the widths (in meters) of a kidney - shaped swimming pool were measured at 3 - meter intervals as indicated…

the widths (in meters) of a kidney - shaped swimming pool were measured at 3 - meter intervals as indicated in the figure below.\nlet x represent the distance from the left end of the pool and let w(x) be the width of the pool at x. use simpsons rule with n = 8 to set up an integral of the form $int_{a}^{b}w dx=s_{n}$ that can be used to estimate the area of the pool in square meters. determine each of the following values.\na = 0\n b = 24\n $delta x=3$\n$x_{0}=0$\n$x_{1}=3$\n$x_{2}=6$\n$x_{3}=9$\n$x_{4}=12$\n$x_{5}=15$\n$x_{6}=18$\n$x_{7}=21$\n$x_{8}=24$

the widths (in meters) of a kidney - shaped swimming pool were measured at 3 - meter intervals as indicated in the figure below.\nlet x represent the distance from the left end of the pool and let w(x) be the width of the pool at x. use simpsons rule with n = 8 to set up an integral of the form $int_{a}^{b}w dx=s_{n}$ that can be used to estimate the area of the pool in square meters. determine each of the following values.\na = 0\n b = 24\n $delta x=3$\n$x_{0}=0$\n$x_{1}=3$\n$x_{2}=6$\n$x_{3}=9$\n$x_{4}=12$\n$x_{5}=15$\n$x_{6}=18$\n$x_{7}=21$\n$x_{8}=24$

Answer

Explanation:

Step1: Recall Simpson's rule formula

$S_n=\frac{\Delta x}{3}[f(x_0) + 4f(x_1)+2f(x_2)+4f(x_3)+\cdots+2f(x_{n - 2})+4f(x_{n - 1})+f(x_n)]$ where $n$ is even, $\Delta x=\frac{b - a}{n}$, and $x_i=a + i\Delta x$. Here, $a = 0$, $b = 24$, $\Delta x=3$, $n = 8$, and $f(x)=w(x)$.

Step2: Identify function values

$w(x_0)=w(0)=6.2$, $w(x_1)=w(3)=7.2$, $w(x_2)=w(6)=6.8$, $w(x_3)=w(9)=5.6$, $w(x_4)=w(12)=5.0$, $w(x_5)=w(15)=4.8$, $w(x_6)=w(18)=4.8$, $w(x_7)=w(21)=4.8$, $w(x_8)=w(24)=4.8$.

Step3: Apply Simpson's rule

$S_8=\frac{3}{3}[6.2+4\times7.2 + 2\times6.8+4\times5.6+2\times5.0+4\times4.8+2\times4.8+4\times4.8+4.8]$ $=6.2 + 28.8+13.6 + 22.4+10+19.2+9.6+19.2+4.8$ $=133.8$

Answer:

$133.8$