/*! 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 59 A dish of lasagna is being heate... [FREE SOLUTION] | 91Ó°ÊÓ

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A dish of lasagna is being heated in a microwave oven. The effective area of the lasagna that is exposed to the microwaves is \(2.2 \times\) \(10^{-2} \mathrm{m}^{2} .\) The mass of the lasagna is \(0.35 \mathrm{kg},\) and its specific heat capacity is \(3200 \mathrm{J} / \mathrm{kg} \cdot \mathrm{C}^{\circ}\) ). The temperature rises by \(72^{\circ} \mathrm{C}\) in 8.0 minutes. What is the intensity of the microwaves in the oven?

Short Answer

Expert verified
The intensity of the microwaves is approximately 295.45 W/m².

Step by step solution

01

Understand the Problem

We need to find the intensity of the microwaves in the oven. Given are the effective area of the lasagna exposed to microwaves, its mass, specific heat capacity, and the temperature increase over time.
02

Calculate the Heat Needed

Use the formula for heat transfer to calculate the heat absorbed by the lasagna:\[ Q = mc\Delta T \]Where:- \(m = 0.35\, \text{kg}\) is the mass,- \(c = 3200\, \text{J/kg°C}\) is the specific heat capacity,- \(\Delta T = 72\, \text{°C}\) is the change in temperature.Substituting these values:\[ Q = 0.35\, \text{kg} \times 3200\, \text{J/kg°C} \times 72\, \text{°C} \]Calculate \( Q \).
03

Convert Time to Seconds

Since the time is given in minutes, convert it to seconds to get the power in watts. The time is 8.0 minutes.\[ 8.0\, \text{minutes} \times 60\, \text{seconds/minute} = 480\, \text{seconds} \].
04

Calculate the Power Absorbed

Use the formula for power with the time converted from minutes to seconds:\[ P = \frac{Q}{t} \]Where:- \(Q\) is the heat calculated in Step 2,- \(t = 480\, \text{seconds}\) from Step 3.Calculate \( P \).
05

Calculate Microwave Intensity

Intensity is given by the power per unit area:\[ I = \frac{P}{A} \]Where:- \(P\) is the power from Step 4,- \(A = 2.2 \times 10^{-2}\, \text{m}^2\) is the area of exposure.Calculate \( I \).

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Key Concepts

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

Specific Heat Capacity
Specific heat capacity is the amount of heat required to change the temperature of 1 kilogram of a substance by 1 degree Celsius. It is an intrinsic property of materials. This means every material has its own specific heat capacity, denoting how much energy it can store. For instance, our lasagna has a specific heat capacity of 3200 J/kg°C. This value tells us that for every kilogram of lasagna, 3200 joules are needed to increase the temperature by one degree Celsius. Next time you're heating up your leftovers, remember that some foods require more energy to warm up just because they have a higher specific heat capacity!
Why does it matter? Understanding specific heat capacity helps in calculating how much energy a substance holds at a given temperature, essential for designing efficient heating systems.
Heat Transfer
The concept of heat transfer is central to understanding how energy moves from one object or substance to another. Heat transfer occurs in three ways: conduction, convection, and radiation. In the case of our microwave lasagna, radiation is the primary method, as microwaves use electromagnetic radiation to heat food.
The formula for calculating the heat transferred is \[ Q = mc \Delta T \] where \( Q \) is the heat energy in joules, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature. This equation helps us determine how much energy the lasagna has absorbed from the microwaves. This kind of energy transfer is crucial for numerous applications beyond just cooking, such as climatology and engineering.
Microwave Heating
Microwave heating is an efficient method used to cook and reheat meals quickly. It uses electromagnetic waves, specifically microwaves, which excite water molecules inside the food. This molecular excitement produces thermal energy within the food, warming or cooking it uniformly. The speed and efficiency of heating depend on the microwave's power output and the food's properties like moisture content and specific heat capacity.
  • Benefits: Quick and uniform heating.
  • Considerations: Foods with higher water content heat faster.
This method of heating harnesses principles of physics like wave-particle interaction, making it not only practical but also a fascinating study of energy transfer in everyday life.
Power Calculation
Understanding power calculation is crucial for determining how much energy is used over time. Power is defined as the rate at which work is done or energy is transferred. In our lasagna problem, we first figure out the heat (energy) absorbed by the food and then calculate the power using the formula: \[ P = \frac{Q}{t} \] where \( P \) is power in watts, \( Q \) is the energy transferred in joules, and \( t \) is the time in seconds.
After obtaining power, we determine intensity, measuring how much energy is transferred per unit area using: \[ I = \frac{P}{A} \] where \( I \) is intensity and \( A \) is the area exposed to the microwaves. These calculations not only help us in the kitchen but are also essential in areas like electrical engineering and sustainable energy, where efficiency is key.

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Most popular questions from this chapter

A hunter is standing on flat ground between two vertical cliffs that are directly opposite one another. He is closer to one cliff than to the other. He fires a gun and, after a while, hears three echoes. The second echo arrives \(1.6 \mathrm{s}\) after the first, and the third echo arrives \(1.1 \mathrm{s}\) after the second. Assuming that the speed of sound is \(343 \mathrm{m} / \mathrm{s}\) and that there are no reflections of sound from the ground, find the distance between the cliffs.

Suppose that a public address system emits sound uniformly in all directions and that there are no reflections. The intensity at a location \(22 \mathrm{m}\) away from the sound source is \(3.0 \times 10^{-4} \mathrm{W} / \mathrm{m}^{2} .\) What is the intensity at a spot that is \(78 \mathrm{m}\) away?

You are flying in an ultralight aircraft at a speed of \(39 \mathrm{m} / \mathrm{s}\). An eagle, whose speed is \(18 \mathrm{m} / \mathrm{s},\) is flying directly toward you. Each of the given speeds is relative to the ground. The eagle emits a shrill cry whose frequency is \(3400 \mathrm{Hz} .\) The speed of sound is \(330 \mathrm{m} / \mathrm{s} .\) What frequency do you hear?

A recording engineer works in a soundproofed room that is \(44.0 \mathrm{dB}\) quieter than the outside. If the sound intensity that leaks into the room is \(1.20 \times 10^{-10} \mathrm{W} / \mathrm{m}^{2},\) what is the intensity outside?

To measure the acceleration due to gravity on a distant planet, an astronaut hangs a \(0.055-\mathrm{kg}\) ball from the end of a wire. The wire has a length of \(0.95 \mathrm{m}\) and a linear density of \(1.2 \times 10^{-1} \mathrm{kg} / \mathrm{m} .\) Using electronic equipment, the astronaut measures the time for a transverse pulse to travel the length of the wire and obtains a value of 0.016 s. The mass of the wire is negligible compared to the mass of the ball. Determine the acceleration due to gravity.

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