Let's solve the problem by understanding the concept of moments about a fulcrum. Here's the detailed step-by-step solution:
1. Determine the Moments of Each Person:
The moment for each person is calculated as their weight multiplied by their distance from the fulcrum.
- For John:
[tex]\[
\text{John's Moment} = 122 \, \text{lbs} \times 3 \, \text{ft} = 366 \, \text{ft-lbs}
\][/tex]
- For Isaiah:
[tex]\[
\text{Isaiah's Moment} = 173 \, \text{lbs} \times 9 \, \text{ft} = 1557 \, \text{ft-lbs}
\][/tex]
- For Kiana:
[tex]\[
\text{Kiana's Moment} = 102 \, \text{lbs} \times 13 \, \text{ft} = 1326 \, \text{ft-lbs}
\][/tex]
2. Calculate the Total Moment:
The total moment is the sum of the individual moments.
[tex]\[
\text{Total Moment} = 366 \, \text{ft-lbs} + 1557 \, \text{ft-lbs} + 1326 \, \text{ft-lbs} = 3249 \, \text{ft-lbs}
\][/tex]
3. Determine the Father's Distance from the Fulcrum:
To balance the seesaw, the moment due to the father's weight should equal the total moment of John, Isaiah, and Kiana. So, we set up the equation:
[tex]\[
\text{Father's Weight} \times \text{Father's Distance} = \text{Total Moment}
\][/tex]
Solving for the father's distance:
[tex]\[
\text{Father's Distance} = \frac{\text{Total Moment}}{\text{Father's Weight}} = \frac{3249 \, \text{ft-lbs}}{295 \, \text{lbs}}
\][/tex]
4. Perform the Division and Round the Result:
[tex]\[
\text{Father's Distance} = 11.01 \, \text{ft} \quad (\text{rounded to two decimal places})
\][/tex]
So, the father must sit 11.01 feet from the fulcrum in order to balance John, Isaiah, and Kiana.
