Chapter 4.1, Problem 72E

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Lawn Sprinkler A lawn sprinkler is constructed in such a way that d θ / d t is constant, where θ ranges between 45 ∘ and1 135 ∘ (see figure). The distance die water travels horizontally is x = v 2 sin 2 θ 32 , 45 ∘ ≤ θ ≤ 135 ∘ 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?Water sprinkler: 45 ∘ ≤ θ ≤ 135 ∘

To determine

To calculate: The value of dxdt and explain why the lawn sprinkler does not spread water

evenly and what part of lawn receives the most water.

Explanation

Given:

The distance water travels horizontally,

x=v2sin2Î¸32 where 450â‰¤Î¸â‰¤1350

dÎ¸dt is constant.

And,

The figure,

Refer to figure in question.

Formula used:

Aâ‹…vâ‹…Î”T=Â MaximumÂ amountÂ ofÂ waterÂ inÂ angle..

For maximum amount of water received: |Aâ‹…vâ‹…dtdx|=|Aâ‹…vdxdt| should be largest.

Calculation:

Consider the equation,

x=v2sin2Î¸32

Differentiate the above equation with respect to t, to get,

dxdt=dxdÎ¸dÎ¸dt=v2cos2Î¸16dÎ¸dt

Lawn sprinkler does not spread the water evenly because the ratio of change of speed is not constant.

If dÎ¸dt is constant then then water will be distributed uniformly over the arc

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started