221,000,000,000,000,000,000 in scientific notation

After a glide reflection, the point x is mapped to the point x' (3,-2). The translation part of the glide reflection is (x,y) -&

Umnica [9.8K]

The coordinates of the original point x, can be obtained by reversing the given transformations individually

The coordinates of the original point x is $\underline{(0, \, 0)}$

Reason:

The type of reflection = Glide reflection

Coordinates of the image of the point x = x'(3, -2)

The translation part of the glide reflection is (x, y) → (x + 3, y)

The line of reflection is y = -1

The coordinates of the original point = Required

Solution:

A glide reflection is also known as a transflection, that involves a symmetric composite transformation of a reflection followed by a translation along the line of reflection

Reflection part;

The distance of the image point from the reflecting line = The object's

point distance from the reflecting line

Therefore, given that the reflecting line is the line y = -1, and the image

point is x'(3, -2), we have;

Distance of image from reflecting line =-1 - (-2) = 1

∴ Distance of object point from reflecting line = 1

y-coordinate of object point = -1 + 1 = 0

Point of image of the object before reflection and after translation = (3, 0)

Translation part;

The translation of the glide reflection is (x, y) → (x + 3, y)

Therefore, the location of the object before translation is ((x + 3) - 3, y), which from the point (3, 0) gives, ;

((3) - 3, 0) → (0, 0)

The coordinates of the original point x is $\underline{(0, \, 0)}$

Learn more here:

brainly.com/question/12890981

8 0

2 years ago

Need help with AP CAL

anzhelika [568]

Answer: Choice C

$\displaystyle \frac{1}{2}\left(1 - \frac{1}{e^2}\right)$

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Explanation:

The graph is shown below. The base of the 3D solid is the blue region. It spans from x = 0 to x = 1. It's also above the x axis, and below the curve $y = e^{-x}$

Think of the blue region as the floor of this weirdly shaped 3D room.

We're told that the cross sections are perpendicular to the x axis and each cross section is a square. The side length of each square is $e^{-x}$ where 0 < x < 1

Let's compute the area of each general cross section.

$\text{area} = (\text{side})^2\\\\\text{area} = (e^{-x})^2\\\\\text{area} = e^{-2x}\\\\$

We'll be integrating infinitely many of these infinitely thin square slabs to find the volume of the 3D shape. Think of it like stacking concrete blocks together, except the blocks are side by side (instead of on top of each other). Or you can think of it like a row of square books of varying sizes. The books are very very thin.

This is what we want to compute

$\displaystyle \int_{0}^{1}e^{-2x}dx\\\\$

Apply a u-substitution

u = -2x

du/dx = -2

du = -2dx

dx = du/(-2)

dx = -0.5du

Also, don't forget to change the limits of integration

If x = 0, then u = -2x = -2(0) = 0
If x = 1, then u = -2x = -2(1) = -2

This means,

$\displaystyle \int_{0}^{1}e^{-2x}dx = \int_{0}^{-2}e^{u}(-0.5du) = 0.5\int_{-2}^{0}e^{u}du\\\\\\$

I used the rule that $\displaystyle \int_{a}^{b}f(x)dx = -\int_{b}^{a}f(x)dx$ which says swapping the limits of integration will have us swap the sign out front.

--------

Furthermore,

$\displaystyle 0.5\int_{-2}^{0}e^{u}du = \frac{1}{2}\left[e^u+C\right]_{-2}^{0}\\\\\\= \frac{1}{2}\left[(e^0+C)-(e^{-2}+C)\right]\\\\\\= \frac{1}{2}\left[1 - \frac{1}{e^2}\right]$

In short,

$\displaystyle \int_{0}^{1}e^{-2x}dx = \frac{1}{2}\left[1 - \frac{1}{e^2}\right]$

This points us to choice C as the final answer.

5 0

2 years ago

Prove: The square of a number that is

Nikolay [14]

Let 3n + 1 denote the "number" in question. The claim is that

(3n + 1)² = 3m + 1

for some integer m.

Now,

(3n + 1)² = (3n)² + 2 (3n) + 1²

… = 9n² + 6n + 1

… = 3n (3n + 2) + 1

… = 3m + 1

where we take m = n (3n + 2).

3 0

2 years ago

What should you do before creating a research question for a presentation? Check all that apply. create an outline to organize m

Sindrei [870]

The things that are to be done before creating a research question for a presentation are: option C, D, and E.

<h3>What is a research question?</h3>

A research question can be defined as an issue, event or subject of inquiry that a researcher is keenly and deeply interested to provide a response or an answer to, when conducting a research.

Basically, the things that are to be done before creating a research question for a presentation are:

Review the research topic and what the prompt asks for.

Think of an argument or an opinion if necessary.

Consider what I know and what I need to know.

Learn more about research question here: brainly.com/question/10129052

#SPJ1

8 0

2 years ago

Read 2 more answers

PRACTICE ANOTHER The administration of a large university wants to take a random sample to measure student opinion of a new food

Reil [10]

Answer:

81 samples

Step-by-step explanation:

According to the empirical rule :

Possible values of the sample mean is within 3 standard deviations of the population mean :

μ ± 3 sd(x) ; sd(x) = standard deviation of sampling distribution.

3 * sd(x) = 1

sd(x) = 1/3

Recall:

Standard deviation of sampling distribution, sd(x)

sd(x) = σ / sqrt(n)

1/3 = 3 / sqrt(n)

Square both sides

1/9 = 9/n

Cross multiply :

n * 1 = 9 * 9

n = 81

3 0

3 years ago