The period [tex]$T$[/tex] (in seconds) of a pendulum is given by

[tex]\[ T = 2 \pi \sqrt{\left(\frac{L}{32}\right)} \][/tex]

where [tex]$L$[/tex] stands for the length (in feet) of the pendulum. If [tex]$\pi = 3.14$[/tex], and the period is 9.42 seconds, what is the length?

The length of the pendulum is [tex]$\qquad$[/tex] feet.

The solution is [tex]$\square$[/tex]



Answer :

To determine the length [tex]\( L \)[/tex] of the pendulum given the period [tex]\( T \)[/tex] and the value of [tex]\( \pi \)[/tex], follow these steps:

1. Given Equation:
[tex]\[ T = 2 \pi \sqrt{\left(\frac{L}{32}\right)} \][/tex]

2. Substitute the Known Values:
[tex]\[ T = 9.42 \quad \text{and} \quad \pi = 3.14 \][/tex]
Thus, the equation becomes:
[tex]\[ 9.42 = 2 \times 3.14 \times \sqrt{\left(\frac{L}{32}\right)} \][/tex]

3. Isolate the Square Root Term:
Divide both sides by [tex]\( 2 \times 3.14 \)[/tex]:
[tex]\[ \frac{9.42}{2 \times 3.14} = \sqrt{\left(\frac{L}{32}\right)} \][/tex]

4. Calculate the Value Inside the Square Root:
First, compute the denominator:
[tex]\[ 2 \times 3.14 = 6.28 \][/tex]
Then,
[tex]\[ \frac{9.42}{6.28} = 1.5 \][/tex]
So,
[tex]\[ 1.5 = \sqrt{\left(\frac{L}{32}\right)} \][/tex]

5. Square Both Sides to Eliminate the Square Root:
[tex]\[ 1.5^2 = \frac{L}{32} \][/tex]
[tex]\[ 2.25 = \frac{L}{32} \][/tex]

6. Solve for [tex]\( L \)[/tex]:
Multiply both sides by 32:
[tex]\[ L = 32 \times 2.25 \][/tex]
[tex]\[ L = 72 \][/tex]

Therefore, the length of the pendulum is [tex]\( \boxed{72} \)[/tex] feet.