Question

according to a certain organizations rules, the maximum sail area s for a boat with length l (in meters) and displacement d (in cubic meters) is determined by the equation given below.
l + 1.25s^{\frac{1}{2}}-9.8d^{\frac{1}{3}}=16.296
find s for a boat with length 20.59 m and displacement 17.78 m³.
the maximum sail area for a boat with length 20.59 m and displacement 17.78 m³ is s = (square) m².
(round the final answer to two decimal places as needed. round all intermediate values to five decimal places as needed.)

Explanation:

Step1: Substitute given values

Substitute $L = 20.59$ and $D=17.78$ into the equation $L + 1.25S^{\frac{1}{2}}-9.8D^{\frac{1}{3}}=16.296$.
$20.59+1.25S^{\frac{1}{2}}-9.8\times(17.78)^{\frac{1}{3}} = 16.296$

Step2: Calculate $9.8\times(17.78)^{\frac{1}{3}}$

$(17.78)^{\frac{1}{3}}\approx2.60877$ (rounded to five - decimal places), then $9.8\times2.60877 = 25.566$ (rounded to five - decimal places).
The equation becomes $20.59+1.25S^{\frac{1}{2}}-25.566 = 16.296$.

Step3: Simplify the left - hand side

$1.25S^{\frac{1}{2}}+(20.59 - 25.566)=16.296$, so $1.25S^{\frac{1}{2}}-4.976 = 16.296$.

Step4: Isolate the term with $S$

Add $4.976$ to both sides of the equation: $1.25S^{\frac{1}{2}}=16.296 + 4.976=21.272$.

Step5: Solve for $S^{\frac{1}{2}}$

Divide both sides by $1.25$: $S^{\frac{1}{2}}=\frac{21.272}{1.25}=17.0176$.

Step6: Solve for $S$

Square both sides: $S=(17.0176)^2\approx289.60$.

Answer:

$289.60$