Question

the formula $t = 2pisqrt{\frac{l}{32}}$ gives the time it takes in seconds, $t$, for a pendulum to make one full swing back and forth, where $l$ is the length of the pendulum, in feet. to the nearest foot, what is the length of a pendulum that makes one full swing in 1.9 s? use 3.14 for $pi$.

Explanation:

Step1: Substitute given values into formula

Given $T = 1.9$ and $\pi=3.14$, the formula $T = 2\pi\sqrt{\frac{L}{32}}$ becomes $1.9=2\times3.14\sqrt{\frac{L}{32}}$.

Step2: Simplify the right - hand side

$2\times3.14 = 6.28$, so the equation is $1.9 = 6.28\sqrt{\frac{L}{32}}$. Then $\sqrt{\frac{L}{32}}=\frac{1.9}{6.28}$.

Step3: Square both sides

$(\sqrt{\frac{L}{32}})^2 = (\frac{1.9}{6.28})^2$, which gives $\frac{L}{32}=(\frac{1.9}{6.28})^2$.

Step4: Solve for L

$L = 32\times(\frac{1.9}{6.28})^2$. Calculate $(\frac{1.9}{6.28})^2=\frac{1.9^2}{6.28^2}=\frac{3.61}{39.4384}\approx0.0915$. Then $L = 32\times0.0915 = 2.928\approx3$.

Answer:

3