The following diagram will help to explain the solution:

We are searching for the surface area of the shaded region.  We can multiply this by the depth (1.5 feet) to find the total volume of this area.

The radius of the outer circle is 44 feet.  Therefore its area is 44²π = 1936π.  The area of the inner circle is 40²π = 1600π.  Therefore the area of the shaded area is 1936π – 1600π = 336π.  The volume is 1.5 times this, or 504π.

The area of a circle can be solved using the equation $\displaystyle A=\pi r^{2}$ 

The area of a circle with radius 4 is $\displaystyle \pi 4^{2}=16\pi$ while the area of a circle with radius 2 is $\displaystyle \pi 2^{2}=4\pi$. $\displaystyle 16\pi \div 4\pi =4$

The circumference of a circle is equal to the diameter of the circle times $\displaystyle \pi$. The diameter is equal to twice the radius so:

$\displaystyle C=2\pi r$

The radius of the first can be solved as follows:

$\displaystyle 80\pi=2\pi r_{1}$

$\displaystyle r{_{1}}=40$

Likewise for the second circle:

$\displaystyle 40\pi=2\pi r_{2}$

$\displaystyle r_{2}=20$

The are of a circle is given by the following formula:

$\displaystyle A=\pi r^2$.

The ratio of the larger area to the smaller area can be found as follows:

$\displaystyle \frac{A_{1}}{A_{2}}=\frac{\pi r_{1}^{2}}{\pi r_{2}^{2}}=\frac{\pi (40)^2}{\pi(20)^2}=\frac{1600\pi}{400\pi}$

Cancelling out $\displaystyle \pi$ and dividing gives the correct answer of 

$\displaystyle 4$

This question asks you to apply the concept of area in finding both the area of a circle and square. Since the cirlce is inscribed in the square, we know that its diameter (two times the radius) is the same length as one side of the square. Since we are given the radius, $\displaystyle 4x$, we can find the area of both the circle and square.

Square:

^{$\displaystyle 2(4x) \cdot 2(4x) = 4 \cdot 16x^{}2 = 64x^{}2$}  

This gives us the area for the entire square.

The bottom half of the square has area $\displaystyle \frac{1}{2} \cdot 64x^{}2 = 32x^{}2$.

Now that we have this value, we must find the area that the circle occupies. The area of a circle is given by $\displaystyle \pi \cdot r^{}2$.

So the area of this circle will be $\displaystyle \pi \cdot 16x^{}2$.

The bottom half of the circle has half that area:

$\displaystyle \rightarrow \pi \cdot 8x^{}2$

Now that we have both our values, we can subtract the bottom half of the circle from the bottom half of the square to give us the shaded region:

$\displaystyle 32 x^{}2 - \pi \cdot 8x^{}2$

To solve, simply use the formula for the area of a circle. Thus,

$\displaystyle A=\pi{r^2}=\pi*7^2=49\pi$

To remember this formula, think about area and how it is in two dimensions, not one or three. So, one of our variables will have to be squared. By process of elimination, it must be r because pi doesn't change so it only makes sense to square the variable that changes between all circles.

The circumference C of a circle with radius r is expressed by:

$\displaystyle C=2\pi r$

Our circle's circumference is $\displaystyle 90\pi$. Therefore,

$\displaystyle 90\pi = 2 \pi r$

$\displaystyle r=45$

Plug this r-value into the formula for the area of a circle.

$\displaystyle A=\pi r^2 = \pi \cdot 45^2 = 2025\pi$

For the sake of simplicity, we will assume that $\displaystyle OA = AB = BC = CD = 1$; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for $\displaystyle r$ in the formula $\displaystyle A = \pi r^{2}$:

$\displaystyle A _{1}= \pi r^{2} = \pi \cdot 1 ^{2} = \pi$ - this is the area of the white circle in the center.

$\displaystyle A _{2}= \pi r^{2} = \pi \cdot 2^{2} =4 \pi$

$\displaystyle A _{3}= \pi r^{2} = \pi \cdot 3^{2} =9 \pi$

$\displaystyle A _{4}= \pi r^{2} = \pi \cdot 4^{2} =16 \pi$ - this is the total area of the figure.

The area of the white ring is the difference between those at of the second-largest and third-largest circles:

$\displaystyle A _{3} - A_{2} =9 \pi - 4 \pi = 5 \pi$

The total area of the white portion of the figure is 

$\displaystyle 5 \pi + \pi = 6 \pi$, 

which is 

$\displaystyle \frac{6 \pi}{16 \pi } \times 100 \% = 37 \frac{1}{2} \%$

of the figure.

If the radius of the circle is $\displaystyle 5$ units, that means that the diameter is $\displaystyle 10$ units. Because the circle is tangent to the sides of the square, we also know that each side of the square is also $\displaystyle 10$ units. 

The area of the square is given by its length times its width, or:

$\displaystyle 10\cdot10=100$

The area of the circle is given by:

$\displaystyle A=\pi r^2 = \pi \cdot 5^2 = 25\pi$

The area that we're looking for lies outside of the circle but inside of the square. To get this area, we subtract the area of the circle from the area of the square. This gives us our answer of $\displaystyle 100-25\pi$ square units. 

If the area of the square is $\displaystyle 36$, then that means that each side is $\displaystyle 6$ since the area of a square is given by $\displaystyle A=s^2$. 

In the figure, the side of the square is equal to the diameter of the semicircle. Therefore, the semicircle's radius is $\displaystyle 3$. 

The area of the semicircle is given by:

$\displaystyle A= \frac{1}{2}(\pi r^2) = \frac{1}{2}\pi (3^2) = \frac{9\pi}{2}$

The area of the entire figure is the sum of the two component areas. 

Our total area is $\displaystyle 36 + \frac{9\pi}{2}$.