A spherical tank has a radius of 6 yards. It is filled with a liquid that costs \( \$ 7.15 \) per cubic yard. What is the total value of the liquid in the tank?

To find the total value of the liquid in the tank, we first need to calculate the volume of the sphere using the formula for the volume of a sphere, which is given by: \[ V = \frac{4}{3} \pi r^3 \] Substituting the radius \( r = 6 \) yards into the formula: \[ V = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi \text{ cubic yards} \] Next, we use \( \pi \approx 3.14 \) for calculations: \[ V \approx 288 \times 3.14 \approx 904.32 \text{ cubic yards} \] Now, we multiply the volume by the cost per cubic yard: \[ \text{Total Value} = 904.32 \times 7.15 \approx 6459.19 \] Therefore, the total value of the liquid in the tank is approximately \( \$6459.19 \). Now, onto some extra knowledge! In ancient times, spherical tanks were used in various cultures for water storage due to their efficient design. The geometric shape not only minimized surface area but also provided structural integrity, making them ideal for holding large volumes of fluids without excessive material. This design can be traced back to civilizations that relied heavily on water management for agriculture and urban life. When it comes to real-world applications, spherical tanks are commonly found in industries such as oil and gas, where they are used to store gases and liquids under pressure. Their shape allows them to withstand external pressures while minimizing the risk of leaks—an essential feature for safety and efficiency in chemical processing and energy production.