a rectangular swimming pool is 42 ft wide by 75 ft long. the accompanying table shows the depth h(x) of the…

a rectangular swimming pool is 42 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 the volume of water in the pool using the trapezoidal rule with n = 15 applied to the integral v = \\( \\int _ { 0 } ^ { 75 } 42 \\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 42 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 the volume of water in the pool using the trapezoidal rule with n = 15 applied to the integral v = \\( \\int _ { 0 } ^ { 75 } 42 \\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 ( \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)=42h(x)).

Step2: Identify the function values

We have (x_0 = 0), (h(x_0)=4); (x_1 = 5), (h(x_1)=6.2); (x_2 = 10), (h(x_2)=7.2); (x_3 = 15), (h(x_3)=7.9); (x_4 = 20), (h(x_4)=8.5); (x_5 = 25), (h(x_5)=9); (x_6 = 30), (h(x_6)=9.5); (x_7 = 35), (h(x_7)=9.9); (x_8 = 40), (h(x_8)=10.3); (x_9 = 45), (h(x_9)=10.7); (x_{10}=50), (h(x_{10})=11.1); (x_{11}=55), (h(x_{11})=11.4); (x_{12}=60), (h(x_{12})=11.7); (x_{13}=65), (h(x_{13})=12.1); (x_{14}=70), (h(x_{14})=12.4); (x_{15}=75), (h(x_{15})=12.7).

Step3: Apply the Trapezoidal Rule

[ \begin{align*} T_{15}&=\frac{5}{2}\times42\left(4 + 2\times6.2+2\times7.2+2\times7.9+2\times8.5+2\times9+2\times9.5+2\times9.9+2\times10.3+2\times10.7+2\times11.1+2\times11.4+2\times11.7+2\times12.1+2\times12.4+12.7\right)\ &=105\left(4+12.4 + 14.4+15.8+17+18+19+19.8+20.6+21.4+22.2+22.8+23.4+24.2+24.8+12.7\right)\ &=105\left(4+(12.4 + 14.4+15.8+17+18+19+19.8+20.6+21.4+22.2+22.8+23.4+24.2+24.8)+12.7\right)\ &=105\left(4+\sum_{i = 1}^{14}2h(x_i)+12.7\right)\ &=105\times(4 + 295.8+12.7)\ &=105\times312.5\ &=32812.5 \end{align*} ]

Answer:

(32813)