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andrezito [222]
2 years ago
7

Sketch the curve represented by the parametric equations. Use arrows to indicate the direction of the curve as t increases. Find

a rectangular-coordinate equation for the curve by eliminating the parameter. Express the vector v with initial point P and terminal point Q in component form. (Assume that each point lies on the gridlines.)

Mathematics
1 answer:
Paul [167]2 years ago
8 0

The <em>lower right</em> image represents the image of the <em>parametric</em> formulas, whose <em>rectangular</em> formula is \frac{x^{2}}{16} + \frac{y^{2}}{25} = 1. The vector in <em>component</em> form is \vec v = (8, -4).

<h3>How to analyze parametric equations and vectors</h3>

<em>Parametric</em> formulas are <em>vectorial</em> expressions in terms of a parameter (t). Planar parametric expression are of the form \vec r(t) = (x(t), y(t)). Ellipses centered at the origin are described by the following expression:

\vec r (t) = (a\cdot \cos t, b \cdot \sin t)     (1)

Where a, b are the lengths of the <em>major</em> and <em>minor</em> semiaxes.

By direct observation of the given <em>parametric</em> equations, we conclude that the ellipse of the lower <em>right</em> image represents the two equations.

The <em>rectangular</em> equation of the ellipse is found by eliminating the parameter:

\cos ^{2}t + \sin ^{2}t = 1  

(\frac{x}{4})^{2} + \left(\frac{y}{5} \right)^{2} = 1

\frac{x^{2}}{16} + \frac{y^{2}}{25} = 1

According to the geometry, vectors can be generated from two points, one of them as the <em>initial</em> point. A vector can be defined as a subtraction between two vectors with <em>initial</em> points at the origin:

\vec v = B(x, y) - A(x, y)     (2)

Where:

  • A(x, y) - Initial point
  • B(x, y) - Final point

If we know that A(x, y) = (1, 8) and B(x, y) = (9, 4), then the equation of the vector is:

\vec v = (9, 4) - (1, 8)

\vec v = (9 - 1, 4 - 8)

\vec v = (8, -4)

To learn more on parametric equations: brainly.com/question/12718642

#SPJ1

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A choreographer uses a number line to position dancers for a ballet. Dancers A and B
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2 years ago
Read 2 more answers
The profit function for the first version of the device was very similar to the profit function for the new version. As a matter
NeTakaya

Answer:

a) - Compressing the P(new) function by a scale of 0.5 about the y axis.

- Moving the P(new) function down by 104 units.

b) The two simplified functions for P(original)

-0.08x² + 10.8x – 200.

-0.16x² + 21.6x – 504.

Step-by-step explanation:

Complete Question

An electronics manufacturer recently created a new version of a popular device. It also created this function to represent the profit, P(x), in tens of thousands of dollars, that the company will earn based on manufacturing x thousand devices: P(x) = -0.16x² + 21.6x – 400.

a. The profit function for the first version of the device was very similar to the profit function for the new version. As a matter of fact, the profit function for the first version is a transformation of the profit function for the new version. For the value x = 40, the original profit function is half the size of the new profit function. Write two function transformations in terms of P(x) that could represent the original profit function.

b. Write the two possible functions from part a in simplified form.

Solution

The equation for the new profit function is

P(x) = -0.16x² + 21.6x – 400

At x = 40, the original profit function is half the size of the new profit function

First, we find the value of the new profit function at x = 40

P(x) = -0.16(40)² + 21.6(40) – 400 = 208

Half of 208 = 0.5 × 208 = 104

P(original at x = 40) = P(new at x = 40) ÷ 2

Since we are told that P(original) is a simple transformation of the P(new)

P(original) = P(new)/2 = (-0.16x² + 21.6x – 400)/2 = -0.08x² + 10.8x – 200 ... (eqn 1)

Or, P(original) = 104

-0.16x² + 21.6x – 400 = 104

P(original) = -0.16x² + 21.6x – 400 - 104 = -0.16x² + 21.6x – 504.

So, the two functions that are simple transformations of P(new) to get P(original) are

-0.08x² + 10.8x – 200

Obtained by compressing the P(new) function by a scale of 0.5 about the y axis.

And

-0.16x² + 21.6x – 504.

Obtained by moving the P(new) function down by 104 units.

Hope this Helps!!!

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3 years ago
"to replace a number with another number that tells about how many or how much". What is ths operation called?
Maru [420]

The operation is known as rounding of a number.

This means basically if i measure the temperature of a city and it comes out to be 22.9 degrees. I will round it of to 30 degrees or the amount of significant figures i am designated with.

Similarly if its 22.85 and i have to round of to 3 significant figures it becomes 22.9 degrees.

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hammer [34]

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<h3>Which units should we use?</h3>

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So we have two types of units in the above sentence, time (in days), and volume (of the gasoline consumed).

Then we conclude that the units that we should use is volume over time, particularly, gallons per day.

So we can represent that measure as:

r = 2.4\frac{gallons}{day}

If you want to learn more about units:

brainly.com/question/1972490

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