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

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}42cdot h(x)dx ).\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}42cdot h(x)dx ).\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 formula for ( \int_{a}^{b}f(x)dx) with (n) sub - intervals is (T=\frac{\Delta x}{2}[f(x_0)+2f(x_1)+2f(x_2)+\cdots+2f(x_{n - 1})+f(x_n)]), 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: Calculate the sum inside the Trapezoidal Rule formula

Let (S = h(x_0)+2h(x_1)+2h(x_2)+\cdots+2h(x_{14})+h(x_{15})). (h(x_0)=4), (h(x_1) = 6.2), (h(x_2)=7.2), (h(x_3)=7.9), (h(x_4)=8.5), (h(x_5)=9), (h(x_6)=9.5), (h(x_7)=9.9), (h(x_8)=10.3), (h(x_9)=10.7), (h(x_{10})=11.1), (h(x_{11})=11.4), (h(x_{12})=11.7), (h(x_{13})=12.1), (h(x_{14})=12.4), (h(x_{15})=12.7). [ \begin{align*} S&=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\ &=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\ &=4+2\times(140.9)+12.7\ &=4 + 281.8+12.7\ &=298.5 \end{align*} ]

Step3: Calculate the volume (V)

Since (V=\frac{\Delta x}{2}\times42\times S) (because (V=\int_{0}^{75}42h(x)dx) and using the Trapezoidal Rule). Substitute (\Delta x = 5) and (S = 298.5) into the formula. (V=\frac{5}{2}\times42\times298.5) First, (\frac{5}{2}\times42=105). Then (V = 105\times298.5=31342.5)

Answer:

(31343)