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The time taken for a water heater to boil water is inversely proportional to the power used in watts. It takes 1000 watts to boil water in 4 minutes. Find the time it takes when using 400 watts.

Verified Answer :

  • Identify the inverse proportionality relationship \(T = \frac{k}{P}\).
  • Calculate the constant \(k\) using the given values: \(k = 4 \times 1000 = 4000\).
  • Apply the constant to find the time for the new power level: \(T = \frac{4000}{400}\).
  • The final time is \(\boxed{10 \text{ minutes}}\).

Explanation

  1. Analysis of the Proportional Relationship

We are given that the time \(T\) taken for a water heater to boil water is inversely proportional to the power \(P\) used. This relationship can be expressed by the equation \(T = \frac{k}{P}\), where \(k\) is a constant of proportionality. We are provided with the initial condition: when the power \(P_1 = 1000\) watts, the time \(T_1 = 4\) minutes.

  1. Finding the Constant of Proportionality

Using the initial condition \(P_1 = 1000\) and \(T_1 = 4\), we can determine the value of the constant \(k\):
\[k = T_1 \times P_1 = 4 \times 1000 = 4000.\]
Now, we have the complete relationship for this specific heater: \(T = \frac{4000}{P}\).

  1. Calculating the Required Time

To find the time \(T_2\) required to boil water when the power is \(P_2 = 400\) watts, we substitute this value into our established formula:
\[T_2 = \frac{4000}{P_2} = \frac{4000}{400} = 10.\]
Thus, it takes 10 minutes to boil water using 400 watts of power.

Examples

The concept of inverse proportionality is widely used in everyday life, such as when planning travel. For example, the time it takes to travel a fixed distance is inversely proportional to the speed at which you are traveling. If you double your speed, the time taken to reach your destination is halved. Understanding this relationship helps in scheduling, efficiency, and resource management in various practical tasks.

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