Question

what is the volume of the doorstop, as shown? 90 in³ 27 in³ 63 in³ 36 in³

Explanation:

Step1: Calculate the volume of the rectangular part

The formula for the volume of a rectangular prism is \(V = l\times w\times h\). Here, \(l = 3\) in, \(w=2\) in, \(h = 3\) in. So \(V_1=3\times2\times3= 18\) \(in^3\).

Step2: Calculate the volume of the triangular - prism part

The formula for the volume of a triangular prism is \(V=\frac{1}{2}\times b\times h\times l\). The base \(b = 3\) in, height of the triangle \(h = 3\) in, and length \(l=6\) in. So \(V_2=\frac{1}{2}\times3\times3\times6 = 27\) \(in^3\).

Step3: Calculate the total volume

The total volume \(V = V_1+V_2\). Substitute \(V_1 = 18\) \(in^3\) and \(V_2=27\) \(in^3\) into the formula. \(V=18 + 27=45\) \(in^3\). Wait, there is a mistake. Let's re - calculate.

Re - calculation:

Step1: Consider the composite shape as a trapezoidal prism

The formula for the volume of a prism is \(V=B\times h\), where \(B\) is the area of the base and \(h\) is the height (the length of the prism, which is \(2\) in).
The base is a trapezoid. The formula for the area of a trapezoid is \(B=\frac{(a + b)h}{2}\), where \(a=(3 + 6)\) in (the two parallel sides of the trapezoid), \(b = 3\) in (the non - parallel side is not relevant here, wrong previous approach), and the height of the trapezoid (the vertical side) is \(3\) in.
\(B=\frac{(3 + 6+3)\times3}{2}=\frac{(12)\times3}{2}=18\) \(in^2\).
The length of the prism (the distance between the two trapezoidal bases) is \(2\) in.
Using the prism volume formula \(V = B\times l\) (where \(l = 3\) in, wrong again. Wait, correct formula:
The shape can be thought of as a composite of two parts:

1. A rectangular part: \(V_1=3\times2\times3 = 18\) \(in^3\)
2. A triangular prism part: The base of the triangular prism is a triangle with base \(b = 6\) in and height \(h = 3\) in, and the length (the same as the width of the rectangular part) \(l = 2\) in.

The volume of the triangular prism \(V_2=\frac{1}{2}\times6\times3\times2=18\) \(in^3\)
Another way:
The shape is a trapezoidal prism. The formula for the volume of a trapezoidal prism is \(V=\frac{(a + b)h}{2}\times l\), where \(a = 3\) in, \(b=(3 + 6)\) in (the two parallel sides of the trapezoid in the cross - section), \(h = 2\) in (the height of the trapezoid in the cross - section is wrong. Wait, correct:
The cross - section is a trapezoid with parallel sides \(a = 3\) in and \(b=(3 + 6)=9\) in, and the height of the trapezoid (the vertical side) \(h_1=3\) in. The length of the prism (the distance along the non - parallel direction) \(l = 2\) in.
\(V=\frac{(3 + 9)\times3}{2}\times2\)
First, calculate \(\frac{(3 + 9)\times3}{2}=\frac{12\times3}{2}=18\), then \(V = 18\times2=36\) \(in^3\)

Answer:

\(36\space in^3\)