Three students simplified the expression 2x- 3(y-2x) +-5times -2y.
Amber -4y + 7y
Butch 4x + 7y
Carl 8x + 7y

Who is correct? Explain the error the other students made

The Taylor polynomial [tex]T_3(x)[/tex] will be written as [tex]2x-8x^2+\dfrac{44x^3}{3}+......[/tex].

Given:

The given function is [tex]f(x) = e^{-4x}sin(2x)[/tex].

It is required to find the Tylor polynomial [tex]t_3(x)[/tex] centered at a=0.

Now, the expansion of the function [tex]e^{-4x}[/tex] can be written as,

[tex]e^{-4x}=\sum\dfrac{(-4x)^n}{n!}\\e^{-4x}=1+(-4x)^1+\dfrac{(-4x)^2}{2!}+\dfrac{(-4x)^3}{3!}+.....\\e^{-4x}=1-4x+\dfrac{16x^2}{2}-\dfrac{64x^3}{6}+.....\\e^{-4x}=1-4x+8x^2-\dfrac{32x^3}{3}+.....[/tex]

Similarly, the expansion of the function [tex]sin(2x)[/tex] will be,

[tex]sin(2x)=\sum\dfrac{(-1)^n(2x)^{2n+1}}{(2n+1)!}\\=\dfrac{2x}{1!}+\dfrac{-(2x)^3}{3!}+.....\\=2x-\dfrac{4x^3}{3}+......[/tex]

So, the function [tex]f(x) = e^{-4x}sin(2x)[/tex] will be written as,

[tex]f(x) = e^{-4x}sin(2x)\\f(x)=(1-4x+8x^2-\dfrac{32x^3}{3}+.....)(2x-\dfrac{4x^3}{3}+......)\\f(x)=2x-8x^2+16x^3-\dfrac{4x^3}{3}+.......\\f(x)=2x-8x^2+\dfrac{(48-4)x^3}{3}+......\\f(x)=2x-8x^2+\dfrac{44x^3}{3}+......[/tex]

Therefore, the Taylor polynomial [tex]T_3(x)[/tex] will be written as [tex]2x-8x^2+\dfrac{44x^3}{3}+......[/tex].

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22,000 - 20% = 17,600

17,600 - 20% = 14,080

$14,080 at the end of the second year .

To find the flux of a vector field through a closed boundary using the divergence theorem, calculate the divergence of the vector field and evaluate the triple integral of the divergence over the solid bounded by the boundary. In this case, the flux is 3 times the volume of the solid.

The student is asking how to use the divergence theorem to find the flux of a vector field through a closed boundary. In this case, the vector field is defined as f(x, y, z) = ⟨x, y, 4z⟩ and the closed boundary is a solid bounded by the paraboloid z = x^2 + y^2 and the plane z = 9.

To use the divergence theorem, we need to calculate the divergence of the vector field, which is the sum of the partial derivatives of f with respect to each variable. In this case, the divergence is 3.

Then, we can use the divergence theorem to find the flux through the closed boundary by evaluating the triple integral of the divergence over the solid bounded by the paraboloid and the plane. In this case, the flux is 3 times the volume of the solid.

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The flux of [tex]\(\vec{F}\)[/tex] through S is 24π.

To apply the divergence theorem, we first compute the divergence of [tex]\(\vec{F}\)[/tex]:

[tex]\nabla \cdot \vec{F} = \frac{\partial}{\partial x} (x) + \frac{\partial}{\partial y} (y) + \frac{\partial}{\partial z} (4z) = 1 + 1 + 4 = 6.[/tex]

The divergence theorem states that the flux of a vector field through a closed surface is equal to the triple integral of its divergence over the region enclosed by the surface.

Thus, we have:

[tex]\iint_S \vec{F} \cdot d\vec{A} = \iiint_W (\nabla \cdot \vec{F}) \, dV = \iiint_W 6 \, dV[/tex]

The region W is bounded below by the paraboloid [tex]\(z = x^2 + y^2\)[/tex], and above by the plane z = 4.

Converting to cylindrical coordinates, we have:

[tex]\iiint_W 6 \, dV = \int_0^{2\pi} \int_0^2 \int_{r^2}^4 6 \cdot r \, dz \, dr \, d\theta = 24\pi.[/tex]

multiply 5 13/16 by 76

5 13/16 * 76 = 441 3/4 revolutions

To find common factors between numbers, list all factors of each number and identify numbers that are in both lists. When multiplying fractions, multiply numerators and denominators then simplify by common factors. Multiplying both sides by the same factor can help in solving equations with fractions.

To find common factors between two or more numbers, you first list out all the factors of each number. Factors are numbers that divide into the original number without leaving a remainder. For instance, if we are looking for common factors of 8 and 12, we list their factors as follows: the factors of 8 are 1, 2, 4, and 8, and the factors of 12 are 1, 2, 3, 4, 6, and 12. After listing out the factors, you look for numbers that appear in both lists. In this example, the common factors of 8 and 12 are 1, 2, and 4.

Another approach mentioned involves multiplying both sides by the same factor to make both sides integers when working with equations. This can be useful when seeking to simplify fractions or solve equations with fractional components.

It is also important to recognize that while multiplying fractions, we multiply the numerators together and the denominators together. Simplifying the result by common factors as needed helps in reducing fractions to their simplest form. For example, if we multiply ½ by ¾, we get a result of ¼ (numerator 1x3=3, denominator 2x4=8) which we can simplify to ¾ by dividing both numerator and denominator by the common factor 3.

y = k/x

7=k/-2

k = 7/-2 = -3.5

y =-3.5/7 =-0.5

y=-0.5

We have the equation here is

16 + 31 = 31

When we simplify the equation to the understandable form, we move all terms or numbers to right and on left side zero will be left.

0 = 31-16-31

We get, 0 = -16

Now we see that both sides of equations are not equal, it means there is no solution so it is an invalid equation.

Answer:

.

Step-by-step explanation:

.

sum means addition

 so x +-20 = 40

x = 40 +20 = 60

x=60

5(3x-7) = 20

15x-35 = 20

15x = 55

x = 3.666666

 so 6(3.666666) -8 = 13.99999 round to 14

Answer:

A line segment that goes from one side of the circle to the other side of the circle and doesn’t go through the center is called chord of the circle.

Step-by-step explanation:

Consider the provided information.

It is given that the line segment goes from one side of the circle to the other side of the circle and doesn’t go through the center.

Diameter: A line segment goes from one side to another side of a circle passes through the center is called the diameter of the circle.

Chord:  A line segment goes from one side to another side of a circle but do not passes through the center is called the chord of the circle.

For better understanding refer the attached figure:

Hence, A line segment that goes from one side of the circle to the other side of the circle and doesn’t go through the center is called chord of the circle.

The probability that the archer hits the target on exactly three out of five shots is 0.3087, or 30.87%, calculated by using the binomial probability formula.

The probability that an archer hits a target on a given shot is 0.7 and the goal is to calculate the probability that the archer hits the target on exactly three out of five shots. This is a binomial probability problem, as each shot can end in either a success (hitting the target) with a probability of 0.7, or a failure (missing the target) with a probability of 0.3.

To calculate the probability of exactly three successes (hits) out of five, we use the binomial probability formula:

P(X=k) = (n choose k) * (p)^k * (1-p)^(n-k)

Where:

n = total number of trials (5 shots)

k = number of successes (3 hits)

p = probability of success on a single trial (0.7)

Applying the formula, we get:

P(3 hits out of 5) = (5 choose 3) * (0.7)^3 * (0.3)^2

= 10 * (0.343) * (0.09)

= 10 * 0.03087

= 0.3087

Therefore, the probability that the archer hits the target on exactly three out of five shots is 0.3087, or 30.87%.

Answer:

[tex]\sqrt[9]{a^{4}}[/tex]

Step-by-step explanation:

To convert a fraction form into a radical form you need to know that the denominator will be the root index and the numerator will be the exponent into the root. For the case of four ninths:

[tex]a^{\frac{4}{9}} = \sqrt[9]{a^{4}} .[/tex]

The answer is: 225.

To find the maximum height that the ball will reach, we need to determine the vertex of the parabola described by the function [tex]\( h(t) = 120t - 16t^2 \)[/tex]. The vertex form of a parabola is[tex]\( h(t) = a(t - h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola. The value of [tex]\( k \)[/tex] will give us the maximum height.

The given function can be rewritten in the form [tex]\( h(t) = -16(t^2 - \frac{120}{16}t) \)[/tex]. To complete the square, we take the coefficient of [tex]\( t \)[/tex], divide it by 2, and square it. This value is then added and subtracted inside the parentheses:

[tex]\( h(t) = -16(t^2 - \frac{120}{16}t + (\frac{120}{32})^2 - (\frac{120}{32})^2) \)[/tex]

[tex]\( h(t) = -16((t - \frac{120}{32})^2 - (\frac{120}{32})^2) \)[/tex]

Now, we expand the squared term and multiply through by -16:

[tex]\( h(t) = -16(t - \frac{120}{32})^2 + 16(\frac{120}{32})^2 \)[/tex]

[tex]\( h(t) = -16(t - 3.75)^2 + 16(3.75)^2 \)[/tex]

The maximum height [tex]\( k \)[/tex] is the constant term when the equation is in vertex form:

[tex]\( k = 16(3.75)^2 \)[/tex]

[tex]\( k = 16 \times 14.0625 \)[/tex]

[tex]\( k = 225 \)[/tex]

Therefore, the maximum height that the ball will reach is 225 feet.

Answer:

Step-by-step explanation:

A man divided $9,000 among his wife, son and daughter.

The wife received twice as much as the daughter.

Let the daughter received d amount.

Then the wife received = 2d

and son received $1,000 more than the daughter.

The son received the amount = 1000+d

So the expression will be = d + 2d +(1000+d) = 9,000

3d + (1000+d) = 9000

4d = 9000 - 1000

4d = 8000

d =  [tex]\frac{8000}{4}[/tex]

d = 2000

Daughter received $2,000

Wife received 2d = 2 × 2000 = $4,000

Son received 1000 + d = 1000 + 2000 = $3,000

If x is the amount the wife received, then the expression represents the amount received by the son :

S = 1000 + (x/2)

Answer:

X = In4-1    C on edge, just took the test