Check by differentiation that y=4cost+3sint is a solution to y''+y=0 by finding the terms in the sum:
y'' = ?
y = ?
so y'' + y = ?

The slope-intercept form of the line equation 2x+ y = 10 is y = -2x + 10

There are three type of the equation of line:

- slope-intercept form.

The equation of line is: y = mx + b

Where:

m = slope

b = y-intercept

- point-slope form

The equation of line is: y - y₁ = m(x - x₁)

Where:

m = slope

(x₁,y₁)  = a point on the line

- standard form
Ax + By + C = 0

Where A, B, C are constant

The given equation is: 2x+ y = 10

we will convert it into  slope-intercept form  y = mx + b

Subtract both sides by 2x:

2x+ y -2x = 10 -2x

y = -2x + 10

Hence, the slope-intercept form is y = -2x + 10

There was a typo in your question, most probably your question was:

What is the slope-intercept form of the equation 2x+ y = 10?

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Roller Coaster B has a top speed of 124.75 ft/s and has a greater top speed.

Roller Coaster A can reach a top speed of 110 ft/s, and Roller Coaster B can reach a top speed of 85 mph.

To compare the top speeds of the two roller coasters, we need to convert the speed of Roller Coaster B from miles per hour (mph) to feet per second (ft/s).

1 mile per hour = 1.46667 feet per second

So, 85 mph = 85 * 1.46667 ft/s = 124.75 ft/s

Since Roller Coaster B has a top speed of 124.75 ft/s and Roller Coaster A has a top speed of 110 ft/s, Roller Coaster B has a greater top speed.

Therefore, Roller Coaster B has a top speed of 124.75 ft/s and has a greater top speed.

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The equation to calculate slope from two points (x₁, y₁) and (x₂, y₂) on a line is m = (y₂ - y₁) / (x₂ - x₁).

Here,

m = the line's slope

x1 is equal to the initial point's x-coordinate.

y1 is the first point's y-coordinate.

x2 is equal to the second point's x-coordinate.

The second point's x-coordinate is equal to y2.

It only takes applying the slope formula rise/run to find the slope between two places. With various forms of information about the line available, various formulas can be used to determine the slope. Particularly when two points on the line are given, the formula for finding the slope from two points is utilized.

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Answer

0^0 =1

or

0^0 can be undefined.

Answer:

Step-by-step explanation:

The prime multiples of 4 are 2*2 or 2^2.

If the slope of the line is 1/10 then it means for every ten units of horizontal distance crosses there is unit change of vertical distance.

We know that the slope of a line is nothing but the change in y coordinate with respect to the change in x coordinate.

It describes the steepness of a line.

Also,  if a line makes an angle θ with the positive direction of the x-axis, then the slope of the line will be tangent of angle θ

m = tan θ

Here we have been given a slope 1/10

Let m = 1/10

From definition of slope of line,

m = tan θ

1/10 = tan θ

θ = arctan(1/10)

θ = arctan(0.1)

θ = 5.71°

This means, when line makes an angle of  5.71° with the positive x-axis then the slope of line is 1/10

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Answer: V=282.74

Step-by-step explanation:

We are given the volume of a cylinder, which is V=πr²h. Luckily, we are given the radius and height, so all we have to do is to plug them into the formula.

V=π(3)²(10)        [exponent]

V=π9(10)            [multiply]

V=90π

If we plug this into the calculator, we get V=282.74.

Basic inequality is a mathematical statement that expresses the relative size of two values. It usually involves the use of an inequality symbol, such as “>” (greater than), “<” (less than), “≥” (greater than or equal to), or “≤” (less than or equal to). An example of a basic inequality is x > 5, which means that x is greater than 5.

The concept of basic inequality is an important one in mathematics, as it helps us to understand relationships between values, and to identify patterns and trends. Inequality is a type of comparison between two or more values, and can be used to determine the size or order of values.

Inequality symbols such as “>” (greater than), “<” (less than), “≥” (greater than or equal to), and “≤” (less than or equal to) are used to indicate the magnitude of the difference between values.

Basic inequality can be used to solve equations, solve for variables, and compare data sets. It can also be used to identify trends in data sets, such as the average, rate of change, and correlations. By understanding basic inequality, we can make predictions and better understand the relationships between values.

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Answer: 24

Step-by-step explanation:

The two triangles IFE and IHG are similar(AA Similarity).

The ratio of line IH : IF = EF : GH = 5 : 12

Therefore, 1.4x + 3 : 6.1x-6.5 = 5:12

Solve this ratio, and you'll get

16.8x+36 = 30.5x-32.5

13.7x = 68.5

x = 5

X is 5, so using X, we can find the length of HI.

30.5-6.5 = 24

Therefore, the answer is 24.

The value for the length of rafters is found using trigonometric function, and that is length = 43.38 inches.

What is trigonometric function?

Simply put, trigonometric functions also referred to as circular functions are the functions of a triangle's angle. This means that these trig functions provide the relationship between the angles and sides of a triangle.

The roof and the ceiling form two back-to-back right triangles .

It bisects the top angle so the top angle of the right triangle would be 35 degrees.

Half the width of 36 foot forms the lower side of the triangle - 36/2 = 18 inch.

The rafter (not counting the 1 foot overhang) is the hypotenuse, call it h.

We can use the sine trigonometric function to find the hypotenuse.

Sin θ = Opposite/Hypotenuse

Sin 35° = 18/h

0.573576 = 18/h

h = 18/0.573576

h = 31.38

Since, 1 foot = 12 inch rafter is extended then -

h = 31.38 + 12 = 32.38 foot

Therefore, the length of rafters is 43.38 inches.

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So on solving the provided question, we can say that the, here  Midsegment theorem has been proved.

Let's draw a line segment connecting the midpoints of two of the sides of a triangle ABC, say AB and BC. We must demonstrate two things in order to establish the Triangle Midsegment Theorem: DE = (1/2)AC. DE is similar to AC.

DE=EF        //Construction

AE=EC        //Given, DE is a midsegment

∠AED ≅ ∠CEF    // Vertical angles

ΔADE ≅ΔCFE     // Side-Angle-Side postulate

∠DAE ≅ ∠FCE   // corresponding angles in congruent triangles, (CPCTC)

AB||CF               // Converse Alternate Interior Angles Theorem

AD=CF              // corresponding sides in congruent triangles, (CPCTC)

AD=DB              //Given, DE is a midsegment

CF=DB              //(7), (8), the transitive property of equality

DFCB is a parallelogram //two opposite sides of a parallelogram, CF & DB, are equal and parallel

DE||BC            //(10), opposite sides of a parallelogram

DF=BC            //(10), opposite sides of a parallelogram

DE=EF           //(4)  corresponding sides in congruent triangles, (CPCTC)

DE= ½DF

DE= ½BC       //(12) , (14), substitution

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So on solving the provided question, we can say that the, here  Midsegment theorem has been proved.

Let's draw a line segment connecting the midpoints of two of the sides of a triangle ABC, say AB and BC. We must demonstrate two things in order to establish the Triangle Midsegment Theorem: DE = (1/2)AC. DE is similar to AC.

DE=EF        //Construction

AE=EC        //Given, DE is a midsegment

∠AED ≅ ∠CEF    // Vertical angles

ΔADE ≅ΔCFE     // Side-Angle-Side postulate

∠DAE ≅ ∠FCE   // corresponding angles in congruent triangles, (CPCTC)

AB||CF               // Converse Alternate Interior Angles Theorem

AD=CF              // corresponding sides in congruent triangles, (CPCTC)

AD=DB              //Given, DE is a midsegment

CF=DB              //(7), (8), the transitive property of equality

DFCB is a parallelogram //two opposite sides of a parallelogram, CF & DB, are equal and parallel

DE||BC            //(10), opposite sides of a parallelogram

DF=BC            //(10), opposite sides of a parallelogram

DE=EF           //(4)  corresponding sides in congruent triangles, (CPCTC)

DE= ½DF

DE= ½BC       //(12) , (14), substitution

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The winning jump in the long jump competition is 5.63m.

Complete Question: Sarah competes in a long jump competition.  Her first jump is 4.25m.  Her best jump is 12% more than this.  However her best best jump is 15% lower than the winning jump.  Work out the length of the winning jump.

We can start by using the information given to set up equations to find the distance of Sarah's second and third jumps, and then the winning jump.

We know that Sarah's first jump is 4.25m and that her second jump is 12% more than this, so we can find her second jump distance by applying the 12% increase:

4.25m + (12/100)*4.25m = 4.25m + 0.51m = 4.76m

We also know that Sarah's third jump is 15% lower than the winning jump in the long jump competition 5.63m. jump, so we can find the winning jump distance by applying the 15% decrease:

Winning Jump = 4.76m / (1- 15/100) = 4.76m / 0.85 = 5.63m

So the winning jump in the long jump competition is 5.63m.

Therefore, The winning jump in the long jump competition is 5.63m.

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Answer:

3

Step-by-step explanation:

We can set this up as a ratio

b = bus s = students

1b:32s

?b:96s

To find out how many more students there are more of than 1 bus can fit, we do 96 divided by 32

96/32 = 3

so we do 1 x 3 for the busses because there are 3 times the number of kids.

3 buses will be needed for 96 students on the field trip.

Step-by-step explanation:

Given : 1 bus for 32 students

To find: How many buses for 96?

Divide 96 from 32 :

96/32 = 3 { 32 × 3 = 96 }

Also to make it easier to understand:

32 students have 1 bus, 32 + 32 + 32 = 96! 96 will have 3 buses.

Hope you understood it, God Bless you, Keep on studying !

The coordinates of the points after dilation is D'(-5/3,1),  E'(1,-2/3), F'(5/3,-1), and C'(-1,-4/3)

Resizing an item uses a transition called dilation. Dilation is used to enlarge or contract the items. The result of this transformation is an image with the same shape as the original. However, there is a variation in the shape's size.

Resizing an item uses a transition called dilation. Dilation is used to enlarge or contract the items. The result of this transformation is an image with the same shape as the original. However, there is a variation in the shape's size.

Given that the scale factor is 1/3. The coordinates of the points after dilation is,

D(-5,3) = D'(-5/3,1)

E(3,-2) = E'(1,-2/3)

F(5,-3) = F'(5/3,-1)

C(-3,-4) = C'(-1,-4/3)

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Answer:

Biblical metaphors are literary devices that can be found throughout the Bible. A biblical metaphor is a figure of speech that exists in or is otherwise connected to the Christian Bible.

It is apparent that the Bible uses metaphors a lot in its terminology (e.g., hands, eyes, feet of God, etc.). Both history and metaphor are included in the Bible.

The biblical account of creation is a complex and contested topic. Some Christians believe that the story of creation is literally true, while others believe that it is a metaphor. There are arguments to be made for both sides of this issue.

Those who believe that the biblical account of creation is literal point to the fact that the Bible is the word of God, and that God cannot lie. They also point to the fact that the scientific evidence does not support the idea of evolution.

Those who believe that the biblical account of creation is metaphorical point to the fact that the Bible is a book of poetry, and that poetry often uses metaphors and symbolism. They also point to the fact that the scientific evidence does support the idea of evolution.

In conclusion, there is no easy answer to the question of whether or not the biblical account of creation is literal or metaphorical. Christians must decide for themselves what they believe. However, it is important to note that both sides of this issue have valid arguments.

In terms of Christian teachings, it is important to remember that the Bible is a complex and multifaceted text. It contains a variety of genres, including poetry, history, prophecy, and law. As such, it is not always easy to interpret the Bible literally. In some cases, it may be more helpful to view the Bible as a metaphor or allegory. This is especially true when the Bible is talking about things that are beyond our understanding, such as the creation of the universe.

Ultimately, it is up to each individual Christian to decide how they interpret the Bible. However, it is important to remember that the Bible is a sacred text, and that it should be treated with respect.

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The simplified form is Negative 3 m Superscript 12 Baseline n Superscript 6 Baseline. Option C

The index form of a number can be defined mathematically as number written as an exponential expression.

It is also be written as a single number raised to another number.

The other names for index forms are the scientific notation and standard forms.

From the information given, we have that;

This is represented as;

-24m⁵n⁴/8m⁻⁷n⁻²

Divide the common terms, we have;

-3m⁵n⁴/m⁻⁷n⁻²

Take the inverse exponents of the denominators

-3m⁵n⁴×m⁷n²

Add the exponents

-3m⁵⁺⁷n⁴⁺²

-3m¹²n⁶

Hence, the expression is -3m¹²n⁶

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Answer:

C on edge 23

Step-by-step explanation:

Answer: Yes it can construct a negative base.

Step-by-step explanation:

A negative base (or a negative radix) each position holds multiples of the appropriate power of the system's base.

Answer:

x = 3 and y = 0     or  (3, 0)

Step-by-step explanation:

First, you use the first equation to find what y equals.

3x + y = 9

y = 9 - 3x

Next, you plug in y to the second equation.

2x - y = 6

2x - (9 - 3x) = 6

2x - 9 + 3x = 6

5x - 9 = 6

5x = 15

x = 3

Then, you plug in x in the first equation.

3x + y = 9

3(3) + y = 9

9 + y = 9

y = 0

Rote learning is NOT a problem- solving strategy in mathematics.

Mathematics is the study of numbers, shape, quantity, and patterns. Teaching methods of mathematics include problem- solving, lecture, inductive, deductive, analytic, synthetic and Discovery Method. The teacher adopts any method according to the needs and interest of students.

Now, According to the question:

Problem Solving Strategy in Mathematics: In a mathematics class, students at the primary level enjoy learning, solving problems, develop mathematical curiosity and become confident in using mathematics to analyze and solve problems.

Problem - solving is a learner- centered approach that emphasizes learner's active involvement in the learning process. In this approach, teachers create a problematic situation for students and then assists them in perceiving, defining and stating the problems in a fear- free classroom environment.

Hence, Rote learning is NOT a problem- solving strategy in mathematics

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Number of cups of flour required to make the same recipe for 3 cups of almond as per the given relation is equal to 7.5 cups of flour.

As given in the question,

For the given recipe cups of flour and cups of almond given is :

2 cups of almond used for 5 cups of flour

The relation is written in the form of expression :

2 cups of almond = 5 cups of flour

⇒ 1 cups of almond = ( 5 / 2 ) cups of flour

⇒ 3 cups of almond =  [ ( 5 / 2 ) × 3 ] cups of flour

⇒ 3 cups of almond =  ( 15 / 2 ) cups of flour

⇒ 3 cups of almond = 7.5 cups of flour

Therefore, the cups of flour required for 3 cups of almond is equal to 7.5 cups of flour.

The above question is incomplete , the complete question is :

A recipe call for 2 cup of almond for 5 cup of flour. Using the same recipe, how many cup of flour will you need for 3 cup of almond?

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To find the area of  a graph and a line, you need to use the integral calculus formula.

Specifically, you will use the definite integral. The definite integral of a function f(x) over an interval [a, b] is given by the formula:

Integral of f(x)dx = F(b) - F(a)

Where F(x) is the antiderivative of f(x). To find the area of a graph, you will first need to find the antiderivative of the function. Then you will substitute the a and b values of the interval into the formula. Finally, subtract F(a) from F(b) to find the area of the graph. To find the area of a line, you will need to use the same formula. However, since the line is a straight line, the antiderivative of the line is simply the equation of the line, so the formula simplifies to:

Integral of f(x)dx = (mx + c)b - (mx + c)a

Where m is the slope of the line, and c is the y-intercept. After plugging in the a and b values of the interval, subtract (mx + c)a from (mx + c)b to find the area of the line.

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Answer:

Step-by-step explanation:

In a right triangle, the sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse (the longest side of a right triangle, also called the hypotenuse). Therefore, to find the length of the side opposite angle θ, we can use the following formula:

opposite = sin(θ) * hypotenuse

In this case, we know that θ = 25 degrees and the adjacent side has a length of 3. Since we are dealing with a right triangle, we can use the Pythagorean theorem to find the hypotenuse:

hypotenuse = sqrt(adjacent^2 + opposite^2) = sqrt(3^2 + opposite^2) = sqrt(9 + opposite^2)

So we can substitute that value into the first equation:

opposite = sin(25) * sqrt(9 + opposite^2)

Solving for opposite, we get:

opposite = sin(25) * sqrt(9 + opposite^2)

opposite = 0.4226 * sqrt(9 + opposite^2)

We can calculate the value of sin(25) which is 0.4226.

opposite = 0.4226 * sqrt(9 + opposite^2)

opposite = 0.4226 * sqrt(opposite^2 + 9)

Square both sides of the equation to get:

opposite^2 = (0.4226 * sqrt(opposite^2 + 9))^2

opposite^2 = 0.1799 * opposite^2 + 1.5874

Subtracting 0.1799 * opposite^2 from both sides of the equation gives:

0 = 1.5874

Solving for the opposite, we get:

opposite = sqrt(1.5874)

opposite = 1.259

So the side opposite to the angle of 25 degrees is 1.259 units long, rounding to at least 3 decimal places.

The type of function that represents function g(x) is given as follows:

Logarithmic.

To define the function in this problem, we must observe a pattern with the inputs and the outputs.

The input and output pairs are given as follows:

From the pattern, we can see that each output is obtained as the power of 3 to which the input is elevated.

This means that the function is a logarithmic function, more specifically the logarithm of base 3 of x.

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Strong negative correlation is 0.89. While the size of the number (0.89) suggests a strong link, the negative (-) symbol before the number implies it is negative.

Correlation coefficients demonstrate how strongly two independent variables, x and y, are connected linearly.

A linear correlation coefficient greater than zero indicates a relationship that is favorable.

A value less than zero denotes a bad relationship. To summarize, when x and y have values of 0, there is no link between them.

A statistic that describes how closely two variables' movements are related to one another is the correlation coefficient (r2).

The most widely used correlation coefficient is the Pearson product-moment correlation, which is used to detect if two variables are linearly related.

This correlation value, however, might not always be an accurate predictor of reliance in a non-linear link.

Hence, Strong negative correlation is 0.89. While the size of the number (0.89) suggests a strong link, the negative (-) symbol before the number implies it is negative.

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The value of function f+g(x) will be -5^x-3x-6 i.e. C.

What is a function?

A function is defined as a relation between a set of inputs having one output each. In simple words, a function is a relationship between inputs where each input is related to exactly one output. Every function has a domain and co-domain or range. A function is generally denoted by f(x) where x is the input. The general representation of a function is y = f(x).

There are several types of functions in math. Some important types are:

Injective function or One to one function: When there is mapping for a range for each domain between two sets.

Surjective functions or Onto function: When there is more than one element mapped from domain to range.

Polynomial function: The function which consists of polynomials.

Inverse Functions: The function which can invert another function.

Now,

Given function are f(x)=-5^x-4

anf g(x)=-3x-2

so f+g(x) will be f(x)+g(x)

f+g(x)=-5^x-4-3x-2

=-5^x-3x-6

Hence,

           The value of function f+g(x) will be the polynomial -5^x-3x-6.

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The two plans will cost the same when Lucy has driven average speed 550miles. ($55 + $0.50 x 550)= $330

Divide the overall distance traveled by the time period to find the average speed. An someone traveling at an average speed of 40 miles divided by 40 minutes, or 1 mile per minute, would be driving 20 miles north and then 20 miles south to arrive at the same location (60 mph)

The overall distance traveled by an object over the total amount of time is the thing's average speed. The average speed v of an object during a motion that covers a total distance of d and a total duration of t can be estimated using the formula v = d t.

(0.6*550)=$330

The difference between both the trip rate is .10$

and the difference between initial fee is 55$

so after 1 mile the difference is .10$

the 55$can be earn after

55*10=550 mile.

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The equation connecting h and v is h = 56/v and the value of h when v = 5 is 11.2

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

Variation = inverse variation

h = 8, v = 7

An inverse variation is represented as

k = vh

Where k is the constant of proportion

Substitute the known values in the above equation, so, we have the following representation

k = 7 * 8

Evaluate

k = 56

So, we have the following representation

vh = 56

Make h the subject

h = 56/v

Substitute the known values in the above equation, so, we have the following representation

h = 56/5

Evaluate

h = 11.2

Hence, the solution is 11.2

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The borderline if the linear inequality's symbol is > (greater than) or (greater than or equal to).

In mathematics, an inequality is a connection that does not have an equal sign between two expressions or values. Inequality follows imbalance, therefore. When two numerical values are not equal, an inequality establishes a connection between them. Difference between equality and inequality Use the not equal symbol, which is most often used, when two values are not equal (). No matter how little or huge the values, various inequalities are employed to contrast them. The two sides can be changed until the variables are all that is left in order to solve several basic inequalities. Unevenness is caused by a number of factors, including: The sum or division of negative values on both sides. The left and right are exchanged.

given

inequality for y

5x – 2y ≤ 10

– 2y ≤ – 5x + 10

y ≥ 5/2x - 5

When dividing by a negative number, remember to switch the inequality sign.

In a linear inequality, you will shade below the boundary line if the sign is (less than) or (less than or equal to).  The borderline if the linear inequality's symbol is > (greater than) or (greater than or equal to).

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The value of m in the equation is 35

Index is the power or exponent which is raised to a number or a variable e.g. in number 3⁵, 5 is the index of 3. The 3 is called the base. The plural form of index is indices.

Product law: To multiply two variables or numbers with the same base, we need to add their powers and raise them to that base":  P⁴ + P⁶ = P¹⁰

[tex](x^{m})^{3}[/tex] = (x¹³)⁵ × (x⁻⁸)⁻⁵

[tex](x^{m})^{3}[/tex] =  x⁶⁵ × x⁴⁰

[tex](x^{m})^{3}[/tex] = x⁶⁵⁺⁴⁰

[tex](x^{m})^{3}[/tex] = x¹⁰⁵

[tex](x^{m})^{3}[/tex] =  (x³⁵)³

Comparing both sides of the equation, m = 35

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