My Math Forum Some advanced precalculus questions

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 April 22nd, 2011, 09:28 PM #1 Newbie   Joined: Apr 2011 Posts: 1 Thanks: 0 Some advanced precalculus questions I got a bunch of homework. But these are the hardest. Others I could have some clues to solve, but for these I'm clueless. Please help 1) If f(x)=3sin x+4sin(x-3) what is the amplitude and period (p) of f(x)? A) a= 1.114, p = 6.283 B) a= 1.114, p = 3.141 C) a =2.28, p= 3.141 D) a=7, p = 6.283 E) a= 7, p= 9.425 2)A Ferris wheel rider's height above the ground is given by the function h(t)= 20+18 cos [ (pi/4)* (t-3)], where h is measured in feet and t in seconds. How much time is required for the Ferris wheel to complete 1 full revolution? A) 3 secs B) 8 secs C) 18 secs D) 20 secs E) 38 secs 3) The graph of the daily temperature at noon of a city is modeled by T= -25 cos [ (2pi/365) * (t-30)] +70, where T represents the temperature in degrees Fareinheit and t represents the day of 365- day year. Whats the range of the temperature from low to high during the year? A) 25-> 70 Celsius B) 30-95 Celsius c) 45-95 Celsius D) 30-90 Celsius E) 35-95 Celsius 4) If f(x)= sin x* sin (-x), then which is the range of f(x)? A) [0,1] B) [-1,0] C) [-1,1] D) [-1,0) E) (0,1] 5) Function D (t) = (4t)/ (0.01t square+5.1). Given the concentration, D, of a drug in bloodstream where D is measured in micrograms per milliliter and t is minutes from the time the drug is taken. What's the maximum concentration of the drug found in bloodstream at any time? A) 0.01 micrograms/ milliliters B) 4 micrograms/ milliliters C) 5.1 micrograms/ milliliters D) 8.856 micrograms/ milliliters E) 22.583 micrograms/ milliliters
 April 23rd, 2011, 04:36 AM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Some advanced precalculus questions 1.) We may use the identity $a\sin(x)+b\sin(x+\alpha)=\sqrt{a^2+b^2+2ab\cos(\al pha)}\sin$$x+\beta$$$ Thus, amplitude is $\sqrt{3^2+4^2+2\cdot3\cdot4\cos(-3)}\approx1.114$ Period is $\frac{2\pi}{1}\approx6.283$ 2.) The period is $\frac{2\pi}{\frac{\pi}{4}}=8$ 3.) The range in degrees Fahrenheit is $70\pm25=[45,95]$ 4.) $f(x)=\sin(x)\sin(-x)=-\sin^2(x)$ range is [-1,0] 5.) Graphing shows D is the only answer which makes sense. I would use a technique from calculus to find the maximum is $20\sqrt{\frac{10}{51}}\approx8.856$.
April 23rd, 2011, 04:52 AM   #3
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
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Hello, belly!

Quote:
 $\text{2) A Ferris wheel rider's height above the ground is given by: }\:h(t)\:=\: 20\,+\,18\cos\left[\frac{\pi}{4}(t-3)\right]$ [color=beige]. . [/color]$\text{where }h\text{ is measured in feet and }t\text{ in seconds.}$ $\text{How much time is required for the Ferris wheel to complete one full revolution?}$ [color=beige]. . [/color]$\text{(A) 3 secs} \;\;\;\;\text{(B) 8 secs} \;\;\;\;\text{(C) 18 secs} \;\;\;\;\text{(D) 20 secs} \;\;\;\;\text{(E) 38 secs}$

$\text{When is the rider at the starting height (20 feet)?}$

[color=beige]. . [/color]$20\,+\,18\cos\left[\frac{\pi}{4}(t-3)\right] \;=\;20\;\;\;\Rightarrow\;\;\;18\cos\left[\frac{\pi}{4}(t-3}\right] \:=\:0$

[color=beige]. . [/color]$\cos\left[\frac{\pi}{4}(t-3)\right] \:=\:0 \;\;\;\Rightarrow\;\;\;\frac{\pi}{4}(t\,-\,3) \:=\:\frac{\pi}{2}\,+\,2\pi n\;\text{ for some integer }n$

$\text{Multiply by }\frac{4}{\pi}:\;\;t\,-\,3 \;=\;\frac{4}{\pi}\left(\frac{\pi}{2}\,+\,2\pi n\right)\;\;\;\Rightarrow\;\;\; t\,-\,3 \;=\;2\,+\,8n$

$\text{Hence: }\;t \;=\;5\,+\,8n$

$\text{The first time that }h=20\text{ is when }n = 0 \;\;\;\Rightarrow\;\;\; t = 5\text{ sec.}$

$\text{The next time that }h=20\text{ is when }n = 1\;\;\;\Rightarrow\;\;\; t = 13\text{ sec.}$

$\text{Therefore, one revolution requires }8\text{ seconds . . . answer (B)}$

MarkFL "eyeballed" the problem! . . . Great1
[color=beige] .[/color]

April 23rd, 2011, 05:31 AM   #4
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
Thanks: 408

Hello again, belly!

Quote:
 $\text{5) Function: }\:D(t) \:=\:\frac{4t}{0.01t^2\,+\,5.1}$ $\text{Given the concentration, }D\text{, of a drug in bloodstream where }D\text{ is measured in }\mu g/ml$ [color=beige]. . [/color]$\text{ and }t\text{ in minutes from the time the drug is taken.}$ $\text{what is the maximum concentration of the drug found in bloodstream at any time?}$ [color=beige]. . [/color]$(A)\; 0.01\:\mu g/ml \;\;\;\;(B)\;4\:\mu g/ml \;\;\;\;(C)\; 5.1\:\mu g/ml \;\;\;\; (D)\;8.856\:\mu g/ml \;\;\;\; (E)\;22.583\:\mu g/ml$

Since this is [color=blue]pre[/color]-Calculus, MarkFL has the best approach . . . graphing.

If we are allowed to use Calculus, we can directly determine the maximum concentration.

$D'(t) \:=\:0 \;\;\;\Rightarrow\;\;\;\frac{(0.01t^2+5.1)\,\cdot\ ,4 \,-\,4t(0.02t)}{(0.01t^2\,+\,5.1)^2}\:=\:0 \;\;\;\Rightarrow\;\;\;\frac{20.4\,-\,0.04t^2}{(0.01t^2\,+\,5.1)^2} \:=\:0$

$\text{Hence: }\;20.4\,-\,0.04t^2\:=\:0 \;\;\;\Rightarrow\;\;\;t^2\:=\:\frac{20.4}{0.04} \:=\:510 \;\;\;\Rightarrow\;\;\;t \:=\:\sqrt{510}$

$\text{And }D(\sqrt{510})\text{ yields MarkFL's answer.}$

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