Question 1

Determine the equation of the graph.
-The formula for the up and down motion of a weight on a spring is given by $s ( t ) = \sin \sqrt { \frac { \mathrm { k } } { \mathrm { m } } } \mathrm { t }$. If the spring constant is 5 , then what mass $\mathrm { m }$ must be used in order to produce a period of 6 seconds?

Accepted Answer

The period $T$ of a sinusoidal function like $s(t) = \sin(\sqrt{\frac{k}{m}}t)$ is given by $T = 2\pi \sqrt{\frac{m}{k}}$. Given $T = 6$ seconds and $k = 5$, we solve for $m$ by rearranging the formula: $m = \frac{T^2 k}{4\pi^2} = \frac{6^2 \cdot 5}{4\pi^2} = \frac{180}{4\pi^2} = \frac{45}{\pi^2}$.

Question 2

Graph the function.
-$y=3-4 \sec \left(x-\frac{\pi}{6}\right)$

Accepted Answer

The answer of Graph the function.
-$y=3-4 \sec \left(x-\frac{\pi}{6}\right)$

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Question 3

Graph the function.
-$$y = - \frac { 1 } { 3 } \csc \left( \frac { 5 } { 6 } x - \frac { \pi } { 6 } \right)$$

Accepted Answer

The answer of Graph the function.
-\[y = - \frac {...

Question 4

Graph the function.
-$$y = \frac { 1 } { 2 } \sec \left( \frac { 3 } { 5 } x + \frac { \pi } { 4 } \right)$$

Accepted Answer

The answer of Graph the function.
-\[y = \frac { 1...

Question 5

Graph the function.
-$$y = \frac { 5 } { 2 } \csc \left( \frac { 4 } { 5 } x - \frac { \pi } { 4 } \right)$$

Accepted Answer

The answer of Graph the function.
-\[y = \frac { 5...

Question 6

Determine the equation of the graph.
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Accepted Answer

The answer of Determine the equation of the graph.
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Question 7

Determine the equation of the graph.
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Accepted Answer

The answer of Determine the equation of the graph.
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Question 8

Determine the equation of the graph.
-A spring with a spring constant of 5 and a 1-unit mass attached to it is stretched $2 \mathrm { ft }$ and released. What is the equation for the resulting oscillatory motion?

Accepted Answer

The equation for the oscillatory motion of a spring-mass system is given by $s(t) = A \sin(\sqrt{\frac{k}{m}} t)$, where $A$ is the amplitude of the oscillation, $k$ is the spring constant, and $m$ is the mass. Here, $A = 2$, $k = 5$, and $m = 1$, so the equation becomes $s(t) = 2 \sin(\sqrt{5} t)$.

Question 9

Graph the function.
-$y=\frac{5}{4} \csc \left(\frac{2}{5} x+\frac{\pi}{5}\right)$

Accepted Answer

The answer of Graph the function.
-$y=\frac{5}{4} \csc \left(\frac{2}{5} x+\frac{\pi}{5}\right)$
 
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Question 10

Determine the equation of the graph.
-Determine the length of a pendulum that has a period of 2 seconds.

Accepted Answer

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

Question 11

Determine the equation of the graph.
-Write the equation that describes the simple harmonic motion of a particle moving uniformly around a circle of radius 8 units, with angular speed 3 radians per second.

Accepted Answer

The equation for simple harmonic motion is $s(t) = A \sin(\omega t)$ or $s(t) = A \cos(\omega t)$, where $A$ is the amplitude (radius) and $\omega$ is the angular speed. Here, $A = 8$ and $\omega = 3$, so the equation is $s(t) = 8 \sin 3t$.

Question 12

Determine the equation of the graph.
-A weight attached to a spring is pulled down 3 inches below the equilibrium position. Assuming that the frequency of the system is $\frac { 7 } { \pi }$ cycles per second, determine a trigonometric model that gives the position of the weight at time $t$ seconds.

Accepted Answer

The amplitude is 3 inches, and since the weight is pulled down, the cosine function should be negative. The angular frequency is $2\pi \times \frac{7}{\pi} = 14$, so the equation is $s(t) = -3 \cos 14t$.

Question 13

Determine the equation of the graph.
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Accepted Answer

The answer of Determine the equation of the graph.
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Question 14

Graph the function.
-$y=\frac{4}{3} \sec \left(\frac{3}{2} x-\frac{\pi}{6}\right)$

Accepted Answer

The answer of Graph the function.
-$y=\frac{4}{3} \sec \left(\frac{3}{2} x-\frac{\pi}{6}\right)$

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Question 15

Graph the function.
-$$y = \csc \left( \frac { 1 } { 2 } x - \frac { \pi } { 5 } \right)$$

Accepted Answer

The answer of Graph the function.
-\[y = \csc \left( \frac...

Question 16

Determine the equation of the graph.
-The position of a weight attached to a spring is $\mathrm { s } ( \mathrm { t } ) = - 2 \cos 7 \mathrm { t }$ inches after $\mathrm { t }$ seconds. What are the frequency and period of the system?

Accepted Answer

The frequency of a cosine function of the form $s(t) = A \cos(Bt)$ is given by $f = \frac{B}{2\pi}$ cycles per second, and the period is the reciprocal of the frequency, $T = \frac{1}{f} = \frac{2\pi}{B}$. For $s(t) = -2\cos(7t)$, $B = 7$, so the frequency is $f = \frac{7}{2\pi}$ cycles per second, and the period is $T = \frac{2\pi}{7}$ seconds.

Question 17

Determine the equation of the graph.
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Accepted Answer

The answer of Determine the equation of the graph.
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Question 18

Determine the equation of the graph.
-Determine the period and frequency of oscillation when a pendulum of length 7 feet is released after being displaced 2 radians. Round constants to 8 decimal places, if necessary.

Accepted Answer

The period of a pendulum is given by $ T = 2 \pi \sqrt { \frac { L } { g } } $, where $ L $ is the length of the pendulum and $ g $ is the acceleration due to gravity. In this case, $ L = 7 $ feet and $ g = 32 $ feet per second squared, so the period is $ T = 2 \pi \sqrt { \frac { 7 } { 32 } } $ seconds. The frequency of a pendulum is given by $ f = \frac { 1 } { T } $, so the frequency in this case is $ f = \frac { 1 } { 2 \pi \sqrt { \frac { 7 } { 32 } } } $ cycles per second.

Question 19

Determine the equation of the graph.
-The position of a weight attached to a spring is $\mathrm { s } ( \mathrm { t } ) = - 3 \cos 5 \pi \mathrm { t }$ inches after $\mathrm { t }$ seconds. When does the weight first reach its maximum height?

Accepted Answer

The weight reaches its maximum height when the cosine function equals 1, which occurs at the beginning of its cycle. Given the equation $s(t) = -3\cos(5\pi t)$, the period of the cosine function is $\frac{2\pi}{5\pi} = \frac{2}{5}$ seconds. The first maximum after $t = 0$ occurs at a quarter of the period, which is $\frac{1}{4} \times \frac{2}{5} = \frac{1}{5}$ seconds.

Question 20

Determine the equation of the graph.
-A rotating beacon is located $13 \mathrm { ft }$ from a wall. The distance from the beacon to the point on the wall where the beacon is aimed is given by
$$a = 13 | \sec 2 \pi t | \text {, }$$
where $t$ is time measured in seconds since the beacon started rotating. Find a for $t = 0.41$ seconds. Ro your answer to the nearest hundredth.

Accepted Answer

To find $a$ for $t = 0.41$ seconds, substitute $t = 0.41$ into the given equation $a = 13 | \sec 2 \pi t |$. Calculate the secant value, take its absolute value, multiply by 13, and round to the nearest hundredth.

Quiz 7: The Circular Functions and Their Graphs

Graph the function. - $y=3-4 \sec \left(x-\frac{\pi}{6}\right)$ $Graph the function. - y=3-4 \sec \left(x-\frac{\pi}{6}\right) A) B) C) D)$

Graph the function. - $y = - \frac { 1 } { 3 } \csc \left( \frac { 5 } { 6 } x - \frac { \pi } { 6 } \right)$ $Graph the function. - y = - \frac { 1 } { 3 } \csc \left( \frac { 5 } { 6 } x - \frac { \pi } { 6 } \right) A) B) C) D)$

Graph the function. - $y = \frac { 1 } { 2 } \sec \left( \frac { 3 } { 5 } x + \frac { \pi } { 4 } \right)$ $Graph the function. - y = \frac { 1 } { 2 } \sec \left( \frac { 3 } { 5 } x + \frac { \pi } { 4 } \right) A) B) C) D)$

Graph the function. - $y = \frac { 5 } { 2 } \csc \left( \frac { 4 } { 5 } x - \frac { \pi } { 4 } \right)$ $Graph the function. - y = \frac { 5 } { 2 } \csc \left( \frac { 4 } { 5 } x - \frac { \pi } { 4 } \right) A) B) C) D)$

Determine the equation of the graph. - $Determine the equation of the graph. - A) y = 2 \csc x B) y = - 2 \sec x C) y = - \sec 2 x D) y = - \csc 2 x$

Determine the equation of the graph. - $Determine the equation of the graph. - A) y = 2 + \csc x B) y = 2 - \csc x C) y = 2 + \sec x D) y = 2 + \sec 2 x$

Determine the equation of the graph. -A spring with a spring constant of 5 and a 1-unit mass attached to it is stretched $2 \mathrm { ft }$ and released. What is the equation for the resulting oscillatory motion?

Graph the function. - $y=\frac{5}{4} \csc \left(\frac{2}{5} x+\frac{\pi}{5}\right)$ $Graph the function. - y=\frac{5}{4} \csc \left(\frac{2}{5} x+\frac{\pi}{5}\right) A) B) C) D)$

Determine the equation of the graph. -Determine the length of a pendulum that has a period of 2 seconds.

Determine the equation of the graph. -Write the equation that describes the simple harmonic motion of a particle moving uniformly around a circle of radius 8 units, with angular speed 3 radians per second.

Determine the equation of the graph. -A weight attached to a spring is pulled down 3 inches below the equilibrium position. Assuming that the frequency of the system is $\frac { 7 } { \pi }$ cycles per second, determine a trigonometric model that gives the position of the weight at time $t$ seconds.

Determine the equation of the graph. - $Determine the equation of the graph. - A) y = \sec 4 x B) y = \csc 4 x C) y = 4 \csc x D) y = 4 \sec x$

Graph the function. - $y=\frac{4}{3} \sec \left(\frac{3}{2} x-\frac{\pi}{6}\right)$ $Graph the function. - y=\frac{4}{3} \sec \left(\frac{3}{2} x-\frac{\pi}{6}\right) A) B) C) D)$

Graph the function. - $y = \csc \left( \frac { 1 } { 2 } x - \frac { \pi } { 5 } \right)$ $Graph the function. - y = \csc \left( \frac { 1 } { 2 } x - \frac { \pi } { 5 } \right) A) B) C) D)$

Determine the equation of the graph. -The position of a weight attached to a spring is $\mathrm { s } ( \mathrm { t } ) = - 2 \cos 7 \mathrm { t }$ inches after $\mathrm { t }$ seconds. What are the frequency and period of the system?

Determine the equation of the graph. - $Determine the equation of the graph. - A) y = - \csc \left( \frac { 1 } { 4 } x \right) B) y = \csc 4 x C) y = - \csc 4 x D) y = \csc \left( \frac { 1 } { 4 } x \right)$

Determine the equation of the graph. -Determine the period and frequency of oscillation when a pendulum of length 7 feet is released after being displaced 2 radians. Round constants to 8 decimal places, if necessary.

Determine the equation of the graph. -The position of a weight attached to a spring is $\mathrm { s } ( \mathrm { t } ) = - 3 \cos 5 \pi \mathrm { t }$ inches after $\mathrm { t }$ seconds. When does the weight first reach its maximum height?

Quiz 7: The Circular Functions and Their Graphs

Graph the function. - y=3−4sec⁡(x−π6) y=3-4 \sec \left(x-\frac{\pi}{6}\right) y=3−4sec(x−6π​)

Graph the function. - y=−13csc⁡(56x−π6) y = - \frac { 1 } { 3 } \csc \left( \frac { 5 } { 6 } x - \frac { \pi } { 6 } \right) y=−31​csc(65​x−6π​)

Graph the function. - y=12sec⁡(35x+π4) y = \frac { 1 } { 2 } \sec \left( \frac { 3 } { 5 } x + \frac { \pi } { 4 } \right) y=21​sec(53​x+4π​)

Graph the function. - y=52csc⁡(45x−π4) y = \frac { 5 } { 2 } \csc \left( \frac { 4 } { 5 } x - \frac { \pi } { 4 } \right) y=25​csc(54​x−4π​)

Determine the equation of the graph. -

Determine the equation of the graph. -

Determine the equation of the graph. -A spring with a spring constant of 5 and a 1-unit mass attached to it is stretched 2ft2 \mathrm { ft }2ft and released. What is the equation for the resulting oscillatory motion?

Graph the function. - y=54csc⁡(25x+π5) y=\frac{5}{4} \csc \left(\frac{2}{5} x+\frac{\pi}{5}\right) y=45​csc(52​x+5π​)

Determine the equation of the graph. -Determine the length of a pendulum that has a period of 2 seconds.

Determine the equation of the graph. -Write the equation that describes the simple harmonic motion of a particle moving uniformly around a circle of radius 8 units, with angular speed 3 radians per second.

Determine the equation of the graph. -

Graph the function. - y=43sec⁡(32x−π6) y=\frac{4}{3} \sec \left(\frac{3}{2} x-\frac{\pi}{6}\right) y=34​sec(23​x−6π​)

Graph the function. - y=csc⁡(12x−π5) y = \csc \left( \frac { 1 } { 2 } x - \frac { \pi } { 5 } \right) y=csc(21​x−5π​)

Determine the equation of the graph. -The position of a weight attached to a spring is s(t) =−2cos⁡7t\mathrm { s } ( \mathrm { t } ) = - 2 \cos 7 \mathrm { t }s(t) =−2cos7t inches after t\mathrm { t }t seconds. What are the frequency and period of the system?

Determine the equation of the graph. -

Determine the equation of the graph. -Determine the period and frequency of oscillation when a pendulum of length 7 feet is released after being displaced 2 radians. Round constants to 8 decimal places, if necessary.

Determine the equation of the graph. -The position of a weight attached to a spring is s(t) =−3cos⁡5πt\mathrm { s } ( \mathrm { t } ) = - 3 \cos 5 \pi \mathrm { t }s(t) =−3cos5πt inches after t\mathrm { t }t seconds. When does the weight first reach its maximum height?

Graph the function. - $y=3-4 \sec \left(x-\frac{\pi}{6}\right)$ $Graph the function. - y=3-4 \sec \left(x-\frac{\pi}{6}\right) A) B) C) D)$

Graph the function. - $y = - \frac { 1 } { 3 } \csc \left( \frac { 5 } { 6 } x - \frac { \pi } { 6 } \right)$ $Graph the function. - y = - \frac { 1 } { 3 } \csc \left( \frac { 5 } { 6 } x - \frac { \pi } { 6 } \right) A) B) C) D)$

Graph the function. - $y = \frac { 1 } { 2 } \sec \left( \frac { 3 } { 5 } x + \frac { \pi } { 4 } \right)$ $Graph the function. - y = \frac { 1 } { 2 } \sec \left( \frac { 3 } { 5 } x + \frac { \pi } { 4 } \right) A) B) C) D)$

Graph the function. - $y = \frac { 5 } { 2 } \csc \left( \frac { 4 } { 5 } x - \frac { \pi } { 4 } \right)$ $Graph the function. - y = \frac { 5 } { 2 } \csc \left( \frac { 4 } { 5 } x - \frac { \pi } { 4 } \right) A) B) C) D)$

Determine the equation of the graph. - $Determine the equation of the graph. - A) y = 2 \csc x B) y = - 2 \sec x C) y = - \sec 2 x D) y = - \csc 2 x$

Determine the equation of the graph. - $Determine the equation of the graph. - A) y = 2 + \csc x B) y = 2 - \csc x C) y = 2 + \sec x D) y = 2 + \sec 2 x$

Determine the equation of the graph. -A spring with a spring constant of 5 and a 1-unit mass attached to it is stretched $2 \mathrm { ft }$ and released. What is the equation for the resulting oscillatory motion?

Graph the function. - $y=\frac{5}{4} \csc \left(\frac{2}{5} x+\frac{\pi}{5}\right)$ $Graph the function. - y=\frac{5}{4} \csc \left(\frac{2}{5} x+\frac{\pi}{5}\right) A) B) C) D)$

Determine the equation of the graph. - $Determine the equation of the graph. - A) y = \sec 4 x B) y = \csc 4 x C) y = 4 \csc x D) y = 4 \sec x$

Graph the function. - $y=\frac{4}{3} \sec \left(\frac{3}{2} x-\frac{\pi}{6}\right)$ $Graph the function. - y=\frac{4}{3} \sec \left(\frac{3}{2} x-\frac{\pi}{6}\right) A) B) C) D)$

Graph the function. - $y = \csc \left( \frac { 1 } { 2 } x - \frac { \pi } { 5 } \right)$ $Graph the function. - y = \csc \left( \frac { 1 } { 2 } x - \frac { \pi } { 5 } \right) A) B) C) D)$

Determine the equation of the graph. -The position of a weight attached to a spring is $\mathrm { s } ( \mathrm { t } ) = - 2 \cos 7 \mathrm { t }$ inches after $\mathrm { t }$ seconds. What are the frequency and period of the system?

Determine the equation of the graph. - $Determine the equation of the graph. - A) y = - \csc \left( \frac { 1 } { 4 } x \right) B) y = \csc 4 x C) y = - \csc 4 x D) y = \csc \left( \frac { 1 } { 4 } x \right)$

Determine the equation of the graph. -The position of a weight attached to a spring is $\mathrm { s } ( \mathrm { t } ) = - 3 \cos 5 \pi \mathrm { t }$ inches after $\mathrm { t }$ seconds. When does the weight first reach its maximum height?