Chapter 2.6, Problem 63E

Solution

a.

To check : The graph $y=g\left(d\right)$ function of the planet mercury, radius= $2.43\times {10}^6$ ,surface= $3.61$ , $g\left(d\right)=\frac{G\times M}{d^2},G\approx 6.674\times {10}^{-11},MassM=3.20\times 10^23$ .

Given information: The function $y=g\left(d\right)$ .

The radius= $2.43\times {10}^6$ ,surface= $3.61$ of the planet is given.

Concept used:

  $y=g\left(d\right)=\frac{G\times M^2}{d^2},$ . On the surface of a planet $m=d$ , the radius of the planet.

The graph of the given planet is given below.

  

This the graph of the $y=g\left(d\right)$ is given above.

b.

To Write: The transformation that can be used to transform $y=\frac{1}{d^2}$ into the graph of $y=g\left(d\right)$

Given information:

The given function is $y=\frac{1}{d^2}$

First do the horizontal stretch by the radius $R=2.43\times {10}^6$ that is $\frac{1}{{\left({}^{\frac{d}{R}}\right)}^2}=\frac{R^2}{d^2}$

After that do the vertical stretch by the factor $g_{surface}=\frac{G\times M}{R^2}$ then $\frac{G\times M}{R^2}\times \frac{R^2}{d^2}=\frac{G\times M}{d^2}$

Thus the transformation formed from $y=\frac{1}{d^2}$ into the graph of $y=g\left(d\right)$ .