Help on a function type question

Juliet and Mercutio are moving at constant speeds in the *xy*-plane. They start moving at the same time. Juliet starts at the point (0, −6)

and heads in a straight line toward the point (10, 5), reaching it in 10 seconds. Mercutio starts at (9, −18) and moves in a straight line. Mercutio passes through the same point on the *x*-axis as Juliet, but 2 seconds after she does.

How long does it take Mercutio to reach the *y*-axis? (Round your answer to three decimal places.)

How would I do this problem? Thank you!

Re: Help on a function type question

You'll want to start by finding the equations of both lines. You are given two points for Juliet explicitly, and it is implied that wherever Juliet has her x-intercept is also on Mercutio's line. You can use this point as the second point for Mercutio. After doing so, you'll want to calculate the distance and rate of speed of Juliet. Use this to find out how long it takes her to reach the x-axis. Add 2 to this number to find Mercutio's rate of speed. After that, determine how long it takes for him to reach the y-axis.

The numbers part is up to you.

Re: Help on a function type question

Quote:

Originally Posted by

**UWM120** Juliet and Mercutio are moving at constant speeds in the *xy*-plane. They start moving at the same time. Juliet starts at the point (0, −6)

and heads in a straight line toward the point (10, 5), reaching it in 10 seconds. Mercutio starts at (9, −18) and moves in a straight line. Mercutio passes through the same point on the *x*-axis as Juliet, but 2 seconds after she does.

How long does it take Mercutio to reach the *y*-axis? (Round your answer to three decimal places.)

How would I do this problem? Thank you!

There are many different ways to approch this problem. All of them should involve drawing a diagram of the situtation. Here ismethod using vectors.

The vector from Juliet's starting point is

$\displaystyle \mathbf{v}=<10-0,5-(-6)>=<10,11>$

Since her total travel time is ten seconds she is moving one-tenth of the length per second. This gives the equation

$\displaystyle J(t)=<0,-6>+t<1,\frac{11}{10}>$ You can check that

$\displaystyle J(0)=<0,-6>$ is her starting point and $\displaystyle J(10)=<10,5>$

So we need to find the time when her y-coordinate is zero. This gives the equation

$\displaystyle -6+\frac{11}{10}t=0 \iff t=\frac{60}{11}$

Now we know that Mercuitio crosses two seconds later this gives

$\displaystyle \frac{60}{11}+2=\frac{82}{11}$

But we still need to find what point they cross at.

$\displaystyle J \left( \frac{60}{11}\right) = <0,-6>+\left( \frac{60}{11}\right)<1,\frac{11}{10}>=<1,0>$

So Mercuitio moves along the direction vector

$\displaystyle <1,0>-<9,-18>=<-8,18>$

So his equation motion is

$\displaystyle M(t)=<9,-18>+\frac{11t}{82}<-8,18>$

He will cross the y-axis when his x-coordinate is zero. This gives the equation

$\displaystyle 9+\frac{11t}{82}(-8)=0 \iff -9=\frac{-88}{82}t \iff t =\frac{82\cdot 9}{88}=\frac{369}{44}$

Re: Help on a function type question