Question

consider the drawing for a swimming pool below. the deep end is in the shape of a rectangular prism, and the shallow end is in the shape of a trapezoidal prism. the deep end of the pool is 5 feet deep and 15 feet long. the shallow end of the pool starts at 3 feet deep and gradually deepens to 5 feet over a length of 8 feet. if the width, w, is 10 feet, what is the total volume, v, of water that would fill the pool? hint: the formula for the area of a trapezoid is ( a=\frac{1}{2}(b_1 + b_2)h ), where ( b_1 ) and ( b_2 ) are the bases, and ( h ) is the height. 1 of 9 questions ( v = 990mathrm{ft}^3 ) ( v = 1070mathrm{ft}^3 ) ( v = 1150mathrm{ft}^3 ) ( v = 1390mathrm{ft}^3 )

Explanation:

Step1: Calculate the volume of the rectangular prism (deep - end)

The formula for the volume of a rectangular prism is \(V = l\times w\times h\). Here, \(l = 15\) ft, \(w = 10\) ft, \(h = 5\) ft.
\(V_{1}=15\times10\times5=750\) \(ft^{3}\)

Step2: Calculate the area of the trapezoidal base

The formula for the area of a trapezoid is \(A=\frac{1}{2}(b_{1}+b_{2})h\). Here, \(b_{1}=3\) ft, \(b_{2}=5\) ft, \(h = 8\) ft.
\(A=\frac{1}{2}(3 + 5)\times8=\frac{1}{2}\times8\times8=32\) \(ft^{2}\)

Step3: Calculate the volume of the trapezoidal prism (shallow - end)

The volume of a prism is \(V=A\times w\). Here, \(A = 32\) \(ft^{2}\), \(w = 10\) ft.
\(V_{2}=32\times10 = 320\) \(ft^{3}\)

Step4: Calculate the total volume

\(V=V_{1}+V_{2}\)
\(V=750+320=1070\) \(ft^{3}\)

Answer:

\(V = 1070\) \(ft^{3}\) (the second option)