To find the length [tex]\( L \)[/tex] of a pendulum that makes one full swing in 1.9 seconds, we'll start by using the given formula for the period of a pendulum:
[tex]\[ T = 2 \pi \sqrt{\frac{L}{32}} \][/tex]
We are given:
- [tex]\( T = 1.9 \)[/tex] seconds
- [tex]\( \pi = 3.14 \)[/tex]
- [tex]\( g = 32 \)[/tex] ft/s[tex]\(^2\)[/tex] (acceleration due to gravity)
We need to solve for [tex]\( L \)[/tex]. We'll start by rearranging the formula to isolate [tex]\( L \)[/tex].
First, square both sides of the given equation to get rid of the square root:
[tex]\[ T^2 = \left(2 \pi \sqrt{\frac{L}{32}}\right)^2 \][/tex]
[tex]\[ T^2 = 4 \pi^2 \left(\frac{L}{32}\right) \][/tex]
Next, solve for [tex]\( L \)[/tex]:
[tex]\[ \frac{T^2}{4 \pi^2} = \frac{L}{32} \][/tex]
Multiply both sides by 32:
[tex]\[ L = 32 \cdot \frac{T^2}{4 \pi^2} \][/tex]
[tex]\[ L = \frac{32 T^2}{4 \pi^2} \][/tex]
Now, substitute the given values [tex]\( T = 1.9 \)[/tex] and [tex]\( \pi = 3.14 \)[/tex]:
[tex]\[ L = \frac{32 \times (1.9)^2}{4 \times (3.14)^2} \][/tex]
Calculate [tex]\( (1.9)^2 \)[/tex]:
[tex]\[ (1.9)^2 = 3.61 \][/tex]
Calculate [tex]\( (3.14)^2 \)[/tex]:
[tex]\[ (3.14)^2 = 9.8596 \][/tex]
So the expression for [tex]\( L \)[/tex] becomes:
[tex]\[ L = \frac{32 \times 3.61}{4 \times 9.8596} \][/tex]
[tex]\[ L = \frac{115.52}{39.4384} \][/tex]
Now, divide to find [tex]\( L \)[/tex]:
[tex]\[ L \approx 2.929 \text{ feet} \][/tex]
Finally, round [tex]\( L \)[/tex] to the nearest foot:
[tex]\[ L \approx 3 \text{ feet} \][/tex]
So, the length of the pendulum, to the nearest foot, is 3 feet.
