Mathematics, 27.10.2020 02:30 brookesquibbs

After a vigorous soccer match, Tina and Michael decide to have a glass of their favorite refreshment. They each run in a straight line along the indicated paths at a speed of 10 ft/sec. Tina starts at (250, 75) and ends up at (-50, 175)
Michael starts at (0, 0) and ends up at (100, 400)
Write parametric equations for the motion of Tina and Michael individually after t seconds.

Answers: 2

Show answers

Answers

Answer from: hallmansean04

The parametric equation that can be used to determine Tina's and

Michael's location are expressed as follows;

Tina's parametric equation is; $\underline{(x, \, y) = (250 - 9.49 \cdot t, \ 75 + 3.16\cdot t)}$ Michael's parametric equation is; $\underline{(x, \, y) = (2.425 \cdot t, \ 9.7\cdot t)}$

Reasons:

The given parameters are;

The speed of the runners = 10 ft./sec

Location Tina starts = (250, 75)

Point Tina ends = (-50, 175)

Location Michael starts = (0, 0)

Point Michael ends = (100, 400)

Parametric equation for Tina's motion;

$The \ \mathbf{slope} \ of \ Tina's \ path \ is \ \dfrac{175 - 75}{-50 - 250} = -\dfrac{100}{300} = -\dfrac{1}{3}$

$The \ \mathbf{direction} \ is \ therefore; \ arctan \left( -\dfrac{1}{3} \right) \approx 161.57^{\circ}$

$The \ speed \ in \ the \ x-direction \ is \ v_x = 10 \times cos \left(arctan \left( -\dfrac{1}{3} \right)\right) \approx - 9.49$

Therefore;

The location in the x-direction after t seconds is x ≈ 250 - 9.49·t

$The \ speed \ in \ the \ x-direction \ is \ v_y = 10 \times sin \left(arctan \left( -\dfrac{1}{3} \right)\right) \approx 3.16$

The location in the y-direction after t seconds is y ≈ 75 + 3.16·t

The parametric equation for the location of Tina after t seconds is

therefore;

$\underline{(x, \, y) = \mathbf{(250 - 9.49 \cdot t, \ 75 + 3.16\cdot t)}}$ .

Parametric equation for Michael's motion;

$The \ direction \ of \ Michael's \ motion \ is \ arctan \left( \dfrac{400}{100} \right) \approx 75.96^{\circ}$

Therefore;

$v_x = 10 \times cos \left(arctan \left(4 \right)\right) \approx 2.425$

x ≈ 0 + 2.425·t = 2.425·t

$v_y = 10 \times sin \left(arctan \left( 4\right)\right) \approx 9.7$

y ≈ 9.7·t

The parametric equation for the location of Michael at time t is therefore;

$\underline{(x, \, y) = \mathbf{(2.425 \cdot t, \ 9.7\cdot t)}}$

Learn more here:

link

Answer from: tordoor7182

The parametric equations for the motion of Tina and Michael are $(x, y) = (250-9.487\cdot t, 75+3.162\cdot t)$ and $(x,y) = (2.42\cdot t, 9.68\cdot t)$ , respectively.

Step-by-step explanation:

Given that both Tina and Michael travels at constant speed in a straight line. We get that vectorial form of the parametric equation of the line is represented by the following expression:

$(x, y) = (x_{o}, y_{o})+\frac{v\cdot t}{r} \cdot (x_{f}-x_{o}, y_{f}-y_{o})$ (Eq. 1)

Where:

, - Current position of the particle, measured in feet.

$x_{o}$ , $y_{o}$ - Initial position of the particle, measured in feet.

$x_{f}$ , $y_{f}$ - Final position of the particle, measured in feet.

- Speed of the particle, measured in feet per second.

- Time, measured in seconds.

- Distance between initial and final position of the particle, measured in feet.

In addition, the distance between initial and final position of the particle is represented by the following Pythagorean expression:

$r = \sqrt{(x_{f}-x_{o})^{2}+(y_{f}-y_{o})^{2}}$ (Eq. 2)

The parametric equation for each person are, respectively:

Tina

$(x_{o},y_{o}) = (250, 75)$ , $(x_{f}, y_{f}) = (-50, 175)$ and $v = 10\,\frac{ft}{s}$

From (Eq. 2):

$r = \sqrt{[(-50)-250]^{2}+(175-75)^{2}}$

$r \approx 316.228\,ft$

In (Eq. 1):

$(x,y) = (250,75)+\frac{2500\cdot t}{79057} \cdot (-300,100)$

$(x,y) = (250, 75) +(-9.487\cdot t, 3.162\cdot t)$

$(x, y) = (250-9.487\cdot t, 75+3.162\cdot t)$

Michael

$(x_{o}, y_{o}) = (0, 0)$ , $(x_{f}, y_{f})=(100,400)$ and $v = 10\,\frac{ft}{s}$

From (Eq. 2):

$r = \sqrt{(100-0)^{2}+(400-0)^{2}}$

$r\approx 412.311\,ft$

In (Eq. 1):

$(x,y) = (0,0)+0.0242\cdot t \cdot (100,400)$

$(x,y) = (2.42\cdot t, 9.68\cdot t)$

The parametric equations for the motion of Tina and Michael are $(x, y) = (250-9.487\cdot t, 75+3.162\cdot t)$ and $(x,y) = (2.42\cdot t, 9.68\cdot t)$ , respectively.

Answer from: Quest

the answer is 40.

proof:

Answer from: Quest

Poppoppoppoppoppoppop

Another question on Mathematics

Mathematics, 21.06.2019 18:00

The given dot plot represents the average daily temperatures, in degrees fahrenheit, recorded in a town during the first 15 days of september. if the dot plot is converted to a box plot, the first quartile would be drawn at __ , and the third quartile would be drawn at __ link to graph: .

Answers: 1

Answer

Mathematics, 21.06.2019 18:30

Kayla wants to find the width, ab, of a river. she walks along the edge of the river 65 ft and marks point c. then she walks 25 ft further and marks point d. she turns 90° and walks until her location, point a, and point c are collinear. she marks point e at this location, as shown. (a) can kayla conclude that δ and δ are similar? why or why not? (b) suppose de = 15 ft. what can kayla conclude about the width of the river?

Answers: 2

Answer

Mathematics, 21.06.2019 19:30

The wheels of a bicycle have a diameter of 70cm. how many meters will the bicycle travel if each wheel makes 50 revolutions

Answers: 2

Answer

Mathematics, 21.06.2019 20:30

Solve each quadratic equation by factoring and using the zero product property. x^2 - 8x + 30 = 3x

Answers: 2

Answer

You know the right answer?

After a vigorous soccer match, Tina and Michael decide to have a glass of their favorite refreshment...