Alan

gradient = (y2 - y1)/x2 - x1)

$${\mathtt{gradient}} = {\frac{\left({\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0}}\right)}{\left({\mathtt{6}}{\mathtt{\,-\,}}{\mathtt{5}}\right)}} \Rightarrow {\mathtt{gradient}} = {\mathtt{3}}$$

Volume of a pyramid is given by one third of the base area multiplied by the height

$${\mathtt{Volume}} = \left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right){\mathtt{\,\times\,}}{{\mathtt{16}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\mathtt{15}} \Rightarrow {\mathtt{Volume}} = {\mathtt{1\,280}}$$

Volume = 1280 m³

.

arcsine is the same as asin which is the same as sin^-1

Press 2nd then asin on the calculator

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}^{\!\!\mathtt{-1}}{\left({\frac{{\mathtt{10}}}{{\mathtt{20}}}}\right)} = {\mathtt{30^{\circ}}}$$

.

0.4 recurring can be expressed as a fraction in the following way:

Let x = 0.44444.....

Multiply by 10

10x = 4.44444....

Subtract x from 10x

9x = 4

So x = 4/9

Now multiply this by 4 to get 16/9

If you want this as a decimal:

$${\frac{{\mathtt{16}}}{{\mathtt{9}}}} = {\mathtt{1.777\: \!777\: \!777\: \!777\: \!777\: \!8}}$$

or 1.77777777....   (i.e. the 7's recurr).

.

y is the seventh root of 8, so y = 8^1/7

$${\mathtt{y}} = {{\mathtt{8}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{7}}}}\right)} \Rightarrow {\mathtt{y}} = {\mathtt{1.345\: \!900\: \!192\: \!632\: \!356\: \!2}}$$

.

The length of the arc is (225/360)*2*pi*510 feet.

Divide this by 20 feet and we get 100.138, so if she plants a tree at 0 feet, then every 20 feet thereafter, she would be able to fit in 101 trees.

.

If you expand and rearrange the first equation you get  x - 2y = 4

If you expand and rearrange the second equation you get x - 2y = -3

Clearly, these are incompatible.  There is no solution!

.

The limit is 2:

.

Imagine the outer squares are hinged at the bottom (where they meet their adjacent squares).  Rotate them about the hinge until they are below the other two squares, so a large square is formed.  

The arc's then form a complete circle of radius 12 inches.  

Inscribed in this circle is a square of side 12*√2 inches.

The shaded area is just the difference between the circle area and the square area.  i.e.

shaded area = pi*144 - 144*2 square inches

.