SOLUTION: For f(x)=1/x^2-2x-8, find the interval(s) where f(x)<0. Express the following statement as a formula with the value of the constant of proportionality determined with the given

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Question 68140: For f(x)=1/x^2-2x-8, find the interval(s) where f(x)<0.
Express the following statement as a formula with the value of the constant of proportionality determined with the given conditions: w varies directly as x and inversely as the square of y. If x=15 and y=5, then w=36.
The electrical resistance R of a wire varies directly as its length L and inversely as the square of its diameter. A wire 20 meters long and 0.6 centimeters in diameter made from a certain alloy has a resistance of 36 ohms. What is the resistance of a piece of wire 60 meters long and 1.2 centimeters in diameter made from the same material?
The period of a simple pendulum is directly proportional to the square root of its length. If a pendulum has a length of 6 feet and a period of 2 seconds, to what length should it be shortened to achieve a 1 swcond period?
Thanks for asssiting me in studying for my final exam.
Found 2 solutions by funmath, stanbon:
Answer by funmath(2933) About Me  (Show Source):
You can put this solution on YOUR website!
For f(x)=1/x^2-2x-8, find the interval(s) where f(x)<0.
f%28x%29=1%2F%28x%5E2-2x-8%29
f%28x%29=1%2F%28%28x-4%29%28x%2B2%29%29
x-4=0 when x=4 and x+2=0 when x=-2
So your possible intervals are (-infinity,-2), (-2,4), or (4, infinity)
for (-infinity,-2) test -3
f%28-3%29=1%2F%28%28-3%29%5E2-2%28-3%29-8%29
f%28-3%29=1%2F%289%2B6-8%29
f%28-3%29=1%2F7 1/7 is not <0 so reject this interval.
for (-2,4) test 0
f%280%29=1%2F%280%5E2-2%280%29-8%29
f%280%29=-1%2F8 -1/8 is <0 so accept this interval.
for (4,infinity) test 5
f%285%29=1%2F%285%5E2-2%285%29-8%29
f%285%29=1%2F%2825-10-8%29
f%285%29=1%2F7 1/7 is not <0 so reject this interval.
Therefore the interval where f(x)<0 is (-2,4)
:
Express the following statement as a formula with the value of the constant of proportionality determined with the given conditions: w varies directly as x and inversely as the square of y. If x=15 and y=5, then w=36.
w=kx%2Fy%5E2
36=k%2815%29%2F%285%29%5E2
36=15k%2F25
36=3k%2F5
%285%2F3%29%2836%29=%285%2F3%29%283k%2F5%29
60=k
Therefore, the formula is: highlight%28w=60x%2Fy%5E2%29
:
The electrical resistance R of a wire varies directly as its length L and inversely as the square of its diameter. A wire 20 meters long and 0.6 centimeters in diameter made from a certain alloy has a resistance of 36 ohms. What is the resistance of a piece of wire 60 meters long and 1.2 centimeters in diameter made from the same material?
R=kL%2Fd%5E2
Convert 20 m to cm's: 20m%2A100cm%2Fm=2000cm
36=k%282000cm%29%2F%28.6cm%29
36%2A.6%2F2000=2000k%2F.6%2A%28.6%2F2000%29
.0108=k
R=.0108L%2Fd%5E2
60m=6000cm
R=.0108%286000cm%29%2F%281.2cm%29%5E2
highlight%28R=45%29 ohms
:
The period of a simple pendulum is directly proportional to the square root of its length. If a pendulum has a length of 6 feet and a period of 2 seconds, to what length should it be shortened to achieve a 1 swcond period?
:
Period=T Length=L
T=k%2Asqrt%28L%29
2=k%2Asqrt%286%29
2%2Fsqrt%286%29=k%2Asqrt%286%29%2Fsqrt%286%29
2%2Asqrt%286%29%2F%28sqrt%286%29%2Asqrt%286%29%29=k
2%2Asqrt%286%29%2F6=k
sqrt%286%29%2F3=k
T=sqrt%286L%29%2F3
1=sqrt%286L%29%2F3
3%281%29=3sqrt%286L%29%2F3
3=sqrt%286L%29
3%5E2=%28sqrt%286L%29%29%5E2
9=6L
9%2F6=6L%2F6
3%2F2=L
L=1.5 ft
Good Luck on Your Final!
Happy Calculating!!!!

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
For f(x)=1/x^2-2x-8, find the interval(s) where f(x)<0
The fraction is negative when the denominator is negative.
The denominator is negative when (x-4)(x+2)<0
Draw a number line and mark -2 and +4
Check a test value in each interval to see where the solution set is:
If x=-100 (x-4)(x+2)is positive so no solutions in (-inf,-2)
If x=0, (x-4)(x+2) is negative so (-2,4) is part of the solution set.
If x=100, (x-4)(x+2) is positive so no solutions in (4,inf(
Final: f(x)is negative when x in in the interval (-2,4)
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Express the following statement as a formula with the value of the constant of proportionality determined with the given conditions: w varies directly as x and inversely as the square of y. If x=15 and y=5, then w=36.
w=kx/y^2 ; 36=k*15/25 = k=60
Therefore w=(60x/y^2)
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The electrical resistance R of a wire varies directly as its length L and inversely as the square of its diameter. A wire 20 meters long and 0.6 centimeters in diameter made from a certain alloy has a resistance of 36 ohms. What is the resistance of a piece of wire 60 meters long and 1.2 centimeters in diameter made from the same material?
R=kL/d^2 ; 36 = k*20 m/0.6^2 cm^2; 36=20k/0.36; k=0.648
EQUATION: R =0.648L/d^2
If L=60 and d=1.2, R=0.648*60/(1.2)^2 ; R= 27
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The period of a simple pendulum is directly proportional to the square root of its length. If a pendulum has a length of 6 feet and a period of 2 seconds, to what length should it be shortened to achieve a 1 swcond period?
Period =k sqrtL
2=ksqrt6
k=2/sqrt6
k=2sqrt6/6 =(1/3)sqrt6
EQUATIOn:
Period = [(1/3)sqrt6]sqrtL
If Period = 1, 1=[(1/3)sqrt6]sqrtL ; 3/sqrt6 = sqrtL; 9/6=L; L=3/2 ft.
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Cheers,
Stan H.