Alan

Oops! 

Picture the triangle made by a line from the centre of the earth through to the top of the lighthouse (which I'll call c); a line from the top of the lighthouse to the ship (which I'll call b); and a line from the ship back to the centre of the earth (which I'll call a).  I would prefer to show a diagram here, but I can't as we can't yet upload files!

Since the angle of depression is 30°, the angle, A, between lines b and c is 90° - 30° or 60°.  We can write the cosine rule for this triangle as a² = b² + c² -2bccos(A).

Now the average radius of the Earth is 6371km (according to Wikipedia), so line a is 6371km long.  Line c is the radius of the Earth plus 325ft (=99.06m) so c is 6,371,000 + 99.06 = 6,371,099.06m.  The only unknown in the cosine rule above is b, and we can solve for b to find two mathematical solutions.  These are 6371km and 198.12m.

Clearly it is the smaller value we want here!  So b = 198.12m.

Now we need angle B, the angle between lines a and c. We get this from the sine rule:  sin(B)/b = sin(A)/a. So:

sin(B)=sin(60°)*198.12/6371000 = 2.693*10^-5 

Angle B = sin^-1(2.693*10^-5) or B = 2.693*10^-5 radians (when an angle is very small its value in radians is approximately equal to its sine).  The distance from the ship to the lighthouse along the curve of the Earth is just B*a where B is in radians.  So $${\mathtt{distance}} = {\mathtt{2.693}}{\mathtt{\,\times\,}}{{\mathtt{10}}}^{\left(-{\mathtt{5}}\right)}{\mathtt{\,\times\,}}{\mathtt{6\,371\,000}} = {\mathtt{distance}} = {\mathtt{171.571\: \!03}}$$ metres

This is 562.9ft to the nearest tenth of a foot.

 Alternatively

 Since the distances are relatively small we could assume the surface of the Earth to be flat near the lighthouse, in which case the distance of the ship from the lighthouse is given by 325*tan(60°)ft or

$${\mathtt{distance}} = {\mathtt{325}}{\mathtt{\,\times\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}{\left({\mathtt{60}}^\circ\right)} = {\mathtt{distance}} = {\mathtt{562.916\: \!512\: \!459\: \!925}}$$  feet

or 562.9ft.  

Using the properties of powers (see the Formulary) and collecting like terms together:

x³*y³*x⁴*y^a = x⁷y^3-a  

If this is to equal x⁷ then the y term must equal 1.  Now y⁰ = 1 so we must have 3-a = 0, or a =3

$$2\frac{2}{5}+2\frac{1}{4} = 2+\frac{2}{5}+2+\frac{1}{4} = 4+\frac{2}{5}+\frac{1}{4}$$

Collect the fractions over a common denominator:

$$\frac{2}{5}+\frac{1}{4} = \frac{2*4}{20} + \frac{1*5}{20} = \frac{8+5}{20} = \frac{13}{20}$$

So, altogether   $$2\frac{2}{5} + 2\frac{1}{4} = 4\frac{13}{20}$$

If I've interpreted the question correctly (a little difficult because of the lack of punctuation!), then:

Let the the original price be P.  The tax is then: tax = 0.08*P.

The discounted price is 0.4P (i.e. 60% less than the original price).  This equals $55, so 0.4P = 55.  Divide both sides by 0.4 to get P = 55/0.4 or P = $137.5.  The tax is therefore: tax = 0.08*137.5, or tax = $11.  The total is then $55 + $11 = $66.

Yes, you are correct.  I answered a similar question a couple of days ago

 http://web2.0calc.com/questions/the-volumes-of-two-mathematically-similar-solids-are-in-the-ratio-27-125-the-surface-area-of-the-smaller-solid-is-36cm-2-work-out-the-surfa#r102051

That is assuming you are talking about perpendicular height.

Yes.  I think it is conventional to assume perpendicular height is meant if the word "height" is used on  its own.  If slant height is meant then both those words would be used. 

The sine function can only have values between ±1 so there is no way it can be 3/2.

Inner brackets first: 8-2x5.  Here do multiplication first: 8 - 10 = -2

The original expression now becomes 1-2(-3+ -2)

Do the remaining bracket first; -3 + -2 = -5

So we now have 1 - 2(-5) = 1 +10 = 11 

The volume of a cone is (1/3)pi.r²h where r is radius and h is height, so: volume = (1/3)*pi*9²*5 m³

or volume = 135*pi m³