Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To solve for the length \( L \) of a pendulum that makes one full swing in 2.2 seconds using the given formula \( T = 2 \pi \sqrt{\frac{L}{32}} \), we need to follow these steps:
1. Substitute the given values into the formula \( T = 2 \pi \sqrt{\frac{L}{32}} \):
[tex]\[ 2.2 = 2 \times 3.14 \times \sqrt{\frac{L}{32}} \][/tex]
2. Isolate the square root term. First, divide both sides by \( 2 \times 3.14 \):
[tex]\[ \frac{2.2}{2 \times 3.14} = \sqrt{\frac{L}{32}} \][/tex]
3. Calculate the value of the left-hand side:
[tex]\[ \frac{2.2}{6.28} \approx 0.35 \][/tex]
Thus,
[tex]\[ 0.35 = \sqrt{\frac{L}{32}} \][/tex]
4. Square both sides to eliminate the square root:
[tex]\[ (0.35)^2 = \frac{L}{32} \][/tex]
[tex]\[ 0.1225 = \frac{L}{32} \][/tex]
5. Isolate \( L \) by multiplying both sides by 32:
[tex]\[ L = 0.1225 \times 32 = 3.92 \][/tex]
The calculated value of \( L \) is approximately 3.92 feet.
Next, we compare this length to the given options (2 feet, 4 feet, 11 feet, 19 feet) to find the closest match:
- Subtract 3.92 from each option:
[tex]\[ |2 - 3.92| = 1.92 \][/tex]
[tex]\[ |4 - 3.92| = 0.08 \][/tex]
[tex]\[ |11 - 3.92| = 7.08 \][/tex]
[tex]\[ |19 - 3.92| = 15.08 \][/tex]
The option 4 feet has the smallest difference from 3.92 feet.
Therefore, the closest length of the pendulum that makes one full swing in 2.2 seconds is:
[tex]\[ \boxed{4 \text{ feet}} \][/tex]
1. Substitute the given values into the formula \( T = 2 \pi \sqrt{\frac{L}{32}} \):
[tex]\[ 2.2 = 2 \times 3.14 \times \sqrt{\frac{L}{32}} \][/tex]
2. Isolate the square root term. First, divide both sides by \( 2 \times 3.14 \):
[tex]\[ \frac{2.2}{2 \times 3.14} = \sqrt{\frac{L}{32}} \][/tex]
3. Calculate the value of the left-hand side:
[tex]\[ \frac{2.2}{6.28} \approx 0.35 \][/tex]
Thus,
[tex]\[ 0.35 = \sqrt{\frac{L}{32}} \][/tex]
4. Square both sides to eliminate the square root:
[tex]\[ (0.35)^2 = \frac{L}{32} \][/tex]
[tex]\[ 0.1225 = \frac{L}{32} \][/tex]
5. Isolate \( L \) by multiplying both sides by 32:
[tex]\[ L = 0.1225 \times 32 = 3.92 \][/tex]
The calculated value of \( L \) is approximately 3.92 feet.
Next, we compare this length to the given options (2 feet, 4 feet, 11 feet, 19 feet) to find the closest match:
- Subtract 3.92 from each option:
[tex]\[ |2 - 3.92| = 1.92 \][/tex]
[tex]\[ |4 - 3.92| = 0.08 \][/tex]
[tex]\[ |11 - 3.92| = 7.08 \][/tex]
[tex]\[ |19 - 3.92| = 15.08 \][/tex]
The option 4 feet has the smallest difference from 3.92 feet.
Therefore, the closest length of the pendulum that makes one full swing in 2.2 seconds is:
[tex]\[ \boxed{4 \text{ feet}} \][/tex]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.