450,000 gallons of water occupy \(\displaystyle \small \small 0.1337 \cdot 450,000 \approx 60,165\) cubic feet. 

Let \(\displaystyle \small h\) be the height of the tank. Then the width of the tank is \(\displaystyle W = 2H\), and its length is \(\displaystyle L = 2W = 4H\). Multiply the dimensions to get the volume:

\(\displaystyle V = LWH = H \cdot 2H\cdot 4H = 8H^{3} = 60,165\)

\(\displaystyle H^{3} = 60,165 \div 8 \approx 7,521\)

\(\displaystyle H = \sqrt[3]{7,521} \approx 19.6\)

\(\displaystyle W = 2H = 2 \cdot19.6 = 39.2\)

\(\displaystyle L = 2W = 2\cdot 39.2 = 78.4\)

Since the tank has four sides and a bottom, but not a top, its surface area is 

\(\displaystyle A = 2LH + 2WH + LW\)

\(\displaystyle A = 2\cdot 78.4\cdot 19.6 + 2\cdot 39.2\cdot 19.6 + 78.4\cdot 39.2\)

\(\displaystyle A = 2\cdot 78.4\cdot 19.6 + 2\cdot 39.2\cdot 19.6 + 78.4\cdot 39.2 \approx 7,683\)

The surface area of the tank is about 7,683 square feet.

The volume of a rectangular prism is equal to its length times its width times its height. We are given the volume, the length, and the width, so using the following formula we can solve for the height of the rectangular prism:

\(\displaystyle V=LWH\)

\(\displaystyle 105=(7)(5)H\)

\(\displaystyle H=\frac{105}{35}=3\)

Using the formula for the volume of a rectangular prism, we can plug in the given values and solve for the width of the prism:

\(\displaystyle V=LWH\)

\(\displaystyle 70=(7)W(2)\)

\(\displaystyle 70=14W\)

\(\displaystyle W=\frac{70}{14}=5\)

We are given the surface area and the length of two sides, so in order to calculate the length of the third side we need the formula for the surface area of a prism in terms of each side length:

\(\displaystyle SA=2LW+2LH+2WH\)

Using the formula, we can simply plug in the given values and solve for the length of the rectangular prism:

\(\displaystyle 92=2L(3)+2L(2)+2(3)(2)\)

\(\displaystyle 92=6L+4L+12\)

\(\displaystyle 10L=80\)

\(\displaystyle L=8\)

The length of an edge of a cube can be used to find either volume or surface area, and vice versa.

I) Gives us the surface area thus, we are able to calculate the length of an edge using the formula,

\(\displaystyle SA=6\cdot s^2\)

\(\displaystyle 294=6s^2 \rightarrow 49=s^2 \rightarrow 7=s\).

II) Gives us the volume thus, we are able to calculate the length of an edge using the formula,

\(\displaystyle V=s^3\)

\(\displaystyle 343=s^3 \rightarrow 7=s\)

Either can be used to find the side length.

\(\displaystyle SA=2lw+2lh+2hw=2*4*6+2*4*5+2*5*6=48+40+60=148\)

Find the surface area of the box by summing the area of all six faces: two 8 by 10, two 8 by 5, two 10 by 5.

\(\displaystyle 2(80)+2(40)+2(50)=340\ in^{2}\)

Since the price is $.10 for each square inch,

\(\displaystyle Total\ cost=340(.10)=\$34\)

Let \(\displaystyle h\) be the height of the prism. Then the width is \(\displaystyle 3h\), and the length is \(\displaystyle 3h+ 16\). Since the sum of the three dimensions is one meter, or 100 centimeters, we solve for \(\displaystyle h\) in this equation:

\(\displaystyle h + 3h + (3h+16) = 100\)

\(\displaystyle 7h+16 = 100\)

\(\displaystyle 7h+16-16 = 100-16\)

\(\displaystyle 7h = 84\)

\(\displaystyle 7h \div 7 = 84 \div 7\)

\(\displaystyle h = 12\)

The height of the prism is 12 cm; the width is three times this, or 36 cm; the length is sixteen centimeters greater than the width, which is 52 cm.

Set  \(\displaystyle l= 52,w=36,h=12\) in the formula for the surface area of a rectangular prism:

\(\displaystyle A = 2lw + 2wh + 2lh\)

\(\displaystyle = 2 \cdot 52 \cdot 36 + 2 \cdot 36 \cdot 12 + 2 \cdot 52 \cdot 12\)

\(\displaystyle = 3,744 + 864 + 1,248\)

\(\displaystyle = 5,856\) square centimeters

We can set the lengths of three sides to be \(\displaystyle x\), \(\displaystyle y\), \(\displaystyle z\), respectively. The volume is 120 means that \(\displaystyle xyz=120\). Also we know the areas of two sides, so we can use \(\displaystyle xy\) to represent the area of 20 and \(\displaystyle xz\) to represent the area of 30. Now the question is to figure out \(\displaystyle yz\). Then we can use

\(\displaystyle xy * xz * yz = x^2 * y^2 * z^2 =(xyz)^2\) to solve \(\displaystyle yz\).

\(\displaystyle 20 * 30 * yz = 120^2\)

\(\displaystyle yz=24\)

The area of a right triangle is equal to half the product of its legs, so each base has area

\(\displaystyle \frac{1}{2} \cdot 12 \cdot 16 = 96\)

The measure of the hypotenuse of each base is determined using the Pythagorean Theorem:

\(\displaystyle c = \sqrt{12^{2}+16^{2}}= \sqrt{144+256} = \sqrt{400}= 20\)

Therefore, the perimeter of each base is 

\(\displaystyle P=12+16+20 = 48\),

and the height of the prism is half this, or \(\displaystyle h = \frac{1}{2} \cdot 48 = 24\).

The lateral area of the prism is the product of its height and the perimeter of a base; this is

\(\displaystyle L= Ph= 24 \cdot 48 = 1,152\)

The surface area is the sum of the lateral area and the two base areas, or

\(\displaystyle S= L+B+B =1,152+96+96 =1,344\).