Question 1

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

Accepted Answer

The frequency $F$ is the reciprocal of the period $P$, so $F = \frac{1}{P} = \frac{1}{2.5} = 0.4$ Hz. The initial position $s(0) = 6$ inches is the amplitude of the cosine function, leading to $s(t) = 6 \cos(2 \pi (0.4) t)$.

Question 2

Solve the problem.
-A weight attached to a spring is pulled down 3 inches below the equilibrium position. Assuming that the frequency of the system is $\frac { 6 } { \pi }$ cycles per second, determine a trigonometric model that gives the position of the weight at time $t$ seconds.
A) $s ( t ) = - 3 \cos 12 t$
B) $s ( t ) = 3 \cos 12 t$
C) $s ( t ) = 3 \pi \cos 6 t$
D) $s ( t ) = - 3 \cos \frac { 6 } { \pi } t$

Accepted Answer

The correct model is $s(t) = -3 \cos(12t)$ because the amplitude is 3 (the initial displacement from equilibrium), and the angular frequency $\omega = 2\pi f$, where $f = \frac{6}{\pi}$ cycles per second, so $\omega = 2\pi \cdot \frac{6}{\pi} = 12$. The negative sign indicates the initial displacement direction (downward).

Question 3

Solve the problem.
-The position of a weight attached to a spring is $s ( t ) = - 3 \cos 7 \pi t$ inches after $t$ seconds. When does the weight first reach its maximum height?
A) After $\frac { \pi } { 7 } \mathrm { sec }$
B) After $\frac { 1 } { 7 } \mathrm { sec }$
C) After $\frac { 7 } { \pi } \mathrm { sec }$
D) After $\frac { 7 } { 4 } \mathrm { sec }$

Accepted Answer

The maximum height is reached when $\cos 7 \pi t = -1$. This occurs at $t = \frac{1}{7}$ seconds.

Question 4

Solve the problem.
-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) $s ( t ) = - 6 \cos 2 \pi t$
B) $s ( t ) = 6 \cos \frac { 1 } { 2 } t$
C) $s ( t ) = 6 \cos 4 \pi t$
D) $s ( t ) = - 6 \cos 4 \pi t$

Accepted Answer

The correct model is $s(t) = -6 \cos 4\pi t$. The amplitude is 6 inches (the initial displacement), and since the period $T = \frac{1}{2}$ second, the frequency $f = \frac{1}{T} = 2$ Hz. The angular frequency $\omega = 2\pi f = 4\pi$ radians per second. The negative sign indicates the initial displacement is in the opposite direction of the positive direction assumed for $s(t)$.

Question 5

Solve the problem.
-A weight attached to a spring is pulled down 4 inches below the equilibrium position. Assuming that the period of the system is $\frac { 1 } { 3 } \mathrm { sec }$, what is the frequency of the system?
A) $\frac { 2 \pi } { 3 }$ cycles per sec
B) 3 cycles per sec
C) $\frac { 2 } { 3 \pi }$ cycles per sec
D) 6 cycles per sec

Accepted Answer

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

Question 6

Solve the problem.
-The position of a weight attached to a spring is $s ( t ) = - 8 \cos 8 \pi t$ inches after $t$ 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 16 inches is first reached after 2 seconds.
B) The maximum height of 8 inches is first reached after $0.13$ seconds.
C) The maximum height of 16 inches is first reached after 4 seconds.
D) The maximum height of 8 inches is first reached after 4 seconds.

Accepted Answer

The maximum height is determined by the amplitude of the cosine function, which is 8 inches. The first maximum occurs at $t = \frac{1}{16}$ seconds, rounding to $0.13$ seconds.

Question 7

Solve the problem.
-A weight attached to a spring is pulled down 9 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) $s ( t ) = 9 \cos 5 t$
B) $s ( t ) = - 9 \cos 10 t$
C) $s ( t ) = - 9 \cos 5 t$
D) $s ( t ) = 9 \cos 10 t$

Accepted Answer

The answer of Solve the problem.
-A weight attached to a...

Question 8

Solve the problem.
-A guitar string is plucked so that it vibrates with a frequency of $F = 60$. Suppose the maximum displacement at the center of the string is $s ( 0 ) = 0.53$. Find an equation of the form $s ( t ) = a \cos b t$ to model this displacement. Round constants to 2 decimal places.
A) $\mathrm { s } ( \mathrm { t } ) = 1.06 \cos 376.99 \mathrm { t }$
B) $\mathrm { s } ( \mathrm { t } ) = 0.53 \cos 376.99 \mathrm { t }$
C) $s ( t ) = 0.53 \cos 9.55 \mathrm { t }$
D) $s ( t ) = 1.06 \cos 9.55 t$

Accepted Answer

The equation for the displacement of a vibrating string can be modeled as $s(t) = a \cos(b t)$, where $a$ is the amplitude and $b$ is the angular frequency. The amplitude $a$ is given as the maximum displacement, which is 0.53. The angular frequency $b$ can be calculated from the frequency $F$ using the formula $b = 2\pi F$. Given $F = 60$, we find $b = 2\pi \times 60 = 376.99$. Therefore, the correct model is $s(t) = 0.53 \cos(376.99 t)$.

Question 9

Solve the problem.
-The position of a weight attached to a spring is $s ( t ) = - 2 \cos 3$ t inches after $t$ seconds. What are the frequency and period of the system?
A) Frequency $= \frac { \pi } { 3 }$ cycles per sec, period $= \frac { 3 } { \pi } \sec$
B) Frequency $= 3$ cycles per sec, period $= \frac { 1 } { 3 } \mathrm { sec }$
C) Frequency $= \frac { 3 } { 2 \pi }$ cycles per sec, period $= \frac { 2 \pi } { 3 } \mathrm { sec }$
D) Frequency $= \frac { 3 } { \pi }$ cycles per sec, period $= \frac { \pi } { 3 } \mathrm { sec }$

Accepted Answer

The general form of a sinusoidal function is $s(t) = A \cos(Bt + C) + D$ or $s(t) = A \sin(Bt + C) + D$, where $\frac{2\pi}{B}$ is the period and $B$ is related to the frequency by $f = \frac{B}{2\pi}$. For $s(t) = -2\cos(3t)$, $B = 3$, so the frequency is $f = \frac{3}{2\pi}$ cycles per second and the period is $T = \frac{2\pi}{3}$ seconds.

Quiz 6: The Circular Functions and Their Graphs

Solve the problem. -Suppose that a weight on a spring has an initial position of $s ( 0 ) = 6$ inches and a period of $P = 2.5$ seconds. Find a function $s ( t ) = a \cos ( 2 \pi F t )$ that models the displacement of the weight.

Solve the problem. -A weight attached to a spring is pulled down 3 inches below the equilibrium position. Assuming that the frequency of the system is $\frac { 6 } { \pi }$ cycles per second, determine a trigonometric model that gives the position of the weight at time $t$ seconds.

Solve the problem. -The position of a weight attached to a spring is $s ( t ) = - 3 \cos 7 \pi t$ inches after $t$ seconds. When does the weight first reach its maximum height?

Solve the problem. -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.

Solve the problem. -A weight attached to a spring is pulled down 4 inches below the equilibrium position. Assuming that the period of the system is $\frac { 1 } { 3 } \mathrm { sec }$ , what is the frequency of the system?

Solve the problem. -A weight attached to a spring is pulled down 9 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.

Solve the problem. -A guitar string is plucked so that it vibrates with a frequency of $F = 60$ . Suppose the maximum displacement at the center of the string is $s ( 0 ) = 0.53$ . Find an equation of the form $s ( t ) = a \cos b t$ to model this displacement. Round constants to 2 decimal places.

Solve the problem. -The position of a weight attached to a spring is $s ( t ) = - 2 \cos 3$ t inches after $t$ seconds. What are the frequency and period of the system?

Quiz 6: The Circular Functions and Their Graphs

Solve the problem. -Suppose that a weight on a spring has an initial position of s(0) =6s ( 0 ) = 6s(0) =6 inches and a period of P=2.5P = 2.5P=2.5 seconds. Find a function s(t) =acos⁡(2πFt) s ( t ) = a \cos ( 2 \pi F t ) s(t) =acos(2πFt) that models the displacement of the weight.

Solve the problem. -A weight attached to a spring is pulled down 3 inches below the equilibrium position. Assuming that the frequency of the system is 6π\frac { 6 } { \pi }π6​ cycles per second, determine a trigonometric model that gives the position of the weight at time ttt seconds.

Solve the problem. -The position of a weight attached to a spring is s(t) =−3cos⁡7πts ( t ) = - 3 \cos 7 \pi ts(t) =−3cos7πt inches after ttt seconds. When does the weight first reach its maximum height?

Solve the problem. -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.

Solve the problem. -A weight attached to a spring is pulled down 4 inches below the equilibrium position. Assuming that the period of the system is 13sec\frac { 1 } { 3 } \mathrm { sec }31​sec , what is the frequency of the system?

Solve the problem. -A weight attached to a spring is pulled down 9 inches below the equilibrium position. Assuming that the frequency of the system is 5π\frac { 5 } { \pi }π5​ cycles per second, determine a trigonometric model that gives the position of the weight at time ttt seconds.

Solve the problem. -The position of a weight attached to a spring is s(t) =−2cos⁡3s ( t ) = - 2 \cos 3s(t) =−2cos3 t inches after ttt seconds. What are the frequency and period of the system?