Question

The formula for the volume of a cone is \(V=1 / 3 \pi r^{2} h\) (where \(V=\) volume, \(\pi=3.14 r=\) radius, and \(h=\) height \() .\) If Mauna Loa is 9 kilometers high and has a radius of roughly 85 kilometers, what is its total volume?

Short answer

The volume of Mauna Loa is approximately 68074.5 cubic kilometers.

Step-by-step solution

Identify the Given Values

We need to determine the values provided in the problem. Here, the height of Mauna Loa is given as \(h = 9\) kilometers, and the radius \(r = 85\) kilometers.

Write Down the Volume Formula

Recall the formula for the volume of a cone, which is: \(V = \frac{1}{3} \pi r^2 h\). This formula will allow us to substitute the given values to compute the volume.

Substitute the Values into the Formula

Now, substitute \(r = 85\), \(h = 9\), and \(\pi \approx 3.14\) into the formula: \[V = \frac{1}{3} \times 3.14 \times (85)^2 \times 9\].

Calculate the Radius Squared

First, compute \(85^2 = 7225\). This step involves multiplying the radius by itself.

Compute the Volume

Substitute \(85^2 = 7225\) back into the formula: \[V = \frac{1}{3} \times 3.14 \times 7225 \times 9\]. First, calculate the product \(3.14 \times 7225 \times 9\).

Simplify the Multiplication

Calculate \(3.14 \times 7225 = 22691.5\). Then, multiply that by 9: \(22691.5 \times 9 = 204223.5\).

Apply the 1/3 Factor

Finally, multiply the result by \(\frac{1}{3}\): \[V = \frac{1}{3} \times 204223.5 = 68074.5\].

Conclude the Solution

This means that the total volume of Mauna Loa, considering its shape as a cone, is approximately 68074.5 cubic kilometers.

Key concepts

Geometric Formulas

Understanding geometric formulas is key to solving many mathematical problems, especially those related to three-dimensional shapes like cones. A cone is a three-dimensional shape with a circular base that tapers to a point, known as the apex. The volume of a cone can be calculated using the formula:\[ V = \frac{1}{3} \pi r^2 h \]- **\(V\)** represents the volume of the cone.- **\(\pi\)** is a constant, approximately equal to 3.14.- **\(r\)** is the radius of the base of the cone.- **\(h\)** is the height of the cone, the perpendicular distance from the base to the apex.This formula is derived from the concept that the volume of a cone is exactly one-third of the volume of a cylinder that has the same base and height. By understanding this relationship, you can easily remember how to calculate a cone's volume. Mastering these geometric formulas helps in solving real-world problems, like determining the volume of large natural structures.

Mathematical Calculations

Mathematical calculations are essential in solving problems involving geometric shapes. For Mauna Loa, calculating the volume involves several steps:1. **Identify Known Values**: Mauna Loa's height is 9 km and its radius is 85 km.2. **Apply the Formula**: Substitute the values into the volume formula for a cone: \[ V = \frac{1}{3} \times 3.14 \times 85^2 \times 9 \]3. **Calculating the Radius Squared**: Multiply the radius by itself to get: - \(85^2 = 7225\)4. **Perform Multiplication**: Compute the product with \(\pi\) and height: - Compute: \(3.14 \times 7225 = 22691.5\) - Then, \(22691.5 \times 9 = 204223.5\)5. **Final Calculation**: Divide by 3 to apply the one-third factor: - \(\frac{204223.5}{3} = 68074.5\)6. **Conclusion**: The total volume of Mauna Loa is approximately 68074.5 cubic kilometers.Breaking down the calculations step-by-step simplifies complex problems, making them easier to understand.

Mauna Loa

Mauna Loa is one of the most massive volcanoes on Earth, resembling a giant cone. Located on the island of Hawaii, Mauna Loa's sheer size is astonishing. When estimating its volume, it's common to model its shape as a perfect cone to simplify calculations. This approximation helps scientists and students alike visualize and quantify such immense geological features using basic geometry. Mauna Loa stands about 9 kilometers high from its base on the ocean floor to its summit. The mountain itself is approximately 85 kilometers wide at its base. These dimensions make it one of the largest subaerial volcanic structures on the planet. Understanding these measurements in conjunction with geometric calculations allows for a predicted volume, providing insight into its geological composition and potential impact on the environment. This approach highlights the intersection of mathematics and earth sciences, showcasing how abstract formulas have concrete applications in understanding the natural world.