Chapter 6.6, Problem 37E

a.

To determine

To Write: The function of the distance the pendulum swings.

Answer to Problem 37E

The function is $y(n)=781.25{(0.8)}^n$

Explanation of Solution

Given information: It is given that the table of values (1, 625), (2, 500), (3, 400)

Formula used: The standard form of the exponential function is $y(n)=a{(b)}^n$

Calculation:

From the table let’s plug in the points onto the equation to find the values of a, b.

Using (1, 625)

  $625=a{(b)}^1$

Using (2,500)

  $500=a{(b)}^2$

Now, solve the equations.

  $\frac{a{b}^2}{ab}=\frac{500}{625}\Rightarrow b=\frac{4}{5}$

Simplify it,

  $b=\frac{4}{5}$

And $a=625$

Now, plug in these values into the formula

  $y(n)=625{\left(\frac{4}{5}\right)}^n$

b.

To determine

To find: After the number of swings the distance is 256mm.

Answer to Problem 37E

After 5 swings the distance becomes 256.

Explanation of Solution

Given information: We are given with $y(n)$ is 256.

Calculation:

Plug in the values of $y(n)$ into the equation then solve for ‘n’

  $256=781.25{(0.8)}^n$

Solve it for ‘n’

Divide both sides by 781.25

  $0.32768={0.8}^n$

Take log on both sides,

  $\log \left(0.32768\right)=n\text{ log}\left(0.8\right)$

Divide both sides by $\log \left(0.8\right)$

  $5=n$

Conclusion: So, after 5 wings the pendulum distance becomes 256 mm.