the parabola is parallel to the line y = 2x - 4 then the gradient of the tangent is 2 (since parallel lines have the same gradient).

Set the differentiated function to 2 to find the x-value of the point where the gradient of the tangent is 2:

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=2[/tex]

[tex]\implies 2-2x=2[/tex]

[tex]\implies -2x=0[/tex]

[tex]\implies x=0[/tex]

Substitute x = 0 into the given function to find the y-value of the point:

[tex]\implies y=4+2\left(0\right)-\left(0\right)^2=4[/tex]

Therefore, the point at which the tangent of the parabola is parallel to the line y = 2x - 4 is:

Substitute the found point and slope into the point-slope formula to create an equation of the tangent:

[tex]\implies y-y_1=m(x-x_1)[/tex]

[tex]\implies y-4=2\left(x-0\right)[/tex]

[tex]\implies y=2x+4[/tex]