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Chapter 8: Problem 55

Shield volcanoes are shaped something like flattened cones. The volume of a cone is equal to the area of its base multiplied by one-third of its height. The largest volcano on Mars, Olympus Mons, is \(27 \mathrm{km}\) high and has a base diameter of 550 km. Compare its volume with that of Earth's largest volcano, Mauna Loa, which is \(9 \mathrm{km}\) high and has a base diameter of \(120 \mathrm{km}\)

Short Answer

Expert verified
Olympus Mons has a volume of approximately 2,135,205 km\textsuperscript{3}; Mauna Loa, 33,930 km\textsuperscript{3}. Olympus Mons is much larger.

Step by step solution

01

Understand the volume formula for a cone

The volume of a cone can be calculated using the formula: \[ V = \frac{1}{3} \times \text{base area} \times \text{height} \]Here, base area can be found using the formula for the area of a circle, which is \( \pi r^2 \), where \( r \) is the radius of the base.
02

Convert diameters to radii

To find the radius, divide the diameter by 2.For Olympus Mons: Radius \( r = \frac{550}{2} = 275 \) km.For Mauna Loa: Radius \( r = \frac{120}{2} = 60 \) km.
03

Calculate the base area for each volcano

Using the formula for the area of a circle, \( \pi r^2 \):For Olympus Mons: \[ \text{Base Area} = \pi \times (275)^2 \approx 237,345 \ \text{km}^2 \]For Mauna Loa: \[ \text{Base Area} = \pi \times (60)^2 \approx 11,310 \ \text{km}^2 \]
04

Calculate the volume of Olympus Mons

Using the volume formula for a cone: \[ V = \frac{1}{3} \times \text{base area} \times \text{height} \]For Olympus Mons: \[ V = \frac{1}{3} \times 237,345 \ \text{km}^2 \times 27 \ \text{km} \approx 2,135,205 \ \text{km}^3 \]
05

Calculate the volume of Mauna Loa

Using the volume formula for a cone: \[ V = \frac{1}{3} \times \text{base area} \times \text{height} \]For Mauna Loa: \[ V = \frac{1}{3} \times 11,310 \ \text{km}^2 \times 9 \ \text{km} \approx 33,930 \ \text{km}^3 \]
06

Compare the volumes

Olympus Mons: \( 2,135,205 \ \text{km}^3 \)Mauna Loa: \( 33,930 \ \text{km}^3 \)Olympus Mons is significantly larger than Mauna Loa in terms of volume.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Shield Volcanoes
Shield volcanoes are types of volcanoes with broad, gentle slopes formed by the eruption of low-viscosity lava that can flow long distances. These types of volcanoes are named for their resemblance to a warrior’s shield.
  • They are formed by effusive eruptions, where lava flow pours out steadily and spreads in all directions.
  • Being composed mostly of basalt, which is a type of low-viscosity lava, helps them to create wide bases with dome-like profiles.
  • Some of the largest shield volcanoes are found on both Earth and Mars.
A famous example on Earth is Mauna Loa in Hawaii. Another legendary shield volcano, found on Mars, is Olympus Mons which is the tallest volcano in the entire solar system.
Volume of a Cone
Calculating the volume of a cone involves using a specific mathematical formula where the volume (V) is found by multiplying the base area (A) by one-third the height (H). This can be written in the formula:

The base area can be determined using the formula for the area of a circle ( . Here’s a simple breakdown:
  • Convert the diameter of the base to the radius by dividing by 2.
  • Calculate the base area using PI * (radius)^ 2 .
  • Use the volume formula for the cone: 1/3 x base area x height
  • This gives you the volume of your cone-shaped object.
For example, Olympus Mons on Mars has a base diameter of 550 km and height of 27 km:
  • First, find the radius: 275 km
  • Then, the base area: Finally, calculate the volume:
    < 1/3 x 237,345 km ^2 x 27 km or approximately 2,135,205 km ^3 <
Following the same steps for Mauna Loa: Kelvin, km Volume, area height everything would give you <3: 11,310 km ^2 x 9 km or approximately 33,930 km ^3
Mathematical Comparisons
When comparing the volumes of two massive structures like Olympus Mons and Mauna Loa, understanding scale, proportion, and large number comparisons is crucial.
  • Start with converting all units to the same scale, such as kilometers in this example.
  • Use precise figures when calculating geometrical properties like base area and radius.
  • Compare the volumes directly by looking at the end results of the calculations.
Here’s a brief breakdown of how these numbers can be compared:
  • Olympus Mons: 2,135,205 km^3 vs. Mauna Loa: 33,930 km ^3.
Olympus Mons has a significantly larger volume than Mauna Loa because it is significantly taller and has a much broader base. Details like these can illuminate not only the immense size differences but also what contributes to these differences naturally occurring formations.

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