The surface area of a cube is six times the square of its sidelength, so we find the sidelength. This is the cube root of volume 1,000, so

\(\displaystyle \sqrt[3]{1000}\) \(\displaystyle =10\) centimeters.

To rewrite this as inches, divide by 2.5:

\(\displaystyle 10\) \(\displaystyle \div\) \(\displaystyle 2.5\) \(\displaystyle = 4\) inches

The surface area of the cube in square inches is 

\(\displaystyle 6\times\) \(\displaystyle 4^{2}\) \(\displaystyle = 96\) square inches.

The volume of a cube is \(\displaystyle s^3\) where \(\displaystyle s\) is the length of one edge, so \(\displaystyle s\) is the cube root of volume:

\(\displaystyle Volume=64\Rightarrow s=\sqrt[3]{64}=4\ in\)

A cube has six faces and the surface area of a cube is \(\displaystyle 6s^2\). So we can write:

Surface area = \(\displaystyle 6s^2=6\times 4^2=6\times 16=96\ in^{2}\)

In order to determine the length of the diagonal of a square we would use the Pythagorean Theorem. First we should find the side length:

\(\displaystyle Side=\sqrt{area}=\sqrt{16}=4\ inches\) 

Now, "square" the length of one side and multiply by 2, then take the square root of that number to get the length of the diagonal:

\(\displaystyle Diagonal=\sqrt{4^2\times 2}=4\sqrt{2}\ inches\)

The area of the farm is:

\(\displaystyle A=200\times 200=40000\) square feet. So the amount of the fertilizer he needs can be calculated as:

\(\displaystyle 40000\div 40=1000\) pounds

Every pound of the fertilizer costs $2, so he needs to spend \(\displaystyle 2\times 1000=2000\) dollars.

We need to use the Pythagorean Theorem in order to solve this problem. We can write:

\(\displaystyle d^2=s^2+s^2=2s^2\) 

where \(\displaystyle d\) is the diagonal length and \(\displaystyle s\) is the side length. The diagonal length of the square is \(\displaystyle d=t\), so we can write:

\(\displaystyle t^2=2s^2\Rightarrow s^2=\frac{t^2}{2}\Rightarrow Area=s^2=\frac{t^2}{2}\)
 

Start by working backwards from the area of the circle to find its diameter.

The area of a circle is \(\displaystyle \pi r^{2}\), and \(\displaystyle \sqrt{12.25}=3.5\) (you can get this by trial and error pretty quickly, since you know it's between 3 and 4, and since 12.25 ends in a 5, you now that a 5 is involved), so the diameter of the circle is 7. This is also the diagonal of the square. Y

ou may remember that the legs of a \(\displaystyle 45-45-90\) right triangle (of which we have two within the square, once we draw the diagonal) are equal to the hypotenuse divided by \(\displaystyle \sqrt{2}\). But even if you forget this, you should recall the Pythagorean Theorem that states that \(\displaystyle A^2+B^{2}=C^{2}\). In this case, \(\displaystyle A\) and \(\displaystyle B\) are equal, so the square of each side is equal to half of \(\displaystyle C^{2}\), or 24.5. And the square of one side of a square is also equal to the area of the square.

Use the perimeter to find the side length of the square.

\(\displaystyle \text{Side}=88\div4=22\) \(\displaystyle cm\)

Now, use the side length to find the area of the square.

\(\displaystyle \text{Area}=\text{Side}^2=22^2=484\) \(\displaystyle cm^2\)

First, use the perimeter to find the side lengths of the square.

\(\displaystyle \text{Side}=36\div4=9\)

Use this information to find the area.

\(\displaystyle \text{Area}=\text{Side}^2=9^2=81\)

Since we know the lengths of one leg and the hypotenuse, we can calculate the length of the other leg using the Pythagorean Theorem. We can use this form:

\(\displaystyle b = \sqrt{ c^{2} - a ^{2}}\)

Setting \(\displaystyle c\) and \(\displaystyle a\) equal to the lengths of the hypotenuse and the leg - 13 and 5, respectively:

\(\displaystyle b = \sqrt{ 13^{2} - 5 ^{2}} = \sqrt{169 - 25} = \sqrt{144} = 12\)

The perimeter is equal to the sum of the lengths of the three sides:

\(\displaystyle 5+ 12 + 13 = 30\)

This is also the perimeter of the square, so the length of each side is one fourth of this, or

\(\displaystyle 30 \div 4 = 7 \frac{1}{2}\)

The area is the square of this, or

\(\displaystyle A = 7 \frac{1}{2} \cdot 7 \frac{1}{2} = 56 \frac{1}{4}\)

First, we need the radius \(\displaystyle r\) of the circle, which can be determined from the area of a circle formula by setting \(\displaystyle A = 4\):

\(\displaystyle \pi r ^{2} = A\)

\(\displaystyle \pi r ^{2} = 4\)

\(\displaystyle \frac{ \pi r ^{2}}{\pi} = \frac{4 }{\pi}\)

\(\displaystyle r ^{2}= \frac{4 }{\pi}\)

\(\displaystyle r = \sqrt{\frac{4 }{\pi} }\)

Simplifying the expression by splitting the radicand and rationalizing the denominator:

\(\displaystyle r = \frac{\sqrt{4}}{\sqrt{\pi}}\)

\(\displaystyle r = \frac{2}{\sqrt{\pi}}\)

\(\displaystyle r = \frac{2 \cdot \sqrt{\pi}}{\sqrt{\pi} \cdot \sqrt{\pi}}\)

\(\displaystyle r= \frac{2 \sqrt{\pi}}{\pi}\)

The circumference of the circle is \(\displaystyle 2 \pi\) multiplied by the radius, or

\(\displaystyle C = 2 \pi r\)

\(\displaystyle C = 2 \pi \cdot \frac{2 \sqrt{\pi}}{\pi} = 4 \sqrt{\pi}\)

This is also the perimeter of the square, so the length of each side is one fourth of this perimeter, or

\(\displaystyle \frac{ 4 \sqrt{\pi}}{4} = \sqrt{\pi}\)

The area of the square is the square of this common sidelength, or 

\(\displaystyle \left ( \sqrt{\pi} \right )^{2} = \pi\).