Solve Distance Formula Problem: Parametric Equations & Slope

In summary, Niall asks how Sally and Hannah's distances change as they walk east, and provides an equation and graph to solve for the distance.
  • #1
MarkFL
Gold Member
MHB
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Here is the question:

Distance formula problem?

Sally and Hannah are lost in the desert. Sally is 5 km north and 3 km east of Hannah. At the same time, they both begin walking east. Sally walks at 2 km/hr and Hannah walks at 3 km/hr.

1. When will they be 10 km apart?
2. When will the line through their locations be perpendicular to the line through their starting locations?

I found that answer for #1 is 11.66. #2 answer is 34/3 but I don't know how to solve for #2.

Here is a link to the question:

Distance formula problem? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
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  • #2
Re: eb's question from Yahoo! Answers regarding distance, parametric equations and slope

Hello eb,

I would place Hannah initially at the origin (0,0), and Sally at (3,5).

1.) I would choose to represent the positions of Hannah and Sally parametrically. The unit of distance is km and the unit of time is hr. So, we may state:

\(\displaystyle H(t)=\langle 3t,0 \rangle\)

\(\displaystyle S(t)=\langle 2t+3,5 \rangle\)

and so the distance between them at time $t$ is:

\(\displaystyle d(t)=\sqrt{((2t+3)-3t)^2+(5-0)^2}=\sqrt{t^2-6t+34}\)

Now, letting $d(t)=10$ and solving for $0\le t$, we find:

\(\displaystyle 10=\sqrt{t^2-6t+34}\)

Square both sides and write in standard form:

\(\displaystyle t^2-6t-66=0\)

Use of the quadratic formula yields one non-negative root:

\(\displaystyle t=3+5\sqrt{3}\approx11.66\)

2.) Initially the slope of the line through their positions is \(\displaystyle \frac{5}{3}\), and so we want to equate the slope of the line through their positions at time $t$ to the negative reciprocal of this as follows:

\(\displaystyle \frac{5-0}{2t+3-3t}=-\frac{3}{5}\)

\(\displaystyle \frac{5}{t-3}=\frac{3}{5}\)

Cross-multiply:

\(\displaystyle 3t-9=25\)

\(\displaystyle 3t=34\)

\(\displaystyle t=\frac{34}{3}\)

To eb and any other guests viewing this topic, I invite and encourage you to post other parametric equation/analytic geometry questions in our http://www.mathhelpboards.com/f21/ forum.

Best Regards,

Mark.
 
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  • #3
Re: eb's question from Yahoo! Answers regarding distance, parametric equations and slope

A new member to MHB, Niall, has asked for clarification:

I'm a bit confused on how you got S(t) = (2t + 3, 5) and H(t) = (3t, 0) could you explain that?

Hello Niall and welcome to MHB! (Rock)

The $y$-coordinate of both young ladies will remain constant as they are walking due east. Since they are walking at a constant rate, we know their respective $x$-coordinates will be linear functions, where the slope of the line is given by their speed, and the intercept is given by their initial position $x_0$, i.e.

\(\displaystyle x(t)=vt+x_0\)

Sarah's speed is 2 kph and her initial position is 3, hence:

\(\displaystyle S(t)=2t+3\)

Hannah's speed is 3 kph and her initial position is 0, hence:

\(\displaystyle H(t)=3t+0=3t\)

Does this explain things clearly? If not, I will be happy to try to explain in another way.
 

FAQ: Solve Distance Formula Problem: Parametric Equations & Slope

1. What is the distance formula for parametric equations?

The distance formula for parametric equations is: d = √[(x2 - x1)^2 + (y2 - y1)^2] where (x1, y1) and (x2, y2) are the coordinates of two points on the parametric curve.

2. How do you find the slope of a parametric equation?

To find the slope of a parametric equation, you can use the derivative of the parametric equation with respect to the parameter t. The slope will be equal to dy/dx, where x and y are the coordinates of the parametric curve in terms of t.

3. Can the distance formula and slope be used for any parametric equation?

Yes, the distance formula and slope can be applied to any parametric equation, as long as it represents a curve in the x-y plane.

4. Is it possible to find the distance and slope of a parametric equation without using calculus?

Yes, it is possible to find the distance and slope of a parametric equation without using calculus. You can use the Pythagorean theorem to calculate the distance and the rise over run method to find the slope.

5. How can the distance and slope of a parametric equation be used in real-world applications?

The distance and slope of a parametric equation have many real-world applications, such as in physics to calculate the motion of objects, in engineering to design curves and trajectories, and in computer graphics to create smooth and realistic animations.

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