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}}-9.80d^{\frac{1}{3}} = 16.296. find s for a boat with length 22.47 m and displacement 19.40 m^{3}. the maximum sail area for a boat with length 22.47 m and displacement 19.40 m^{3} is s = m^{2} (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 $L = 22.47$ and $D=19.40$ into the equation $L + 1.25S^{\frac{1}{2}}-9.80D^{\frac{1}{3}}=16.296$.
So we get $22.47+1.25S^{\frac{1}{2}}-9.80\times19.40^{\frac{1}{3}} = 16.296$.

Step2: Calculate $9.80\times19.40^{\frac{1}{3}}$

First, calculate $19.40^{\frac{1}{3}}\approx2.70577$. Then $9.80\times2.70577 = 26.51655$.
The equation becomes $22.47+1.25S^{\frac{1}{2}}-26.51655 = 16.296$.

Step3: Simplify the left - hand side

$22.47-26.51655+1.25S^{\frac{1}{2}}=-4.04655 + 1.25S^{\frac{1}{2}}$.
So, $-4.04655+1.25S^{\frac{1}{2}}=16.296$.

Step4: Isolate the term with $S$

Add $4.04655$ to both sides of the equation:
$1.25S^{\frac{1}{2}}=16.296 + 4.04655=20.34255$.

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

Divide both sides by $1.25$: $S^{\frac{1}{2}}=\frac{20.34255}{1.25}=16.27404$.

Step6: Solve for $S$

Square both sides: $S=(16.27404)^2\approx264.84$.

Answer:

$264.84$