Math 141
Name: __________________________________
Homework 7
1. The human ear is sensitive to an amazing range of sound levels. For example, the sound emitted by
a jackhammer is equivalent in magnitude to about "!!,!!! human voices, while a single human voice is
approximately as loud as "!!,!!! buzzing mosquitoes. In terms of power, a jackhammer emits about
1 watt of sonic energy, a human voice emits about !Þ!!!!" watts, and a buzzing mosquito emits about
!Þ!!!!!!!!!" (or "!"! ) watts.
For this reason, the “loudness” of a sound is usually described in terms of decibels, which are a
logarithmic unit. The decibel level P of a sound is related to the power T according to the formula
P œ "! logŒ
T

T!
where T! œ "!"# watts is a constant representing the softest sound audible to human ears.
(a) Based on the information above, find the decibel levels of the sounds emitted by a jackhammer, a
human voice, and a buzzing mosquito. What is the decibel level of a sound having power T! ?
(b) Make a table showing the decibel levels for sounds having powers of !Þ!!!!" watts, !Þ!!!" watts,
!Þ!!" watts, !Þ!" watts, !Þ" watts, " watt, "! watts, and "!! watts.
(c) If the power of a sound is doubled (say, by having two jackhammers instead of one), how much does
the decibel level increase?
(d) Find a formula for the power T as a function of the decibel level P. Use your formula to find the
power of a "#&-decibel sound.
(e) The band Disaster Area wants to make an awesome "'! decibels of sound at their next rock concert.
(They want to be able to brag that they are “as loud as a jet engine.”) Unfortunately, no legally
available speaker can emit more than "%& decibels of sound. How many speakers will they need?
2. A man with a video camera is standing 4000 ft. from the base of a rocket launching pad, recording
the launch:
Because the rocket is ascending, the man must constantly adjust the angle of the camera to match the
angle of elevation of the rocket.
(a) Find a formula relating the height 2 of the rocket and the angle ) to which the camera must be set.
(b) Use your answer to part (a) to find a formula relating
.2
.)
and
.
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(c) Suppose that the rocket is ascending at a rate of 600 ftÎsec. How quickly is the angle ) increasing at
the instant that ) œ !Þ& radians?
3. Little Joey is riding the Ferris wheel at the county fair:
The wheel has a radius of %! feet, and the center of the wheel sits %& feet above the ground. Let 2
represent Joey's height above the ground, and let ) be measure of the angle shown in the above picture.
(a) Determine the values of 2 for ) œ !°, ) œ *!°, ) œ ")!°, and ) œ #(!°.
(b) Sketch a graph of 2 as a function of ).
(c) Find a formula for 2 as a function of ). Does your formula agree with your answers to parts (a)
and (b)?
(d) Use your answer to part (c) to find a formula relating
.2
.)
and
.
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Now suppose that the Ferris wheel is rotating counterclockwise, with one full rotation every 20 seconds.
(e) What is
.)
? Your answer should be expressed in radians per second.
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(f) Use your answers to parts (d) and (e) to find a formula for
Little Joey ascending when ) œ !?
(g) Sketch a graph of
.2
as a function of ).
.>
.2
as a function of ). How quickly is
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