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GG 103 Examples of Math Problems

First Problem

Calculate the area of the Big Island of Hawaii as a percentage of the total area of Hawaii

Second Problem

Use the geothermal gradient to calculate the temperature in the crust at a depth of 16 km

Third Problem

Calculate the spreading rate for a Hawaiian Islands as it drifts away from the Hawaiian Hot Spot

Fourth Problem

Calculate the subsidence (sinking) rate of a Hawaiian volcano

Fifth Problem

Convert the rate of flow of a pyroclastic flow from mi/hr to ft/s

Sixth Problem

Calculate the volume of a Hawaiian shield volcano

Seventh Problem

Calculate the rate of growth in m/1000 yr for a shield volcano

Eighth Problem

Calculate the time required for a pahoehoe flow to reach the ocean from Kilauea Caldera

Ninth Problem

Estimate the time required for the Nu'uanu landslide to flow up to 120 km from O'ahu

Tenth Problem

Determine the absolute age of a rock sample

Eleventh Problem

Convert the average annual rainfall in Hawai'i from cm/yr to in/yr

Twelfth Problem

Calculate the average annual erosion rate in cm/yr, if a 1700 m deep valley formed in 60,000 yr

Thirteenth Problem

Use Darcy's Law equation to calculate the average flow volume for a portion of the subsurface

Fourteenth Problem

How long would it take a tsunami wave traveling at 700 km/hr to reach Hawaii from the subduction zones near California?


Calculate the area of the Big Island of Hawaii as a percentage of the total area of Hawaii

To calculate the percentage, divide the area of the Big Island of Hawaii by the total area of Hawaii (i.e. the entire state), and multiple the fraction by 100%.

The area of the Big Island is 10,451 km2.
The total area of Hawaii is 16,749 km2.

10,451 km2/16,749 km2 x 100% = 62.4%

To receive credit for the problem in class and on the quiz, you must show your work.

Use the geothermal gradient to calculate the temperature in the crust at a depth of 16 km

The geothermal gradient in the crust averages 30oC/km.

16 km x 30oC/km = 480oC

To receive credit for the problem in class and on the quiz, you must show your work.

Calculate the spreading rate for a Hawaiian Islands as it drifts away from the Hawaiian Hot Spot

To calculate the rate of drift for a Hawaiian Island, the distance of the island from the hot spot and the age of the island (time) is required. A rate is determined by the distance divided by the time: distance/time or, simply, d/t.

Rate = d/t

For example, the distance between Honolulu and Hilo is 300 km, and the age of O'ahu is approximately 2.5 million years. Calculate the average rate of drift for O'ahu.

d/t = 300km/2,500,000 yr = 0.00012 km/yr

To better visualize this rate, it should be converted to cm/yr. First convert kilometers to meters, then meters to centimeters.

300 km * 1000m/1 km = 300,000 m

300,000 m * 100 cm/1 m = 30,000,000 cm

Finally,

d/t = 30,000,000 cm/2,500,000 yr = 12 cm/yr

To receive credit for the problem in class and on the quiz, you must show your work.

Calculate the subsidence (sinking) rate of a Hawaiian volcano

To calculate the isostatic sinking rate of a Hawaiian volcano, divide the distance that the volcano sank by the age of the volcano. For example, if it took Mauna Loa 800,000 yr to sink 300 m:

d/t = 300 m/800,000 yr

To better visualize this rate, it should be converted to cm/1000 yr. First convert meters to centimeters:

300 m * 100cm/1 m = 30,000 cm

Finally, divide the distance in centimeters by the age of the volcano (time):

d/t = 300,000 cm/800,000 yr = 0.0375 cm/yr

To better visualize this rate, multiple the annual rate by 1000 to yield 1 cm/1000 yr

0.00375 cm/yr x 1000 = 37.5 cm/1000 yr

To receive credit for the problem in class and on the quiz, you must show your work.

Convert the rate of flow of a pyroclastic flow from mi/hr to ft/s

To convert mi/hr => ft/s, convert mi => ft and hr => s.

1 mi = 5280 ft

1 hr = 60 min

1 min = 60 s

Give a rate of 80 mi/hr:

80 mi/hr x 5280 ft/mi x hr/60 min x min/60 s = 117 ft/s

To receive credit for the problem in class and on the quiz, you must show your work.

Calculate the rate of growth in m/1000 yr for a shield volcano

To calculate the the growth rate for a shield volcano, you must know the total height and age of the volcano.

Mauna Loa is 9,000 m tall, from the sea floor, and at least 600,000 years old.

To calculate the rate in m/1000 yr, you can determine the rate in m/yr, then multiple by 1000.

Rate = 9000 m/600,000 yr = 0.015 m/yr

0.015 m/yr x 1000 = 15 m/1000 yr

To receive credit for the problem in class and on the quiz, you must show your work.

Calculate the volume of a Hawaiian shield volcano

Given the annual rate of lava production and the age of a shield volcano, you can calculate its volume.

If Mauna Loa erupts an average of 0.1 km3/yr and is at least 600,000 years old:

0.1 km3/yr x 600,000 yr = 60,000 km3

To receive credit for the problem in class and on the quiz, you must show your work.

Estimate the time required for the Nu'uanu landslide to flow up to 120 km from O'ahu

The debris field from the collapses of the Ko'olau shield volcano extends for up to 120 km to the northeast of O'ahu. The landslide could have traveled at an average speed of approximately 30 mi/hr.

To estimate the time required for the Nu'uanu landslide to flow 120 km from O'ahu, you need to convert the flow rate from mi/hr to km/hr.

1 mi = 1.6 km

30 mi/hr x 1.6 km/mi = 48 km/hr

The rate formula:

rate = d/t

Therefore,

t = d/rate

t = 120 km/(48 km/hr) = 2.5 hr

To receive credit for the problem in class and on the quiz, you must show your work.

Calculate the time required for a tsunami wave to travel from the Big Island to Hawaii

Flank collapse on the Big Island would generate a destructive tsunami wave. Tsunami waves can travel at the speeds of jet planes, up to 700 km/hr. At this speed, calculate the travel time to O'ahu for a wave generated on the Kilauea side of Hawai'i? The distance is approximately 275 mi.

You need to convert the distance to kilometers - 1 mi = 1.6 km

275 mi x 1.6 km/mi = 440 km

Rearrange the rate equation to solve for time:

rate = d/t

t = d/rate

t = 440 km/700 km/hr

t = 0.63 hr

Convert the value into minutes:

t = 0.63 hr x 60 min/hr

t = 38 min

To receive credit for the problem in class and on the quiz, you must show your work.

Calculate the time required for a pahoehoe flow to reach the ocean from Kilauea Caldera

The shoreline is approximately 25 km from Kilauea Caldera. Calculate the time required for a pahoehoe flow erupted in Kilauea Caldera to reach the coast if it flows at an average of 10 mi/hr.

You need to convert the rate to kilometers per hour - 1 mi = 1.6 km

10 mi/hr x 1.6 km/mi = 16 km/hr

Rearrange the rate equation to solve for time:

rate = d/t

t = d/rate

t = 25 km/16 km/hr

t = 1.56 hr

To receive credit for the problem in class and on the quiz, you must show your work.

Determine the absolute age of a rock sample

Use the concept of half-lives to determine the absolute of a rock sample.

A sample of granite has a U-235/Pb-207 ratio of 1/31. U-235 is the radioactive parent isotope, and Pb-207 is the daughter isotope.

The half-life of U-235 is 713 million years.

Determine the absolute age of this sample of granite:

P/D after one half-life = 50/50 or 1/1

P/D after two half-lives = 25/75 or 1/3

P/D after three half-lives = 12.5/87.5 or 1/7

P/D after four half-lives = 6.25/93.75 or 1/15

P/D after five half-lives = 3.125/96.875 or 1/31

Therefore, five half-lives have passed.

5 half-lives x 713 million years/half-life = 3.6 billion years

To receive credit for the problem in class and on the quiz, you must show your work.

Convert the average annual rainfall in Hawai'i from cm/yr to in/yr

Given that the average annual precipitation in Hawai'i is 190 cm/yr, convert this value to inches per year.

2.54 cm = 1 in

190 cm/yr x 1 in/2.54 cm = 75 in/yr

To receive credit for the problem in class and on the quiz, you must show your work.

Calculate the average annual erosion rate in cm/yr, if a 1700 m deep valley formed in 60,000 yr.

Given that the average annual precipitation in Hawai'i is 190 cm/yr, convert this value to inches per year.

rate = distance/time

rate = 1700 m/60,000 yr

rate = 0.0283 m/yr

To better visualize this rate, convert m/yr => cm/yr

0.0283 m/yr x 100 cm/m = 2.8 cm/yr

To receive credit for the problem in class and on the quiz, you must show your work.

Use Darcy's Law equation to calculate the average flow volume for a portion of the subsurface

Geologist use Darcy's Law to calculate the flow volume rate for subsurface units.
Darcy's Law:

Q = KAh/L

Calcalate Q given the following measurements:

K = 2.0 x 10-3 cm/s
A = 1 x 107 m2
h = 7 m
L = 15 km

First, convert all of the values to meters:

K = 2.0 x 10-3 cm/s x 1 m/100 cm = 2.0 x 10-5 m/s

L = 15 km x 1000 m/km = 15,000 m

Q = KAh/L

Q = (2.0 x 10-5 m/s)(1 x 107 m2)(7 m)/15,000 m = 0.09 m3/s

To receive credit for the problem in class and on the quiz, you must show your work.

How long would it take a tsunami wave traveling at 700 km/hr to reach Hawaii from the subduction zones near California?

If California is 2400 miles from Hawaii, how long would it take a fast tsunami wave to reach Hawaii?

First convert the miles to kilometers, 1 mi = 1.6 km:

2400 mi x 1.6 km/mi = 3840 km.

Now you know the rate at which a tsunami wave can travel, 700 km/hr, and the distance in kilometers to California, 3840 km. Rearrange the distance/time = rate equation to solve for time:

d/t = rate

d/rate = t

3840 km/700 km/hr = 5.5 hr

To receive credit for the problem in class and on the quiz, you must show your work.

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