a rectangular swimming pool is 44 ft wide by 75 ft long. the accompanying table shows the depth ( h(x) ) of…

a rectangular swimming pool is 44 ft wide by 75 ft long. the accompanying table shows the depth ( h(x) ) of the water in 5 - ft intervals from one end of the pool to the other. estimate volume of water in the pool using the trapezoidal rule with ( n = 15 ) applied to the integral ( v=int_{0}^{75} 44 cdot h(x) d x ).\n\nthe volume of the pool is ( square mathrm{ft}^{3} ).\n(round to the nearest integer as needed.)

a rectangular swimming pool is 44 ft wide by 75 ft long. the accompanying table shows the depth ( h(x) ) of the water in 5 - ft intervals from one end of the pool to the other. estimate volume of water in the pool using the trapezoidal rule with ( n = 15 ) applied to the integral ( v=int_{0}^{75} 44 cdot h(x) d x ).\n\nthe volume of the pool is ( square mathrm{ft}^{3} ).\n(round to the nearest integer as needed.)

Answer

Explanation:

Step1: Recall the Trapezoidal Rule formula

The Trapezoidal Rule for approximating (\int_{a}^{b}f(x)dx) is (T_n=\frac{\Delta x}{2}\left(f(x_0)+2f(x_1)+2f(x_2)+\cdots+2f(x_{n - 1})+f(x_n)\right)), where (\Delta x=\frac{b - a}{n}). Here, (a = 0), (b = 75), (n = 15), so (\Delta x=\frac{75-0}{15}=5), and (f(x)=44h(x)).

Step2: List the values of (h(x))

From the table: (h(0)=4), (h(5)=6.2), (h(10)=7.2), (h(15)=7.9), (h(20)=8.5), (h(25)=9), (h(30)=9.5), (h(35)=9.9), (h(40)=10.3), (h(45)=10.7), (h(50)=11.1), (h(55)=11.4), (h(60)=11.7), (h(65)=12.1), (h(70)=12.4), (h(75)=12.7)

Step3: Calculate (T_{15})

[ \begin{align*} T_{15}&=\frac{5}{2}\times44\left[h(0)+2\left(h(5)+h(10)+h(15)+h(20)+h(25)+h(30)+h(35)+h(40)+h(45)+h(50)+h(55)+h(60)+h(65)+h(70)\right)+h(75)\right]\ &=110\left[4 + 2(6.2+7.2+7.9+8.5+9+9.5+9.9+10.3+10.7+11.1+11.4+11.7+12.1+12.4)+12.7\right]\ &=110\left[4+2\times(6.2 + 7.2+7.9+8.5+9+9.5+9.9+10.3+10.7+11.1+11.4+11.7+12.1+12.4)+12.7\right]\ &=110\left[4+2\times(140.9)+12.7\right]\ &=110\left[4 + 281.8+12.7\right]\ &=110\times298.5\ &=32835 \end{align*} ]

Answer:

(32835)