Question 1

Determine the equation of the graph.
-A weight attached to a spring is pulled down 2 inches below the equilibrium position. Assuming that the period of the system is $\frac { 1 } { 3 }$ sec, what is the frequency of the system?
A) $\frac { 2 \pi } { 3 }$ cycles per sec
B) $\frac { 2 } { 3 \pi }$ cycles per sec
C) 3 cycles per sec
D) 6 cycles per sec

Accepted Answer

The frequency $f$ of a system is the reciprocal of the period $T$, so $f = \frac{1}{T}$. Given $T = \frac{1}{3}$ sec, the frequency $f = \frac{1}{\frac{1}{3}} = 3$ cycles per sec.

Question 2

Determine the equation of the graph.
-A weight attached to a spring is pulled down 8 inches below the equilibrium position. Assuming that the frequency of the system is $\frac { 5 } { \pi }$ cycles per second, determine a trigonometric model that gives the position of the weight at time $t$ seconds.
A) $s ( t ) = 8 \cos 10 t$
B) $s ( t ) = - 8 \cos 5 t$
C) $s ( t ) = 8 \cos 5 t$
D) $s ( t ) = - 8 \cos 10 t$

Accepted Answer

The answer of Determine the equation of the graph.
-A weight...

Question 3

Determine the equation of the graph.
-Suppose that a weight on a spring has an initial position of $s ( 0 ) = 6$ inches and a period of $\mathrm { P } = 2.5$ seconds. Find a function $\mathrm { s } ( \mathrm { t } ) = \mathrm { a } \cos ( 2 \pi \mathrm { Ft } )$ that models the displacement of the weight.
A) $s ( t ) = 12 \cos ( 2 \pi ( 2.5 ) t )$
B) $\mathrm { s } ( \mathrm { t } ) = 12 \cos ( 2 \pi ( 0.4 ) \mathrm { t } )$
C) $s ( t ) = 6 \cos ( 2 \pi ( 2.5 ) t )$
D) $s ( t ) = 6 \cos ( 2 \pi ( 0.4 ) t )$

Accepted Answer

The amplitude is 6, and the frequency $F = \frac{1}{P} = \frac{1}{2.5} = 0.4$, so the correct equation is $s(t) = 6 \cos(2 \pi (0.4) t)$.

Question 4

Determine the equation of the graph.
-A weight attached to a spring is pulled down 6 inches below the equilibrium position. Assuming that the period of the system is $\frac { 1 } { 2 }$ second, determine a trigonometric model that gives the position of the weight at time $t$ seconds.
A) $s ( t ) = 6 \cos 4 \pi t$
B) $s ( t ) = - 6 \cos 4 \pi t$
C) $\mathrm { s } ( \mathrm { t } ) = 6 \cos \frac { 1 } { 2 } \mathrm { t }$
D) $s ( t ) = - 6 \cos 2 \pi t$

Accepted Answer

The amplitude is 6, and since the weight is pulled down, the cosine function should be negative. The period is $\frac{1}{2}$ second, so the frequency is $4\pi$.

Question 5

Determine the equation of the graph.
-The position of a weight attached to a spring is $\mathrm { s } ( \mathrm { t } ) = - 5 \cos 8 \pi t$ inches after $\mathrm { t }$ seconds. What is the maximum height that the weight reaches above the equilibrium position and when does it first reach the maximum height? Round values to two decimal places, if necessary.
A) The maximum height of 5 inches is first reached after $0.13$ seconds.
B) The maximum height of 10 inches is first reached after 2 seconds.
C) The maximum height of 10 inches is first reached after 4 seconds.
D) The maximum height of 5 inches is first reached after 4 seconds.

Accepted Answer

The maximum height is determined by the amplitude of the cosine function, which is 5 inches. The first maximum occurs at $t = \frac{1}{16}$ seconds, which rounds to 0.06 seconds, not 0.13. However, given the options, A is the closest to the correct timing despite the rounding error.

Question 6

Determine the equation of the graph.
-A guitar string is plucked so that it vibrates with a frequency of $\mathrm { F } = 60$. Suppose the maximum displacement at the center of the string is $\mathrm { s } ( 0 ) = 0.54$. Find an equation of the form $\mathrm { s } ( \mathrm { t } ) = \mathrm { a } \cos \mathrm { bt }$ to model this displacement. Round constants to 2 decimal places.
A) $s ( t ) = 1.08 \cos 376.99 t$
B) $s ( t ) = 1.08 \cos 9.55 t$
C) $s ( t ) = 0.54 \cos 9.55 t$
D) $\mathrm { s } ( \mathrm { t } ) = 0.54 \cos 376.99 \mathrm { t }$

Accepted Answer

The answer of Determine the equation of the graph.
-A guitar...

Quiz 7: The Circular Functions and Their Graphs

Determine the equation of the graph. -A weight attached to a spring is pulled down 2 inches below the equilibrium position. Assuming that the period of the system is 13\frac { 1 } { 3 }31​ sec, what is the frequency of the system?

Determine the equation of the graph. -A weight attached to a spring is pulled down 6 inches below the equilibrium position. Assuming that the period of the system is 12\frac { 1 } { 2 }21​ second, determine a trigonometric model that gives the position of the weight at time ttt seconds.

Determine the equation of the graph. -A weight attached to a spring is pulled down 2 inches below the equilibrium position. Assuming that the period of the system is $\frac { 1 } { 3 }$ sec, what is the frequency of the system?

Determine the equation of the graph. -A weight attached to a spring is pulled down 6 inches below the equilibrium position. Assuming that the period of the system is $\frac { 1 } { 2 }$ second, determine a trigonometric model that gives the position of the weight at time $t$ seconds.