Let's solve the problem step-by-step.
Firstly, understand the given formula for the hull speed of a boat, which is:
[tex]\[ v = 1.34 \sqrt{l}, \][/tex]
where \( l \) is the hull length in feet and \( v \) is the hull speed in knots.
### Step 1: Relate the Lengths of Nina Pinta and Santa Monica
We are given that the Santa Monica has a hull length that is 10 feet shorter than that of the Nina Pinta. Let's denote the hull length of the Nina Pinta by \( l_n \). Therefore, the hull length of the Santa Monica, \( l_s \), is:
[tex]\[ l_s = l_n - 10. \][/tex]
### Step 2: Express the Hull Speed of Santa Monica
Using the hull speed formula \( v = 1.34 \sqrt{l} \), we can find the hull speed of the Santa Monica by substituting its hull length:
[tex]\[ v_s = 1.34 \sqrt{l_s} = 1.34 \sqrt{l_n - 10}. \][/tex]
Given the form \( v_s = a \sqrt{b} \), it can be seen that:
[tex]\[ a = 1.34 \][/tex]
and
[tex]\[ b = l_n - 10. \][/tex]
### Step 3: Determine the Restrictions on \( l_n \)
For the hull length of the Santa Monica, \( l_s = l_n - 10 \), to be physically meaningful, it must be positive:
[tex]\[ l_n - 10 > 0 \][/tex]
[tex]\[ l_n > 10. \][/tex]
Thus, the restrictions on \( l_n \) are:
[tex]\[ l_n > 10. \][/tex]
### Final Expression and Restrictions:
So, the complete detailed solution is:
- The value of \( a \) is 1.34.
- The expression for \( b \) is \( l_n - 10 \).
- The restriction on \( l_n \) is \( l_n > 10 \).
Thus, the results can be summarized as follows:
[tex]\[ v_s = 1.34 \sqrt{l_n - 10}, \][/tex]
with the restriction:
[tex]\[ l_n > 10. \][/tex]
So:
- \( a = 1.34 \)
- \( b = l_n - 10 \)
- [tex]\( l_n > 10 \)[/tex]
