THE CALCULUS CRUSADERSRelated Rates: Crusaders Unite Question

THE SITUATION

Zeph, Bench and Jamie find a clue that brings them closer to the block of wood. Zeph and Bench meet up at a park, but Jamie is missing in action.

After she calls Bench on his phone, they agree to meet at her house.

THE QUESTION

Bench and Zeph are at the park (point P), j = 40 metres north of Jamie’s house (point H). Together they walk at a speed of dj/dt = 12 metres per minute south. Jamie is standing p metres from her house, where p, with respect to time t, is defined as:

What is the rate of change of the angle at j?

THE DIAGRAM

THE SOLUTION

To evaluate p(t), we need to complete the square.

THE SOLUTION

To evaluate p(t), we need to complete the square.

We can substitute t + 2 as u by letting t + 2 = u, which means that du = dt.

THE SOLUTION

To evaluate p(t), we need to complete the square.

We can substitute t + 2 as u by letting t + 2 = u, which means that du = dt.

Similarly, we can let v = u/2, which means 2dv = du.

THE SOLUTION

Remember from our list of derivatives that

THE SOLUTION

Remember from our list of derivatives that

In order to make the equation fit the pattern, like stated previously, we must complete the square...

THE SOLUTION

Recall that we made substitutions for values, previously.

THE SOLUTION

Recall that we made substitutions for values, previously.

Also notice that the limits change accordingly as a result of the substitution.

THE SOLUTION

We now know how far Jamie is standing from her house, which is the length of the adjacent side in the diagram.

THE SOLUTION

Now it’s just as simple as using trigonometric ratios and the values of j and p to solve for θ.

THE SOLUTION

We can avoid using the Quotient Rule (differentiating by Quotient Rule is monstrous)

THE SOLUTION

We can avoid using the Quotient Rule (differentiating by Quotient Rule is monstrous)

Instead, we can differentiate implicitly, both side with respect to t.

THE SOLUTION

We then result with this:

THE SOLUTION

We then result with this:

We can then substitute dj/dt = -12 and θ = 1.568328887 into the equation, and solve for dθ/dt.

No more confusion! Zeph, Bench and Jamie have gathered and organized themselves and are clear to continue with the mission!

HOME AT LAST!