Answered

The hull speed of a boat is approximated by the function

$v = 1.34 \sqrt{l},$

where $l$ is the hull length in feet and $v$ is the hull speed in knots.



Suppose the Santa Monica has a hull length that is 10 ft shorter than that of the Nina Pinta. What expression represents the hull speed of the Santa Monica in terms of the length, $l_n$, of the Nina Pinta?



Function: $v_s = a \sqrt{b}$



Where $a = $ □ and $b = l_n$ □ .



What are the restrictions on $l_n$?

$l_n >$ □

Verified Answer :

  • The hull speed of the Santa Monica is expressed as \(v_s = 1.34\sqrt{l_n - 10}\).
  • By matching the form \(v_s = a\sqrt{b}\), we find \(a = \boxed{1.34}\) and \(b = l_n \boxed{- 10}\).
  • The restriction on the length of the Nina Pinta is \(l_n > \boxed{10}\).

Explanation

  1. Analyzing the Problem Data

The hull speed of a boat is given by the formula \(v = 1.34\sqrt{l}\), where \(l\) is the hull length in feet and \(v\) is the speed in knots. We are given two boats: the Nina Pinta and the Santa Monica. Let \(l_n\) represent the hull length of the Nina Pinta and \(l_s\) represent the hull length of the Santa Monica. The problem states that the Santa Monica is \(10\) ft shorter than the Nina Pinta, which means \(l_s = l_n - 10\).

  1. Finding the Expression for Hull Speed

To find the hull speed of the Santa Monica, \(v_s\), we substitute its length \(l_s\) into the given hull speed formula:
\[v_s = 1.34\sqrt{l_s} = 1.34\sqrt{l_n - 10}.\]
We are asked to express this in the form \(v_s = a\sqrt{b}\). Comparing \(v_s = 1.34\sqrt{l_n - 10}\) with the required form, we identify the constants and expressions:
\[a = 1.34\]
\[b = l_n - 10\]
This allows us to write the speed as \(v_s = 1.34\sqrt{l_n - 10}\).

  1. Determining Restrictions on \(l_n\)

Finally, we must consider the physical constraints on the hull length \(l_n\). Both the Nina Pinta and the Santa Monica must have a positive hull length.

  1. For the Nina Pinta: \(l_n > 0\).
  2. For the Santa Monica: \(l_s = l_n - 10 > 0\), which implies \(l_n > 10\).

Since \(l_n > 10\) automatically satisfies the condition \(l_n > 0\), the overall restriction on the length of the Nina Pinta is \(l_n > 10\).

Examples

In nautical engineering, understanding how a boat's hull length affects its theoretical maximum speed is crucial. If you are comparing two vessels, knowing the difference in their lengths allows you to determine how much slower or faster one might be compared to the other. For instance, if you are planning to modify a boat by extending or shortening its hull, this formula helps in predicting the change in its performance capabilities. This basic algebraic model serves as a foundation for more complex hydrodynamic designs.

Related Questions