Calculate the area of the water plane using Simpson's First Rule, given the interval and ordinates.

Question

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To find the area of the water plane, we will use Simpson's First Rule. This rule is applicable because there is an odd number of ordinates (7 ordinates). We assume the given breadths are half-breadths, which is standard practice in naval architecture for calculating waterplane area.

The common interval ( $h$ ) is 12 metres.
The ordinates ( $y_n$ ) are $0, 7.6, 8.7, 9.2, 9.5, 9.4, 8.5$ metres.

List the ordinates and their corresponding Simpson's Multipliers (SM).
For 7 ordinates, the Simpson's Multipliers are $1, 4, 2, 4, 2, 4, 1$.

Calculate the product of each ordinate and its Simpson's Multiplier.

Ordinate ( $y_n$ ) (m)	Simpson's Multiplier (SM)	Product ( $y_n \times SM$ ) (m)
0	1	$0 \times 1 = 0$
7.6	4	$7.6 \times 4 = 30.4$
8.7	2	$8.7 \times 2 = 17.4$
9.2	4	$9.2 \times 4 = 36.8$
9.5	2	$9.5 \times 2 = 19.0$
9.4	4	$9.4 \times 4 = 37.6$
8.5	1	$8.5 \times 1 = 8.5$
	Sum of Products	$\sum (y_n \times SM) = 149.7$

Apply Simpson's First Rule formula to find the area of one side of the water plane.
The formula for Simpson's First Rule is:
$A_{one\_side} = \frac{h}{3} \sum (y_n \times SM)$
Substitute the values:
$A_{one\_side} = \frac{12}{3} \times 149.7$ $A_{one\_side} = 4 \times 149.7$ $A_{one\_side} = 598.8 \, \text{m}^2$
Calculate the total water plane area.
Since the given ordinates are half-breadths, the total water plane area is twice the area of one side.
$A_{total} = 2 \times A_{one\_side}$ $A_{total} = 2 \times 598.8$ $A_{total} = 1197.6 \, \text{m}^2$
The area of the water plane is $\text{1197.6 m}^2$ .

Calculate the area of the water plane using Simpson's First Rule, given the interval and ordinates.

Answer

Step-by-step solution

List the ordinates and their corresponding Simpson's Multipliers (SM).

Calculate the product of each ordinate and its Simpson's Multiplier.

Apply Simpson's First Rule formula to find the area of one side of the water plane.

Calculate the total water plane area.

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