### Video Transcript

A person of mass 66 kilograms walked up an inclined plane to a height of 450 meters in five minutes. Determine their average power throughout the walk.

Letβs begin by sketching a diagram to model the scenario. We are told that a person walks up an inclined plane to a height of 450 meters in five minutes. We are told that they have a mass of 66 kilograms and need to calculate the average power throughout the walk. We know that we can calculate the power π by multiplying a force πΉ by a velocity π£. We will begin by calculating the average velocity π£ by dividing the total displacement by the total time.

As there are 60 seconds in a minute and five multiplied by 60 is 300, the total time is 300 seconds. We can calculate the total displacement π using our knowledge of right angle trigonometry. We will begin by letting the angle of inclination be πΌ. The sine ratio tells us that the sine of angle πΌ is equal to the opposite over the hypotenuse. In this question, sin πΌ is equal to 450 over π. And this means that the total displacement π is equal to 450 over sin πΌ. The average velocity is therefore equal to 450 over sin πΌ divided by 300. Dividing the numerator and denominator by 150, we have three over sin πΌ divided by two. And this simplifies to three over two sin πΌ. The average power throughout the walk is therefore equal to some force πΉ multiplied by three over two sin πΌ.

We will now use Newtonβs second law, πΉ equals ππ, to determine the value of the force πΉ. We know that at any point on their journey, the person will exert a downward force equal to their weight. This is equal to their mass of 66 kilograms multiplied by the acceleration due to gravity π. Once again, using our knowledge of right angle trigonometry, we see that the force acting down the plane will be equal to 66π multiplied by sin πΌ.

We are trying to calculate the force πΉ acting up the plane, and assuming that the person is moving at a constant velocity, we know that the acceleration π is equal to zero meters per second squared. Taking the positive direction to be the direction of travel, the sum of our forces is πΉ minus 66π multiplied by sin πΌ. This is equal to a mass of 66 kilograms multiplied by an acceleration of zero. The right-hand side of our equation is equal to zero. And adding 66π multiplied by sin πΌ to both sides, we have πΉ is equal to 66π multiplied by sin πΌ.

The average power π is therefore equal to 66π sin πΌ multiplied by three over two sin πΌ. We can divide the numerator and denominator by sin πΌ and also by two, giving us 33π multiplied by three. This is equal to 99π. And taking π to be 9.8 meters per second squared, we have a power π equal to 970.2 watts. Whilst we could leave our answer in these units, we could also give it in horsepower, recalling that one horsepower is equal to 735 watts. Dividing 970.2 by 735 gives us 1.32 horsepower. This is the average power throughout the walk.