Question

find the total volume of the barn by finding the volume of the trapezoidal prism and the volume of the rectangular prism. then add the volumes to find the total volume of the barn.
( v_{\text{trap}} = square , \text{ft}^3 )
( v_{\text{rect}} = square , \text{ft}^3 )
( v_{\text{total}} = square , \text{ft}^3 )
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Explanation:

Step1: Calculate volume of trapezoidal prism

The formula for the volume of a trapezoidal prism is \( V_{\text{trap}}=\text{Area of trapezoid}\times\text{length} \). The area of a trapezoid is \( \frac{(a + b)}{2}\times h \), where \( a = 3\) ft, \( b = 15\) ft, \( h = 4\) ft, and the length of the prism is \( 12\) ft. First, find the area of the trapezoid: \( \frac{(3 + 15)}{2}\times4=\frac{18}{2}\times4 = 9\times4 = 36\) square feet. Then multiply by the length: \( 36\times12 = 432\) cubic feet.
\( V_{\text{trap}}=\frac{(3 + 15)}{2}\times4\times12 = 432 \)

Step2: Calculate volume of rectangular prism

The formula for the volume of a rectangular prism is \( V_{\text{rect}}=l\times w\times h \), where \( l = 15\) ft, \( w = 12\) ft, \( h = 19\) ft. So, \( 15\times12\times19 = 180\times19 = 3420\) cubic feet.
\( V_{\text{rect}}=15\times12\times19 = 3420 \)

Step3: Calculate total volume

Add the volumes of the trapezoidal prism and the rectangular prism: \( V_{\text{total}}=V_{\text{trap}}+V_{\text{rect}} = 432 + 3420 = 3852\) cubic feet.
\( V_{\text{total}}=432 + 3420 = 3852 \)

Answer:

\( V_{\text{trap}} = \boldsymbol{432} \) \( ft^3 \)
\( V_{\text{rect}} = \boldsymbol{3420} \) \( ft^3 \)
\( V_{\text{total}} = \boldsymbol{3852} \) \( ft^3 \)