Question 68140
For f(x)=1/x^2-2x-8, find the interval(s) where f(x)<0.
{{{f(x)=1/(x^2-2x-8)}}}
{{{f(x)=1/((x-4)(x+2))}}}
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(-3)=1/((-3)^2-2(-3)-8)}}}
{{{f(-3)=1/(9+6-8)}}}
{{{f(-3)=1/7}}} 1/7 is not <0 so reject this interval.
for (-2,4) test 0
{{{f(0)=1/(0^2-2(0)-8)}}}
{{{f(0)=-1/8}}} -1/8 is <0 so accept this interval.
for (4,infinity) test 5
{{{f(5)=1/(5^2-2(5)-8)}}}
{{{f(5)=1/(25-10-8)}}}
{{{f(5)=1/7}}} 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/y^2}}}
{{{36=k(15)/(5)^2}}}
{{{36=15k/25}}}
{{{36=3k/5}}}
{{{(5/3)(36)=(5/3)(3k/5)}}}
{{{60=k}}}
Therefore, the formula is: {{{highlight(w=60x/y^2)}}}
:
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}}}
Convert 20 m to cm's: {{{20m*100cm/m=2000cm}}}
{{{36=k(2000cm)/(.6cm)}}}
{{{36*.6/2000=2000k/.6*(.6/2000)}}}
{{{.0108=k}}}
{{{R=.0108L/d^2}}}
60m=6000cm
{{{R=.0108(6000cm)/(1.2cm)^2}}}
{{{highlight(R=45)}}} 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*sqrt(L)}}}
{{{2=k*sqrt(6)}}}
{{{2/sqrt(6)=k*sqrt(6)/sqrt(6)}}}
{{{2*sqrt(6)/(sqrt(6)*sqrt(6))=k}}}
{{{2*sqrt(6)/6=k}}}
{{{sqrt(6)/3=k}}}
{{{T=sqrt(6L)/3}}}
{{{1=sqrt(6L)/3}}}
{{{3(1)=3sqrt(6L)/3}}}
{{{3=sqrt(6L)}}}
{{{3^2=(sqrt(6L))^2}}}
{{{9=6L}}}
{{{9/6=6L/6}}}
{{{3/2=L}}}
L=1.5 ft
Good Luck on Your Final!
Happy Calculating!!!!