/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 79 In \(1986,\) an enormous iceberg... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 79

In \(1986,\) an enormous iceberg broke away from the Ross Ice Shelf in Antarctica. It was an approximately rectangular prism \(160 \mathrm{km}\) long, \(40.0 \mathrm{km}\) wide, and \(250 \mathrm{m}\) thick. (a) What is the mass of this iceberg, given that the density of ice is \(917 \mathrm{kg} / \mathrm{m}^{3}\) ? (b) How much heat transfer (in joules) is needed to melt it? (c) How many years would it take sunlight alone to melt ice this thick, if the ice absorbs an average of \(100 \mathrm{W} / \mathrm{m}^{2}, 12.00 \mathrm{h}\) per day?

Short Answer

Expert verified
To begin, we first determine the volume of the iceberg which is V = (160,000 m)(40,000 m)(250 m). The mass of the iceberg can then be found using the density of ice, 917 kg/m鲁, and the volume of the iceberg, m = (917 kg/m鲁) 脳 V. Next, we calculate the heat transfer needed to melt the iceberg using the heat of fusion of ice (334,000 J/kg) and the mass of the iceberg, Q = m 脳 (334,000 J/kg). To find how many years it would take sunlight alone to melt the iceberg, we first determine the area of the iceberg exposed to sunlight, A = (160,000 m)(40,000 m), and the energy absorbed per day by the iceberg, E_absorbed = (100 W/m虏) 脳 A 脳 (43,200 s). Then, the time required to melt the iceberg is found by dividing the total heat transfer needed by the energy absorbed per day, t_melt = Q / E_absorbed. Finally, we convert the time required to melt the iceberg from days to years by dividing by 365.

Step by step solution

01

Calculate the Volume of the Iceberg

First, we need to find the volume of the iceberg. The volume of a rectangular prism can be calculated using the formula: \[V = l 脳 w 脳 h\] where \(l\) is the length, \(w\) is the width, and \(h\) is the height of the prism. Given, the iceberg's dimensions are: Length (l) = 160.0 km (convert to meters: 160.0 脳 1000 = 160,000 m) Width (w) = 40.0 km (convert to meters: 40.0 脳 1000 = 40,000 m) Thickness (h) = 250 m Now, calculate the volume of the iceberg: \[V = (160,000 \,\text{m})(40,000 \,\text{m})(250 \,\text{m})\]
02

Calculate the Mass of the Iceberg

Now that we have the volume of the iceberg, we can calculate its mass using the ice's density: \[m = \rho 脳 V\] where \(m\) = mass of the iceberg \(\rho\) = density of ice (917 kg/m鲁) \(V\) = volume of the iceberg First, find the product of density and volume: \[m = (917 \,\mathrm{kg/m}^3) 脳 V\]
03

Calculate the Heat Transfer Needed to Melt the Iceberg

To calculate the heat transfer to melt the iceberg, we use the formula: \[Q = m 脳 L_f\] where \(Q\) = heat transfer, \(m\) = mass of the iceberg, and \(L_f\) = heat of fusion of ice (334,000 J/kg) Substitute the mass \(m\) calculated in step 2: \[Q = m 脳 (334,000 \,\mathrm{J/kg})\]
04

Calculate Total Energy Absorbed by the Iceberg per Day

To find the energy absorbed in one day, we need to multiply the energy absorption rate by the area of the iceberg exposed to sunlight and by the time it is exposed to sunlight: \[E_{absorbed} = P 脳 A 脳 t\] where \(E_{absorbed}\) = energy absorbed, \(P\) = 100 W/m虏 (absorption rate), \(A\) = area, and \(t\) = time (12.00 h per day, convert to seconds: 12 脳 3600 = 43,200 s) Calculate the area of the iceberg exposed to sunlight (consider the top surface): \[A = l 脳 w = (160,000 \,\mathrm{m})(40,000 \,\mathrm{m})\] Now, calculate the total energy absorbed per day by the iceberg: \[E_{absorbed} = (100 \,\mathrm{W/m^2}) 脳 A 脳 (43,200 \,\mathrm{s})\]
05

Calculate the Time Required for Sunlight to Melt the Iceberg

Now that we have the energy absorbed per day (\(E_{absorbed}\)) and the total heat transfer needed to melt the iceberg (\(Q\)), we can calculate the time required in days: \[t_{melt} = \frac{Q}{E_{absorbed}}\] Substitute the values obtained from steps 3 and 4 in the equation above to find out how many days are needed to melt the iceberg completely. Once the number of days is found, divide it by 365 to find the number of years it would take to melt the iceberg solely by sunlight.

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Key Concepts

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

Density and Volume Calculations
The mass of an object is deeply connected to its volume and density. In this exercise, the iceberg is modeled as a rectangular prism, where we need to find its volume first. The formula for volume of a rectangular prism is given by:\[ V = l \times w \times h \]where
  • \(l\) is the length of the iceberg (in this case, 160 km converted to meters, which is 160,000 m),
  • \(w\) is the width (40 km, which is 40,000 m),
  • \(h\) is the height/thickness (250 m).
By plugging these values into the formula, we calculate the volume in cubic meters.
Next, to find the mass, we use the relationship between mass, volume, and density:\[ m = \rho \times V \]The density of ice is given as 917 kg/m鲁. By multiplying the calculated volume with this density, we obtain the mass of the iceberg in kilograms.
Heat Transfer
Heat transfer is the process of energy moving from one body or substance to another due to a temperature difference. To melt the iceberg, we need to determine how much heat energy must be transferred to it. The formula used to calculate this energy involves the mass of the iceberg and the heat of fusion of ice, which is the energy required to turn solid ice into liquid water at its melting point:\[ Q = m \times L_f \]where
  • \(Q\) is the total heat energy needed,
  • \(m\) is the mass of the iceberg, already calculated,
  • \(L_f\) is the heat of fusion for ice, 334,000 J/kg.
By substituting the mass value into this formula, you calculate \(Q\), the total joules of energy needed to completely melt the iceberg. This calculation highlights the enormity of energy required for phase changes, even for naturally cold substances like ice.
Energy Absorption
Energy absorption is crucial in processes like melting, where the iceberg absorbs energy to change its state. Here, we calculate how much energy the iceberg absorbs from sunlight each day. The formula used is:\[ E_{absorbed} = P \times A \times t \]where
  • \(E_{absorbed}\) is the energy absorbed per day,
  • \(P\) is the power or rate of energy per unit area (100 W/m虏),
  • \(A\) is the area of the surface exposed to sunlight, calculated from the surface area \(l \times w\),
  • \(t\) is the time in seconds the sunlight strikes, 12 hours converted to 43,200 seconds.
By computing this, we see how much energy is absorbed daily. Comparing this daily absorption to the total energy required for melting allows us to estimate the number of years needed if melting was only reliant on sunlight.

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