an athlete is training to run a marathon. she plans to run 2 miles the first week. she increases the distance by 8% each week. which function models how far she will run in the nth week?

Answer: the answer is a

Step-by-step explanation:

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The approximation sin (x) = x - x^3/6 has an absolute error below 0.000001 for the interval [-0.164375,+0.164375].

In mathematics, the alternating series is an infinite series of a term in the form of

[tex]{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}}{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}} \\{\displaystyle \sum _{n=0}^{\infty }(-1)^{n+1}a_{n}}{\displaystyle \sum _{n=0}^{\infty }(-1)^{n+1}a_{n}}[/tex]

The expansion series of sin(x)

[tex]\rm sin (x) = \frac{x}{1!} -\frac{x^{3} }{3!} +\frac{x^{5} }{5!} -\frac{x^{7} }{7!} +..[/tex]  (a)

the given approximation [tex]\rm sin (x) = \frac{x}{1!} -\frac{x^{3} }{3!}[/tex]     (b)

So, the approximation error term =(b)-(a)

       [tex]\rm = -[ +\frac{x^{5} }{5!} -\frac{x^{7} }{7!} +\frac{x^{9} }{9!}-....]\\\rm = -\frac{x^{5} }{5!} +\frac{x^{7} }{7!} -\frac{x^{9} }{9!}+....[/tex]

The alternative series theorem states that we need to ensure that the absolute value of the first term [tex]\rm \frac{x^{5} }{5!}[/tex]less than the error limit, For that the sum will be below the error limit, which means

[tex]\rm \frac{x^{5} }{5!}[/tex] <  0.000001

Solving for x, we get

x [tex]\rm x^{5} = 5! \times 0.000001\\ = 0.000120\\\rm = 0.000120^{1/5} \\ = 0.164375[/tex]

Therefore, the approximation   [tex]\rm sin (x) = \frac{x}{1!} -\frac{x^{3} }{3!}[/tex] has an absolute error below 0.000001 for the interval [-0.164375,+0.164375].

by checking

[tex]\rm sin(0.164375)-(0.164375)^{3/3!} \\= 9.993513\times10^{-7} < 0.000001[/tex]

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Answer:

Answers are

Y

2+s

4x^3+y

Step-by-step explanation:

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The first, second, and third are the polynomials.

Polynomial is an algebraic expression that consists of variables and coefficients. Variable are called unknown. We can apply arithmetic operations such as addition, subtraction, etc. But not divisible by variable

Given

1.  y

2.  x² - x +√2

3.  2 + s

4.  4x³ + y

To find

Which expressions are polynomials?

1.  y is a polynomial with variable y.

2.  x² - x +√2 is a polynomial with variable x.

3.  2 + s is a polynomial with variable s.

4.  4x³ + y is not a polynomial because the polynomial has only one variable in it.

Thus, the first, second, and third are the polynomials.

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Answer:

The measure of each interior angle of the polygon is [tex]168\°[/tex]

Step-by-step explanation:

we know that

The sum of the internal angles in a polygon is equal to

[tex]S=(n-2)180\°[/tex]

where

n is the number of sides of polygon

In this problem we have

[tex]n=30\ sides[/tex]

substitute

[tex]S=(30-2)180\°=5,040\°[/tex]

To find the measure of each internal angle divide the total sum of internal angles by the number of sides

so

[tex]5,040\°/30=168\°[/tex]

Answer:168

Step-by-step explanation:

N is the number of sides. Formula is (n-2)×180/n. So (30-2)= 28×180=5040÷30= 168

Using Cramer's Rule, the value of x in the given system of linear equations is 1.

We can solve the given system of equations using Cramer's Rule. Cramer's Rule states that in a system of linear equations, if the determinant of the coefficient matrix is non-zero, then the system has a unique solution.

First, let's find the determinant of the coefficient matrix:

| 1   3 |

| 3   1 |

Det = (1*1) - (3*3) = -8

Since the determinant is non-zero (-8 ≠ 0), the system has a unique solution.

Next, let's find the determinant of the x-column matrix:

| 16   3 |

|  8   1 |

Det(x) = (16*1) - (3*8) = -8

Finally, we can find the value of x by dividing the determinant of the x-column matrix by the determinant of the coefficient matrix: x = Det(x) / Det = (-8) / (-8) = 1.

The equation for the plane that contains the given line and is perpendicular to the provided plane is 8x + 5y - 7z + 17 = 0.

To find an equation for the plane that contains the line given by v = (−1, 1, 2) + t(7, 6, 2) and is perpendicular to the plane described by 8x + 5y − 7z + 8 = 0, we first need to understand that the normal vector of this plane will be parallel to the normal vector (8, 5, -7) of the given plane since both planes are perpendicular to each other.

Knowing that the line lies on the required plane, we can use the point (−1, 1, 2) that the line passes through, along with the parallel normal vector, to write the equation of our plane. The general form of the equation of a plane is Ax + By + Cz + D = 0, where (A, B, C) is the normal vector to the plane. Substituting our point into the equation, we calculate the constant D.

Thus, the equation of our plane will be 8x + 5y − 7z + D = 0. Inserting the point (−1, 1, 2) into this equation gives us 8(-1) + 5(1) - 7(2) + D = 0, which simplifies to -8 + 5 - 14 + D = 0. Solving for D yields D = 17. Hence, the final equation of the plane is 8x + 5y − 7z + 17 = 0.

Answer:

12 and 5

Step-by-step explanation:

Answer:

Step-by-step explanation:

A geometric sequence or geometric progression is a sequence where each term after the first one can be found by multiplying a certain factor.

So, basically, to find the geometric sequence, we just need to divide the second term by the first term and observe if it results a constant factor.

By doing that, we observe that the second equence is a geometric sequence with a reasonf of 1/2, let's demonstrate it

[tex]8\times \frac{1}{2}= 4 (second \ term)\\4 \times \frac{1}{2}=2 (third \ term)\\ 2 \times \frac{1}{2}= 1 (fourth \ term)\\ 1 \times \frac{1}{2}= \frac{1}{2} (fiftg \ term)\\ \frac{1}{2} \times \frac{1}{2}= \frac{1}{4} (sixth \ term)[/tex]

As you can observe, we perfectly built the sequence is its ratio or factor.

Therefore, choice B is a geometric sequence.

The element with 35 protons is bromine (Br). Its isotopes with 44 and 46 neutrons have the symbols ^79Br and ^81Br, respectively, with the mass numbers derived from adding the number of neutrons to the atomic number 35.

To address Exercise 1.8.2, we first need to identify the element with 35 protons. In the periodic table, the element with 35 protons is bromine (Br). This element is defined by the number of protons in its nucleus, which is also known as its atomic number. Therefore, bromine is the element we are looking for.

Next, we need to write the symbols for bromine's isotopes with 44 and 46 neutrons. The total number of nucleons (protons plus neutrons) gives us the mass number of an isotope. For bromine with 44 neutrons, the mass number is 35 (protons) + 44 (neutrons) = 79. Therefore, the symbol for this isotope of bromine is ^79Br.

For the isotope with 46 neutrons, the mass number is 35 (protons) + 46 (neutrons) = 81. So, the symbol for this isotope of bromine is ^81Br. Each isotope differs in the number of neutrons, and therefore, in the mass number, while the number of protons (the atomic number) remains constant.

The element with 35 protons is bromine (Br). Its isotopes with 44 and 46 neutrons are written as ^79Br and ^81Br, respectively, where the numbers represent the mass numbers of each isotope.

To answer Exercise 1.8.2, we must first identify the element with 35 protons. The element with 35 protons is bromine, which is represented by the chemical symbol Br. To write the symbols for its isotopes with 44 and 46 neutrons, we must add the atomic number (protons) and the number of neutrons to get the mass number for each isotope.

For the isotope with 44 neutrons:

For the isotope with 46 neutrons:

Isotopes of an element have the same number of protons but different numbers of neutrons, hence different mass numbers. The isotopes of bromine are thus ^79Br and ^81Br, referring to the atomic mass which is composed of the protons and neutrons together.

A diameter is a line segment that has endpoints on a circle and passes through the center of the circle. The correct option is: diameter

A diameter is a line segment that has endpoints on a circle and passes through the center of the circle. It is a crucial geometric element in the study of circles and plays a fundamental role in various geometric properties.

Key points about the diameter:

Definition: As mentioned, a diameter is a line segment that connects any two points on the circle's circumference and passes through the center of the circle. It is the longest chord of the circle.

Length: The length of the diameter is exactly twice the length of the radius. If "r" represents the radius, then the length of the diameter is "2r."

Properties: The diameter divides the circle into two equal halves, each known as a semicircle. It is also the largest possible chord of a circle.

In summary, a diameter is a significant geometric concept related to circles, and it is characterized by its property of passing through the center of the circle and being twice the length of the radius.

The correct option is: diameter

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