A lawn sprinkler is constructed in such a way that d θ/ d t is constant, where θranges between 4

A lawn sprinkler is constructed in such a way that d θ/ d t...

A lawn sprinkler is constructed in such a way that $d \theta / d t$ is constant, where $\theta$ ranges between $45^{\circ}$ and $135^{\circ}$ (see figure). The distance the water travels horizontally is $x=\frac{v^{2} \sin 2 \theta}{32}, \quad 45^{\circ} \leq \theta \leq 135^{\circ}$ where $v$ is the speed of the water. Find $d x / d t$ and explain why this lawn sprinkler does not water evenly. What part of the lawn receives the most water? For more information on the "calculus of lawn sprinklers," see the article "Design of an Oscillating Sprinkler" by Bart Braden in Mathematics Magazine. To view this article, go to MathArticles.com.

A lawn sprinkler is constructed in such a way that $d \theta / d t$ is constant, where $\theta$ ranges between $45^{\circ}$ and $135^{\circ}$ (see figure). The distance the water travels horizontally is
$x=\frac{v^{2} \sin 2 \theta}{32}, \quad 45^{\circ} \leq \theta \leq 135^{\circ}$
where $v$ is the speed of the water. Find $d x / d t$ and explain why this lawn sprinkler does not water evenly. What part of the lawn receives the most water?
For more information on the "calculus of lawn sprinklers," see the article "Design of an Oscillating Sprinkler" by Bart Braden in Mathematics Magazine. To view this article, go to MathArticles.com.

Key Concepts

Projectile Motion

This concept involves the motion of an object thrown into the air, subject only to the acceleration due to gravity. The formula x = (v² sin2?)/32 arises from decomposing the initial velocity into horizontal and vertical components and finding the distance traveled under gravitational pull. Understanding projectile motion provides the physical basis behind the spray of water from the sprinkler.

Chain Rule in Differentiation

The chain rule is a fundamental differentiation technique used when differentiating a composite function. Since the horizontal distance x is expressed as a function of ?, which itself is a function of time t (with d?/dt constant), the chain rule allows us to compute dx/dt by differentiating x with respect to ? and then multiplying by d?/dt.

Nonuniform Distribution Due to Nonlinear Mapping

Even though the sprinkler rotates with constant angular speed, the horizontal displacement is determined by a nonlinear function (sin2?) of the angle. This nonlinearity causes the rate at which x changes with respect to ? (and hence time) to vary. As a result, in certain regions the water moves faster (leaving less time for deposition) and in others it moves slower, leading to an uneven distribution of water on the lawn.

Relation between Rate of Change and Water Distribution

The derivative dx/dt indicates how quickly the water moves horizontally at a given instant. Variations in dx/dt imply that the same angular sweep covers different horizontal distances at different times. Regions where dx/dt is smaller (i.e., where the rate of horizontal travel is slow) will receive a higher concentration of water because the water spends more time in those areas.

Transcript

00:01 So this question asks us to find out where the sprinklers watering the most and why this creek weiss watering the long unevenly.

00:09 So this is essentially word problem involving absolute extreme.

00:13 When you see that this is saying it's asking for where it's morning the most, we know it's looking for an extreme.

00:18 So in this case, we know we definitely have to take the derivative of the given function between the close interval of five degrees and 1 35 degrees.

00:27 So in this case, the v, or the speed of the water, is constant, so we can treat it as a constant when we're taking the dirt root of of ex.

00:36 So while we do this, we get me squared over 32 times, the derivative of the sign of tooth data.

00:45 So the directive of sine function is coincides.

00:48 We get co signing to athena times to do to train changeable times, the derivative of data.

00:57 Since that it's not a constant, so we have to consider that your view of it now in the question it said that this derivative d data d t is a constant but note that the function of data itself is not a constant, so we still have to include it so we can actually replace deep data e t.

01:23 With the constant c just to make it a little cleaner.

01:27 So i'm just going to group all the constants together.

01:30 When i'm writing this out, it's a dx dt equals see, which is dictated.

01:42 Dee tee times me squared over 16.

01:47 And this happens because the two there's a two under numerator and 3200 denominators, so that reduces the fraction a tiny bit.

01:58 And then we multiply this by co sign to data.

02:03 Now it wants to know where it's watering the most, so we have to find a absolute extreme.

02:08 So to do that, we must set this equal to zero on before we get into this at all.

02:14 So that's why it's watering evenly.

02:16 We can see from this function.

02:18 Dx over dt equals c v squared over 16 times kristen to data that this is not a constant function...

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