4.3 Related Ratesap Calculus

AP Calculus LHopitals Rule and Related rates. AP Calculus Learning Objectives Explored in this Section. Calculate derivatives; Interpret the meaning of a derivative. AP Calculus AB – “Related Rates” Solving Related Rates Problems 1. Name the variables and constants. Determine what you are asked to find. Write an equation(s) that relates the variables. Implicitly differentiate with respect to time. Evaluate and solve for the unknown. PCHS AP CALCULUS. Home Distance Learning Info Assignments & Videos Mr. Zimora's Corner. Related Rates WS 3: File Size: 153 kb: File Type: pdf: Download File. Setting up Related-Rates Problems. In many real-world applications, related quantities are changing with respect to time. For example, if we consider the balloon example again, we can say that the rate of change in the volume, (V ), is related to the rate of change in the radius, (r ).

Calculus Related Rates Problem:
At what rate does the angle change as a ladder slides away from a house?

Calculus

A 10-ft ladder leans against a house on flat ground. The house is to the left of the ladder. The base of the ladder starts to slide away from the house at 2 ft/s. At what rate is the angle between the ladder and the ground changing when the base is 8 ft from the house?

Calculus Solution

To solve this problem, we will use our standard 4-step Related Rates Problem Solving Strategy.

1. Draw a picture of the physical situation.
See the figure. We’ve labeled the angle $theta$ that the ladder makes with the ground, since the problem is asking us to find the rate at which that angle changes, $dfrac{dtheta}{dt}$, at a particular moment — when $x = 8$. Recall also that $dfrac{dx}{dt} = 2$ ft/s. We’ll use these values at the end of our solution.



2. Write an equation that relates the quantities of interest.
B. To develop your equation, you will probably use . . . a trigonometric function (like $cos{theta}$ = adjacent/hypotenuse).
This is the hardest part of Related Rates problem for most students initially: you have to know how to develop the equation you need, how to pull that “out of thin air.” By working through these problems you’ll develop this skill. The key is to recognize which of the few sub-types of problem it is; we’ve listed each on our Related Rates page.

In this problem, the diagram above immediately suggests that we’re dealing with a right triangle. Furthermore, we need to related the rate at which $theta$ is changing, $dfrac{dtheta}{dt}$, to the rate at which x is changing, $dfrac{dx}{dt}$, and so we first need to write down an equation that somehow relates $theta$ and x. Such a relation must be trigonometric.

4.3 Related Ratesap Calculus

Specifically, we notice that x is the side of the triangle that is adjacent to the angle. Furthermore, the hypotenuse of the triangle remains constant throughout the problem, since the ladder’s length is always 10 ft. Hence at every moment:
$$cos{theta} = frac{x}{10}$$
That’s it. That’s the key relationship that will allow us to complete the solution.

3. Take the derivative with respect to time of both sides of your equation. Remember the chain rule.


begin{align*}
frac{d}{dt} cos{theta} &= frac{d}{dt} left( frac{x}{10} right)
&= frac{1}{10} frac{d}{dt}(x)
-sin{theta}, frac{dtheta}{dt} &= frac{1}{10}frac{dx}{dt}
end{align*}

4.3 Related Ratesap Calculus
Open to read why the d(theta)/dt and dx/dt are there.
Are you wondering why the $dfrac{dtheta}{dt}$ and $dfrac{dx}{dt}$ appear? The answer is the Chain Rule.

While the derivative of $cos theta$ with respect to angle $theta$ is
$$dfrac{d}{dtheta}cos theta = -sin theta,$$
the derivative of $cos theta$ with respect to time t is
$$dfrac{d}{dt}cos theta = -sin theta ,dfrac{dtheta}{dt}.$$
Similarly, while the derivative of x with respect to x is
$$dfrac{d}{dx}x = 1,$$
the derivative of x with respect to time t is
$$dfrac{d}{dt}x = dfrac{dx}{dt}.$$
(Recall that that rate is $dfrac{dx}{dt} = 2$ ft/s in this problem.)

Remember that $theta$ and x are both functions of time t: the angle changes as time passes and the ladder’s x-position changes as the ladder slides down the wall. We could have captured this time-dependence explicitly by writing our relation as
$$ cos theta(t) = dfrac{x(t)}{10}$$
to remind ourselves that both $theta$ and x are functions of time t. Then when we take the derivative,
begin{align*}
frac{d}{dt}cos theta(t) &= frac{d}{dt}left( dfrac{x(t)}{10}right)
left(-sin theta(t)right) dfrac{dtheta(t)}{dt}&= dfrac{1}{10}dfrac{dx(t)}{dt}
end{align*}

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[Recall $dfrac{dx(t)}{dt} = 2$ ft/s, and we’re looking for $dfrac{dtheta(t)}{dt}$ at the moment when x = 8 ft.]

Most people find that writing the explicit time-dependence $theta(t)$ and x(t) annoying, and so just write $theta$ and x instead. Regardless, you must remember that both $theta$ and x depend on t, and so when you take the derivative with respect to time the Chain Rule applies and you have the $dfrac{dtheta}{dt}$ and $dfrac{dx}{dt}$ terms.

4. Solve for the quantity you’re after.

Let’s solve the preceding equation for $dfrac{dtheta}{dt}$:
begin{align*}
-sin{theta}, frac{dtheta}{dt} &= frac{1}{10}frac{dx}{dt}
frac{dtheta}{dt} &= -frac{1}{sin{theta}}frac{1}{10}frac{dx}{dt}
end{align*}
To complete the solution, we need to know the value of $sin theta$ at the moment when x = 8 ft.

Begin subproblem to find $sin theta$ at the moment of interest.
You can use any of these three approaches:
Approach #1:
Looking back at the figure, we see that
$$ sin theta = dfrac{y}{10}$$
Next, recognize that at this instant the triangle is a “3-4-5 right triangle,” with the actual proportions 6-8-10. Hence y = 6 ft at this instant, and so
$$sintheta = dfrac{y}{10} = dfrac{6}{10} = dfrac{3}{5}$$
Approach #2:
Looking back at the original figure, we see that
$$ sin theta = dfrac{y}{10}$$
So we need to know the value of y when x = 8 ft. The Pythagorean theorem as applied to the triangle lets us solve for y at this instant:
begin{align*}
x^2 + y^2 &= 10^2 [4px]8^2 + y^2 &= 10^2 [4px]64 + y^2 &= 100 [4px]y^2 &= 100 – 64 = 36 [4px]y &= 6
end{align*}
Hence at this instant
$$sintheta = dfrac{y}{10} = dfrac{6}{10} = dfrac{3}{5}$$
Approach #3.
Recall the trig identity $sin^2 theta + cos^2 theta = 1$:
begin{align*}
sin^2 theta + cos^2 theta &= 1 [4px]sin^2 theta + left(frac{8}{10} right)^2 &= 1 [4px]sin^2 theta + left(frac{4}{5} right)^2 &= 1 [4px]sin^2 theta + frac{16}{25} &= 1 [4px]sin^2 theta &= 1 – frac{16}{25} = frac{9}{25} [4px]sin theta = frac{3}{5}
end{align*}
End subproblem.



We can now substitute values into our preceding equation:
$$frac{dtheta}{dt} = -frac{1}{sin{theta}}frac{1}{10}frac{dx}{dt}$$
We have $sin theta = dfrac{3}{5}$ and $dfrac{dx}{dt} =2$:
begin{align*}
frac{dtheta}{dt} & = -frac{1}{sin{theta}}frac{1}{10}frac{dx}{dt}
&= , -frac{1}{3/5}frac{1}{10}(2)
&=, – frac{5}{3}frac{1}{10}(2)
&= , – frac{1}{3} text{ rad/s} quad cmark
end{align*}
That’s the answer. The negative value indicates that the angle is decreasing at the ladder slides down the wall, as we expect.


Caution: IF you are using a web-based homework system and the question asks,

At what rate does the angle decrease?

then the system has already accounted for the negative sign and so to be correct you must enter a POSITIVE VALUE: $boxed{dfrac{1}{3}} , dfrac{text{rad}}{text{s}} quad checkmark$

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