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$An object is attached to a coiled spring. The object begins at its rest position at t=0 seconds. It is then propelled downward. Write an equation for the distance of the object from its rest position after t seconds, if the amplitude is 2 , inches and the period is 3.5 seconds. The equation for the distance d of the object from its rest position is ◻. (Type an exact answer, using pi as needed. Use integers or fractions for any numbers in the equation.)$

An object is attached to a coiled spring. The object begins at its rest position at $t=0$ seconds. It is then propelled downward. Write an equation for the distance of the object from its rest position after $t$ seconds, if the amplitude is $2$ , inches and the period is $3$ . $5$ seconds. $\newline$ The equation for the distance $\mathrm{d}$ of the object from its rest position is $\square$ . $\newline$ (Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the equation.)

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Precalculus

Identify discrete and continuous random variables

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Q. An object is attached to a coiled spring. The object begins at its rest position at

t=0

seconds. It is then propelled downward. Write an equation for the distance of the object from its rest position after

t

seconds, if the amplitude is

2

, inches and the period is

3

5

seconds.

\newline

The equation for the distance

\mathrm{d}

of the object from its rest position is

\square

\newline

(Type an exact answer, using

\pi

as needed. Use integers or fractions for any numbers in the equation.)

Write Equation: To write the equation for the distance of the object from its rest position, we need to use the formula for simple harmonic motion, which is $d(t) = A \cdot \cos(2 \cdot \pi \cdot t / T)$ , where $A$ is the amplitude and $T$ is the period.
Substitute Amplitude: The amplitude $A$ is given as $2$ inches, so we will substitute $A = 2$ into the equation.
Substitute Period: The period $T$ is given as $3.5$ seconds, so we will substitute $T = 3.5$ into the equation.
Substitute Values: Now we substitute the values into the equation: $d(t) = 2 \cdot \cos(2 \cdot \pi \cdot t / 3.5)$ .
Calculate Coefficient: We can simplify the equation by calculating the coefficient of $t$ inside the cosine function. The coefficient is $2 \times \pi / 3.5$ .
Simplify Coefficient: The simplified coefficient is $(2 \times \pi) / 3.5$ , which we can leave in terms of $\pi$ for an exact answer.
Final Equation: The final equation for the distance $d$ of the object from its rest position after $t$ seconds is $d(t) = 2 \cdot \cos\left(\frac{2 \cdot \pi}{3.5} \cdot t\right)$ .