Question Video: Determining the Power of an Engine as a Function of Time Mathematics

The work done by an engine at time 𝑡 is given by the relation 𝑊(𝑡) = 14𝑒^(2𝑡) J. Find the power of the engine as a function of time.

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Video Transcript

The work done by an engine at time 𝑡 is given by the relation 𝑊 of 𝑡 is equal to 14 𝑒 to the power of two 𝑡 joules. Find the power of the engine as a function of time.

In this question, we are given an expression for the work done by an engine in terms of time 𝑡. It is equal to 14 multiplied by 𝑒 to the power of two 𝑡, where the work is given in standard units of joules. We know that the power supplied by a force is the time derivative of the work done. This means that we need to differentiate our expression to find an expression for the power in terms of time 𝑡.

We can do this by using the general rule for differentiating exponential functions. If 𝑦 is equal to 𝑎 multiplied by 𝑒 to the power of 𝑛𝑥, then d𝑦 by d𝑥 is equal to 𝑎𝑛 multiplied by 𝑒 to the power of 𝑛𝑥. Differentiating 14 multiplied by 𝑒 to the power of two 𝑡 gives us 14 multiplied by two multiplied by 𝑒 to the power of two 𝑡. This simplifies to 28𝑒 to the power of two 𝑡, and the standard units of power are watts.

The power of the engine as a function of time is 28𝑒 to the power of two 𝑡 watts.

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