Chapter 12.2, Problem 41STP

a.

To determine

To find the points of intersection of the orbits of Pluto and the comet.

Answer to Problem 41STP

  $(39.24,4.386)$

Explanation of Solution

Given:

The orbit of the Pluto modeled by the equation $\frac{x^2}{{39.5}^2}+\frac{y^2}{38.3}=1$ and the comet’s path is modeled by the equation $x=y^2+20$

Calculation:

Graphing the above two equations to find the point of intersection as follows-

  

Clearly from the above graph the point of intersection is $(39.24,4.386)$

b.

To determine

To find whether the comet will necessarily hit the Pluto.

Answer to Problem 41STP

Not sure.

Explanation of Solution

Given:

The orbit of the Pluto modeled by the equation $\frac{x^2}{{39.5}^2}+\frac{y^2}{38.3}=1$ and the comet’s path is modeled by the equation $x=y^2+20$

Calculation:

As it is visible from the graph, it can’t be sure that it will necessarily hit, it may or may not hit the Pluto.

c.

To determine

To find where do the graphs $y=2x+1$ and $2x^2+y^2=11$ intersect.

Answer to Problem 41STP

  $(1,3)$

Explanation of Solution

Given:

The orbit of the Pluto modeled by the equation $\frac{x^2}{{39.5}^2}+\frac{y^2}{38.3}=1$ and the comet’s path is modeled by the equation $x=y^2+20$

Calculation:

Let us graph the two equations $y=2x+1$ and $2x^2+y^2=11$

  

Clearly from the above graph the point of intersection is $(1,3)$

d.

To determine

To find the coordinates of point that lie on the graphs of both $x^2+y^2=25$ and $2x^2+3y^2=66$

Answer to Problem 41STP

  $(3,4),(-3,4),(-3,-4),(3,-4)$

Explanation of Solution

Given:

The orbit of the Pluto modeled by the equation $\frac{x^2}{{39.5}^2}+\frac{y^2}{38.3}=1$ and the comet’s path is modeled by the equation $x=y^2+20$

Calculation:

Let us graph the two equations $x^2+y^2=25$ and $2x^2+3y^2=66$

  

The following are the points that lie on both the graphs-

  $(3,4),(-3,4),(-3,-4),(3,-4)$