(a) What is the resistance of the heater element in a \(1500-\mathrm{W}\) hair dryer that plugs into a \(120-\mathrm{V}\) outlet? (b) What is the current through the hair dryer when it is turned on? (c) At a cost of \(\$ 0.10\) per kW .h, how much does it cost to run the hair dryer for 5.00 min? (d) If you were to take the hair dryer to Europe where the voltage is \(240 \mathrm{V},\) how much power would your hair dryer be using in the brief time before it is ruined? (e) What current would be flowing through the hair dryer during this time?

Short Answer

Expert verified
Based on the calculations above, the hair dryer has the following properties: (a) The resistance of the heater element is 9.60 ohms. (b) The current through the hair dryer is 12.5 A when used in the US. (c) The cost to run the hair dryer for 5.00 minutes is $0.0125. (d) The power the hair dryer would be using in Europe is 6000 W, a brief time before getting ruined. (e) The current flowing through the hair dryer in Europe is 25 A.

Step by step solution

01

(a) Calculate the resistance of the heater element

To calculate the resistance of the heater element, we can use the formula: \(P = \frac{V^2}{R}\) Where \(P\) is the power (\(1500\ \mathrm{W}\)), \(V\) is the voltage (\(120\ \mathrm{V}\)), and \(R\) is the resistance. Rearranging the formula to solve for \(R\), we get: \(R = \frac{V^2}{P}\) Now, plug in the values and solve for \(R\): \(R = \frac{(120\ \mathrm{V})^2}{1500\ \mathrm{W}} = 9.60\ \Omega\)
02

(b) Calculate the current through the hair dryer

To calculate the current through the hair dryer, we can use Ohm's law: \(I = \frac{V}{R}\) Where \(I\) is the current, \(V\) is the voltage (\(120\ \mathrm{V}\)), and \(R\) is the resistance (calculated in part (a)). Now, plug in the values and solve for \(I\): \(I = \frac{120\ \mathrm{V}}{9.60\ \Omega} = 12.5\ \mathrm{A}\)
03

(c) Calculate the cost to run the hair dryer for 5.00 minutes

First, we need to determine how much energy the hair dryer uses in 5.00 minutes. We can use the formula: \(E = P \times t\) Where \(E\) is the energy, \(P\) is the power, and \(t\) is the time. Since the hair dryer uses \(1500\ \mathrm{W}\) of power, we need to convert it to kilowatts: \(P = 1.5\ \mathrm{kW}\) Also, convert the time to hours: \(t = \frac{5.00\ \mathrm{min}}{60\ \mathrm{min/h}} = 0.08333\ \mathrm{h}\) Now, plug in the values and solve for \(E\): \(E = 1.5\ \mathrm{kW} \times 0.08333\ \mathrm{h} = 0.125\ \mathrm{kWh}\) The cost of running the hair dryer for 5.00 minutes is: \(Cost = 0.125\ \mathrm{kWh} \times \$0.10\ \mathrm{per\ kWh} = \$0.0125\)
04

(d) Calculate the power the hair dryer would be using in Europe

To calculate the power the hair dryer would be using in the brief time before it gets ruined in Europe, we can use the formula: \(P = \frac{V^2}{R}\) Where \(P\) is the power, \(V\) is the voltage in Europe (\(240\ \mathrm{V}\)), and \(R\) is the resistance (calculated in part (a)). Now, plug in the values and solve for \(P\): \(P = \frac{(240\ \mathrm{V})^2}{9.60\ \Omega} = 6000\ \mathrm{W}\)
05

(e) Calculate the current flowing through the hair dryer in Europe

To calculate the current flowing through the hair dryer in Europe, we can use Ohm's law: \(I = \frac{V}{R}\) Where \(I\) is the current, \(V\) is the voltage in Europe (\(240\ \mathrm{V}\)), and \(R\) is the resistance (calculated in part (a)). Now, plug in the values and solve for \(I\): \(I = \frac{240\ \mathrm{V}}{9.60\ \Omega} = 25\ \mathrm{A}\)

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