Alan

if we assume there are 365 days in a year then there are 86500/(15*365) orbits per day, so the astronaut did approximately 330*86500/(15*365) orbits.

$${\frac{{\mathtt{330}}{\mathtt{\,\times\,}}{\mathtt{86\,500}}}{\left({\mathtt{15}}{\mathtt{\,\times\,}}{\mathtt{365}}\right)}} = {\mathtt{5\,213.698\: \!630\: \!136\: \!986\: \!301\: \!4}}$$

approximately 5200 orbits (or 5210 orbits or 5214 orbits, depending on how approximate you want to be!)

To make a 15% profit he must sell the lot for 1.15*$72000

$${\mathtt{total}} = {\mathtt{1.15}}{\mathtt{\,\times\,}}{\mathtt{72\,000}} \Rightarrow {\mathtt{total}} = {\mathtt{82\,800}}$$

The price per radio is therefore $82800/48

$${\mathtt{radioprice}} = {\frac{{\mathtt{82\,800}}}{{\mathtt{48}}}} \Rightarrow {\mathtt{radioprice}} = {\mathtt{1\,725}}$$ 

price per radio = $1725

Pricey radios!

It's sin^-1 Melody:

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}^{\!\!\mathtt{-1}}{\left({\frac{{\mathtt{11}}{\mathtt{\,\times\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}{\left({\mathtt{13}}^\circ\right)}}{{\mathtt{6.8}}}}\right)} = {\mathtt{21.339\: \!374\: \!187\: \!23^{\circ}}}$$

I guess you just want this to be expanded?

(m+2)(m-2) = m² - 2m + 2m - 2² = m² - 2² = m² - 4

The equation -(1/2) = 6x + 10 represents a vertical line on a graph at a fixed value of x (at x = -7/4), so for the line y=mx to be parallel to this you must have m = ∞. 

However, if you meant the first equation to be -(1/2)y = 6x + 10, then this can be rearranged as y = -12x -20 so for y = mx to be parallel to this m = -12 (i.e. the gradients must be the same).

2

0.000002204 = 2204/1000,000,000 = 551/250,000,000

 Anonymous wrote: A natural logarithm is NOT in base 10. The base is e (epsilon). 

He/she is almost right (though it's sad he/she felt the need to sneer), but the word "epsilon" isn't really appropriate.  The e in question is just the number e = 2.718281828459...etc.  The natural logarithm is also sometimes called the Napierian logarithm after John Napier (1550-1617) who invented logarithms.

Logarithms to base 10 are sometimes referred to as common logarithms.  

The deceleration force, F,  is F = 0.183*mass*g (g is acceleration of gravity).  This equals -mass*a, where 'a' is acceleration (deceleration) of the car, so

a = -0.183*9.81m/s². (The negative sign just means it is slowing down.)

With constant acceleration we have the kinetic equation v² = u² +2as, where v = final velocity (0), u = initial velocity (51.8km/hr = 51.8*10³/3600 m/s) and s is distance travelled.  So:

0 = (51.8*10³/3600)² - 2*0.183*9.81*s

 s = (51.8*10³/3600)²/2*0.183*9.81

$${\mathtt{s}} = {\frac{{\left({\frac{{\mathtt{51.8}}{\mathtt{\,\times\,}}{{\mathtt{10}}}^{{\mathtt{3}}}}{{\mathtt{3\,600}}}}\right)}^{{\mathtt{2}}}}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{0.183}}{\mathtt{\,\times\,}}{\mathtt{9.81}}\right)}} \Rightarrow {\mathtt{s}} = {\mathtt{57.663\: \!954\: \!885\: \!109\: \!463\: \!3}}$$ metres

or s ≈ 57.7 metres

The force needed to get the cup to slip is 0.42*cupmass*gravitational acceleration. This must equal cupmass*accelerationof plane. Therefore:

acceleration of plane = 0.42*9.81 m/s² (assuming plane isn't so high gravitational acceleration is significantly reduced!), as the mass of the cup cancels out.

 $${\mathtt{AccnOfPlane}} = {\mathtt{0.42}}{\mathtt{\,\times\,}}{\mathtt{9.81}} \Rightarrow {\mathtt{AccnOfPlane}} = {\mathtt{4.120\: \!2}}$$ m/s²