What are the first 3 triangular numbers?

The log2 function is a mathematical function that returns the logarithm of a given number to the base 2.

It is used to calculate the power to which a number must be raised to get a particular result.

For example, the log2 of 16 is 4, since 2^4 = 16. It can also be expressed as log2 16 = 4. Log2 is used in many areas of mathematics, including number theory, probability theory, and calculus.

It is also used in computer science and engineering to solve complex problems. Furthermore, it is used in cryptography to calculate the key sizes of encryption algorithms. In general, it is an essential tool for solving mathematical and scientific problems.

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a) The quadratic function is factored as follows: f(x) = 4(x - 0.5)(x + 2.5).

b) The x-intercepts of the quadratic function are given as follows: x = -2.5 and x = 0.5.

c) The end behavior of the quadratic function is given as follows: as x -> ± ∞, f(x) -> ∞.

d) The graph is constructed at the end of the answer, considering the x-intercepts, the y-intercept, the end behavior and the vertex.

The quadratic function for this problem is defined as follows:

f(x) = 4x² + 8x - 5.

Hence the coefficients are given as follows:

a = 4, b = 8, c = -5.

The discriminant is given as follows:

D = 8² - 4 x 4 x (-5)

D = 144.

Hence the x-intercepts are given as follows:

Considering the intercepts and the leading coefficient of a = 4, the function is factored as follows:

f(x) = 4(x - 0.5)(x + 2.5).

As the leading coefficient is positive, the parabola is concave up, meaning that the end behavior is that the function increases to infinity both at the left tail and at the right tail.

The y-intercept is given as follows:

f(0) = 4(0)² + 8(0) - 5

f(0) = -5.

The coordinates of the vertex are given as follows:

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On solving the provided question, we can say that - by the prime factors we have x^2 + 1 +2x = X= -1, -1

Any natural number other than 1 with just itself and the number 1 as its divisors is considered a prime factor. Actually, 2, 3, 5, 7, 11, and so on are some of the first prime numbers. an amount that has been multiplied to produce another amount. By dividing 15 by 3 and 5, for instance, we get 3 5 = 15. main elements: Prime factors include all factors that are prime but are not composite. A few prime factors of 30 are 2, 3, and 5. Only 2 and 3 are prime factors of 12, so listing 2 twice as 2 2 3 (or 22 3) is necessary to factorize 12. The sum of 2 + 3 is not 12.

here we have the equation,

[tex]x^2 + 1 + 2x\\[/tex]

(x+1)(x+1)

x = -1, -1

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Answer:

[tex]108\pi[/tex] or approx. [tex]339.3 m^{2}[/tex]

Step-by-step explanation:

The question asks us to find the total surface area of the given hemisphere. The question gave us the formula for total surface area of a sphere, [tex]SA_{tot} =4\pi r^2[/tex]. But we need it for a hemisphere, so multiply the equation by [tex]\frac{1}{2}[/tex].

[tex]SA} = (\frac{1}{2}) 4\pi r^2[/tex] => [tex]SA} =2\pi r^{2}[/tex]

We also need the area of the base, which is circle. [tex]A_{circle}=\pi r^{2}[/tex].

Now add these equations together to get our equation.

[tex]SA_{tot} =2\pi r^2 +\pi r^{2}[/tex] => [tex]SA_{tot} =3\pi r^2[/tex]

Now we have the proper equation for total surface area of a hemisphere, [tex]SA_{tot} =3\pi r^{2}[/tex]. Plug in the value of [tex]r[/tex] which is given in the problem, [tex]r=6m[/tex].

=> [tex]SA_{tot} =3\pi (6)^{2}[/tex] => [tex]SA_{tot} =3\pi (36)[/tex] => [tex]SA_{tot} =108\pi m^{2}[/tex] or approx. [tex]339.3 m^{2}[/tex]

The steps to solve the equation  (5y - 1)/2 = 7 are assigned and attached

The expression in the problem is (5y - 1)/2 = 7

Clearing the fraction

The fraction is cleared by applying the Multiplication Property of Equality

(5y - 1)/2 = 7

5y - 1 = 14

Adding 1 to both sides of the equation is done using Addition Property of Equality

5y - 1 + 1 = 14 + 1

5y = 15

Making y the subject of formula Division Property of Equality is applied

5y / 5 = 15 / 5

y = 5

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At a significance level of 0.05, the sample data is not consistent with the null hypothesis that the proportion of population who would respond "yes" is 0.5. The P-value of 0.0005 is less than 0.05, and also the sample proportion 0.4616 is not in the interval (0.477, 0.523) which we found as 95% Confidence interval.

Therefore, we reject the null hypothesis and conclude that the population proportion of "yes" responses is different from 0.5.

The P-value for a score test for H0: π = 0.5 can be found using the z-score and a standard normal table. The z-score is calculated as

[tex]z = \frac{x-0.5}{0.5\sqrt{\frac{x-0.5}{n} } }[/tex], that is

z = (x - 0.5) / (√(0.5(1 - 0.5) / n)

where x is the sample proportion of "yes" responses (842 / (842 + 982) = 0.4616), π is the population proportion of "yes" responses, and n is the sample size (842 + 982 = 1824).

[tex]z = \frac{0.4616-0.5}{0.5\sqrt{\frac{0.4616-0.5}{1824} } }[/tex]

= (0.4616 - 0.5)/ (√(0.5(1 - 0.5) / 1824)

This gives a z-score of -3.28.

To find the P-value, we can use the standard normal table to find the probability of observing a z-score less than -3.28. This P-value is approximately 0.0005, which is less than the commonly used significance level of 0.05. Therefore, we would reject the null hypothesis that π = 0.5.

To construct a 95% confidence interval for π, we can use the formula for a normal approximation interval:

π ± z×(√(π(1-π) / n)) that is

[tex]\pi \frac{+}{} z\frac{\pi (1 -\pi )}{n}[/tex]

Where π = 0.5, z = 1.96 (for a 95% confidence level), and n = 1824.

This gives a 95% confidence interval of (0.477, 0.523)

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Answer:

(20/3,0) is your x-intercept

Step-by-step explanation:

To find x-intercept, let y = 0

[tex] \displaystyle{3x = 20}[/tex]

Solve for x, divide both sides by 3:

[tex] \displaystyle{ \dfrac{3x}{3} = \dfrac{20}{3}} \\ \\ \displaystyle{ x = \dfrac{20}{3}}[/tex]

Hence, the x-intercept is (20/3,0). We write in order pair since it's x-intercept and x = 20/3 at y = 0

Answer:

x= 1/4 - 1/4 x= 1/4 + 1/4

Step-by-step explanation:

Answer:

Type: (+ ve)

Number of solutions: 2

Step-by-step explanation:

We are given an equation (-8x² + 4x - 1 = 0)

Now, change the sign

-8x² + 4x - 1 = 0

8x² - 4x + 1 = 0

Here,

Now, discriminant

D = b² - 4ac

D = (8)² - (4)(8)(1)

D = 64 - 32

D = 32 > 0

Here, 32 is greater than 0 so, it has two solutions.

The solution of the equation 3(2m - 1) ≤ 4m + 7 is option A m ≤ 5.

A mathematical statement known as an inequality in algebra uses the inequality symbol to show the relationship between two expressions. An inequality symbol denotes that the expressions on each side are not equal.

The given inequality is 3(2m - 1) ≤ 4m + 7

Remove the brackets by multiplying 3 with each term inside it:

6m - 3 ≤ 4m + 7

Subtract 4m from both sides of the equation.

6m - 3 - 4m ≤ 4m + 7 - 4m

2m -3 ≤ 7

Add 3 to both sides of the equation.

2m - 3 + 3 ≤ 7 + 3

2m ≤ 10

Divide both sides of the equation with 2.

2m/2 ≤ 10/2

m ≤ 5

Hence the solution of the equation is m ≤ 5, that is option A.

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By the Number line, the true statements are,

⇒ 1 < 3

⇒ - 3 < 1

Number line is a horizontal line where numbers are marked at equal intervals one after another form smaller to greater.

We have to given that;

In Number line, Plot the numbers - 3, 1 and 3.

Now, After plotting the numbers in Number line, we get;

⇒ - 3 < 1 < 3

Thus, The correct statements are,

⇒ 1 < 3 and - 3 < 1

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Answer:14

Step-by-step explanation:You can use distance formula to find the length of each side, and then add them together to approximate the perimeter.

The inverse of the logarithmic function f x log9x is f-1(x)=9^x.

The inverse of a logarithmic function is the exponential function. To find the inverse of the logarithmic function f x log9x, we can use the following equation: f-1(x)=9^x. This equation states that the inverse of the logarithmic function f x log9x is the exponential function f-1(x)=9^x.

The inverse of the logarithmic function f x log9x is f-1(x)=9^x, which is the exponential function.This means that for any value of x, the inverse of the logarithmic function f x log9x is equal to 9 raised to the power of x. For example, if x=2, then the inverse of the logarithmic function f x log9x is 9^2, or 81. Similarly, if x=3, then the inverse of the logarithmic function f x log9x is 9^3, or 729. In general, the inverse of the logarithmic function f x log9x is 9 raised to the power of x, for any value of x.

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Answer: f –1(x) = 9x

Step-by-step explanation:

The value of x is 15 units and the value of y is 12 units from the given figure.

Two triangles are similar if the angles are the same size or the corresponding sides are in the same ratio. Either of these conditions will prove two triangles are similar.

From the figure, AB is parallel to CD.

∠AOB=∠COD (Vertically opposite angles)

∠OAB=∠ODC (Alternate angles)

By AA similarity ΔAOB similar to ΔCOD

Now, 9/x =y/20 =3/5

9/x=3/5

x=15 units

y/20=3/5

y/4=3/1

y=12 units

Therefore, the value of x is 15 units and the value of y is 12 units from the given figure.

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Answer:

D

Step-by-step explanation:

Because the account that gets debited depends on the specific transaction

A negatively skewed distribution, sometimes referred to as a left-skewed distribution, is a kind of distribution where more values are clustered on the right side (tail) of the scatterplot but the left tail of the distribution graph is longer.

The mean of positively skewed data will be lower than median in a distribution that has a negative skewness, which is the exact reverse of what would be expected. The distribution exhibits zero skewness if the data is graphed symmetrically, regardless of how long or thick the tails are.

More values are plotted on the graph's right side, the distribution's tail is longer on the left, and the distribution is said to be negatively skewed.

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Answer:

perpendicular

Step-by-step explanation:

If Andrew takes same amount of time to kayak 4 miles upstream and 9 miles downstream in a river with a 6 mph current, then the speed of Andrew’s kayak in still water is 15.6 miles per hour.

Therefore the answer is 15.6 mph.

We can begin solving the problem by using the equation:

Speed (relative to water) = Speed (in still water) + Speed (of the water)

Let x be the speed of Andrew's kayak in still water.

When kayaking upstream, his speed relative to the water is x - 6 mph (since the current is moving in the opposite direction at 6 mph).

When kayaking downstream, his speed relative to the water is x + 6 mph (since the current is moving in the same direction at 6 mph).

Since the problem states that it takes Andrew the same amount of time to kayak 4 miles upstream as it takes him to kayak 9 miles downstream, we can set up the following equation knowing time = distance/ speed :

4/ (x - 6) = 9/ (x + 6)

Simplifying and solving for x we get

4(x + 6) = 9(x - 6)

4x + 24 = 9x - 54

9x - 4x = 54 + 24

5x = 78

x = 15.6

So, Andrew's kayak speed in still water is 15.6 miles per hour.

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Answer:

55.2049%

Step-by-step explanation:

R², the coefficient of determination, is the proportion of the variation in the dependent variable that is predictable from the independent variable.

In this case, if we square our correlation coefficent r=0.743, we get R²=0.552049, which means that 55.2049% of the variation in the number of yearly boaters can be explained by differences in lake sizes

The answers are given as 1) 165 m 2)3 seconds 3) 9 seconds 4)120 m

A curve is defined as a smoothly- flowing continuous line that has bent. It does not have any sharp turns. The way to identify the curve is that the line bends and changes its direction at least once. Curve of the graph is a pictorial representation of a particular function.

Given :  A object is thrown from the top of the building thus by observing the graph we derive the following information

1) Maximum height = 165 m

2) The object reaches maximum height at t=3 seconds

3) The time it took to reach the ground is 9 seconds

4) The building is 120 m tall

Hence, the answers are  1) 165 m 2)3 seconds 3) 9 seconds 4)120 m

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Answer:

as a factor 2(x-1) this should help or be the correcr answer

Answer:

f(3) = 2

Step-by-step explanation:

See the attached graph.  f(3) = ____ is asking us to find the value of the function, f, at the point x = 3.  At x = 3, the value of y is 2.

The simplified form of the algebraic expression [tex]2c-7c[/tex] is [tex]-15[/tex]

The given algebraic expression is [tex]2c-7c[/tex] and we need to use [tex]c=3[/tex]

Let's substitute the value of [tex]c[/tex] in the given algebraic expression, that is

                      [tex]=2(3)-7(3)\\=6-21\\=-15[/tex]

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Answer:

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

Given expression

Simplify it:

Find its value by plugging in the value of c:

Triangle ABD is a right-angled triangle. The value of s which is the hypotenuse of the triangle ABD is 17 units.

A triangle is a three-edged polygon with three vertices. It is a fundamental form in geometry. Triangle ABC denotes a triangle with vertices A, B, and C. In Euclidean geometry, any three non-collinear points define a unique triangle and, by extension, a unique plane. Triangles are three-sided polygons with three vertices. The angles of the triangle are formed by connecting the three sides end to end at a point. The total of the triangle's three angles equals 180 degrees.

Here,

Given information:

From the given figure, the following information can be extracted:

Triangle ABD is a right-angled triangle.

Base AB is 8 units, Height BD is 15 units.

s is the hypotenuse of the triangle ABD.

Use the Pythagoras theorem to solve for the value of s or AD,

AD²=AB²+BD²

s²=8²+15²

s²=289

s=17 units

Therefore, the value of s which is the hypotenuse of the triangle ABD is 17 units.

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I’ve added my work attached :)
n = 250

Using the information in the table the domain is 4 and the range  [-1,1]

The term "domain" refers to a group of potential input values, and the domain of a graph is made up of all the input values visible on the x-axis. The y-axis displays the range, which is a collection of possible output values.

The domain is typically listed in the left column and the range is typically listed in the right column in a table like the one below. The domain and range of a relation or function, respectively, can be thought of as the input values (x) and the output values (y), respectively.

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Answer:

7.0

Step-by-step explanation:

from the first figure which is not a 0

7.3-7.0(1 significant figure)

Answer:

[tex]\frac{d}{c}[/tex]

Step-by-step explanation:

[tex]\frac{cxcxdxdxd}{cxcxcxdxd}[/tex]  You can cross out 2 c's from the top and the bottom and 2 d's from the top and the bottom.  It now looks like this

[tex]\frac{d}{c}[/tex]

Answer:

Step-by-step explanation:

y=2x-5+2

then you do minus 2 on both sides

Hope this helped!

The length of the hypotenuse of a 45 90 45 triangle whose leg measures 9 in is 12.73 in.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the square of the length of the other two sides (a and b).

In this case, a = b = 9 in, so:

c2 = 92 + 92

c2 = 81 + 81

c2 = 162

c = √162

c = 12.73 in

The length of the hypotenuse of a 45 90 45 triangle whose leg measures 9 in is 12.73 in, calculated using the Pythagorean theorem.The length of the hypotenuse of a 45 90 45 triangle whose leg measures 9 in is 12.73 in.

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The complex numbers are solved using De Moivre's Theorem and then are matched with their respective solutions.

What is De Moivre's Theorem?

De Moivre's formula, also known as the theorem and identity of de Moivre, says that for any real number x and integer n, holds -

(cos x + i sin x)^n=cos nx + i sin nx

where i is the imaginary unit (i^2 = 1).

Also, [r(cos x + i sin x)]^n = r^n(cos nx + i sin nx) or [r cis x]^n = [r^n cis nx].

The first complex number is (1+i)^5.

Solve using De Moivre's formula -

(1+i)^5 = [√2{(1/√2)+(i/√2)}^5]

=(√2)^5[cos(π/4)+i sin(π/4)]^5

=(√2)^5[cos(5π/4)+i sin(5π/4)]

=(√2)^5[(-1/√2)-(i/√2)]

=4(-1-i)

=-4-4i

The second complex number is (-1+i)^6.

Solve using De Moivre's formula -

(-1+i)^6=(1-i)^6

=[√2{(1/√2)-(i/√2)}^6]

=(√2)^6[cos(-π/4)+i sin(-π/4)]^6

=(√2)^6[cos(-6π/4)+i sin(-6π/4)]

=8(0+1i)

=8i

The third complex number is 2[cos(20)+i sin(20)]^3.

Solve using De Moivre's formula -

2[cos(20)+i sin(20)]^3=2^3[cos(60)+i sin(60)]

=8[(1/2)+{(√3)i/2}]

=4+4√3i

The last complex number is 2[cos(π/4)+i sin(π/4)]^4.

Solve using De Moivre's formula -

2[cos(π/4)+i sin(π/4)]^4=2^4[cos(4π/4)+i sin(4π/4)]

=16[cos(π)+i sin(π)]

=16(-1+i(0)]

=-16

Therefore, the complex numbers are solved using the theorem.

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