related rates vid
TRANSCRIPT
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!