Question

Not sure if you're allowed to ask multiple questions on here at once, but somebody please explain all of this to me.

Answer 1

1. First, convert the model distance to base SI.

`30\,\mathrm{cm} = 3*10^(-1)\,\mathrm m`

Then the ratio of model distance to actual distance is

`(30\,\rm cm)/(1.50*10^(11)\,\rm m) = (3*10^(-1)\,\mathrm m)/(1.50*10^(11)\,\rm m) = (2)/(10^(12)) = (2)/(2*5*10^(11)) = (1)/(5*10^(11))`

or 1 : 5 × 10¹¹.

2. The model/actual distance ratio should be the same for each planet.

Mercury:

`12.6\,\mathrm{cm} = 1.26*10^(-1)\,\mathrm m`

`(12.6\,\rm cm)/(5.80*10^(10)\,\rm m) = (1.26*10^(-1)\,\rm m)/(5.80*10^(10)\,\rm m) \approx (1)/(4.6*10^(11))`

or 1 : 4.6 × 10¹¹.

Venus:

`21.6\,\mathrm{cm} = 2.16*10^(-1)\,\rm m`

`(21.6\,\rm cm)/(1.08*10^(11)\,\rm m) = (2.16*10^(-1)\,\rm m)/(1.08*10^(11)\,\rm m) = (2)/(10^(12)) = (2)/(2*5*10^(11)) = \frac1{5*10^(11)}`

or 1 : 5 × 10¹¹.

Mercury's distance is the incorrect one. The correct model distance should be
`x` such that

`(x\,\rm cm)/(5.80*10^(10)\,\rm m) = (1)/(5*10^(11))`

Solve for
`x`.

`x\,\mathrm{cm} = (5.80*10^(10)\,\rm m)/(5*10^(11))`

`x\,\mathrm{cm} = (1.16\,\rm m)/(10^1)`

`x\,\mathrm{cm} = 1.16*10^(-1)\,\rm m`

`x\,\mathrm{cm} = \boxed{11.6\,\rm cm}`

3.The actual distance for Mars is
`y` such that

`(34.4\,\rm cm)/(y\,\rm m) = (1)/(5*10^(11))`

Solve for
`y`.

`34.4*5*10^(11) \,\mathrm{cm} = y\,\rm m`

`y\,\mathrm m = 172 *10^(11)\,\rm cm`

`y\,\mathrm m = 172 *10^(13)\,\rm m`

`y\,\mathrm m = \boxed{1.72} *10^{\boxed{11}}\,\rm m`

4. Divide the distance by the speed of light to recover the time.

`(7.92*10^(11)\,\rm m)/(3.0*10^8(\rm m)/(\rm s)) = 2.64*10^3\,\mathrm s`

Convert to minutes.

`2.64*10^3\,\mathrm s \cdot (1\,\rm min)/(60\,\rm s) = 0.044*10^3\,\mathrm{min} = \boxed{44}\,\rm min`