Question

surface area & volume
what is the volume of the prism shown below?
(prism image with dimensions 5 cm, 4 cm, 6 cm, 2 cm)
\boxed{\text{cm}^3}

Explanation:

Step1: Identify the base shape

The prism has a trapezoidal base? Wait, no, looking at the dimensions: the front face is a trapezoid? Wait, no, actually, the base of the prism (the two - dimensional face that is repeated) seems to be a trapezoid? Wait, no, let's check the formula for the volume of a prism. The volume of a prism is given by \( V=B\times h \), where \( B \) is the area of the base and \( h \) is the height (the distance between the two bases).

Looking at the diagram, the base of the prism (the front face) is a trapezoid? Wait, no, the dimensions are 4 cm, 5 cm, 2 cm, and 6 cm. Wait, actually, the base is a trapezoid with bases 2 cm and 5 cm? No, wait, maybe it's a trapezoidal prism where the two parallel sides of the trapezoid are 2 cm and 5 cm, the height of the trapezoid (the distance between these two parallel sides) is 4 cm, and the length of the prism (the distance along the direction perpendicular to the trapezoidal base) is 6 cm.

Wait, the formula for the area of a trapezoid is \( B=\frac{(a + b)}{2}\times h_{trapezoid} \), where \( a \) and \( b \) are the lengths of the two parallel sides, and \( h_{trapezoid} \) is the height of the trapezoid. Then the volume of the prism is \( V = B\times l \), where \( l \) is the length of the prism (the distance between the two trapezoidal bases).

So, \( a = 2\) cm, \( b = 5\) cm, \( h_{trapezoid}=4\) cm, and \( l = 6\) cm.

Step2: Calculate the area of the trapezoidal base

First, calculate the area of the trapezoid: \( B=\frac{(2 + 5)}{2}\times4\). Let's compute that: \( \frac{7}{2}\times4=7\times2 = 14\) \( cm^{2}\).

Step3: Calculate the volume of the prism

Now, the volume of the prism is \( V=B\times l \), where \( l = 6\) cm. So \( V=14\times6 = 84\) \( cm^{3}\). Wait, wait, maybe I made a mistake. Wait, another way: maybe the base is a triangle? Wait, no, the diagram shows a prism with a front face that has a height of 4 cm, a top side of 5 cm, a bottom side of 2 cm, and the length is 6 cm. Wait, maybe the cross - section is a trapezoid. Wait, let's re - check.

Wait, alternatively, maybe the base is a trapezoid with bases 4 cm and... No, maybe I misidentified the base. Wait, the formula for the volume of a prism is also equal to the area of the base times the length. Let's look at the dimensions again. The prism has a length of 6 cm, and the cross - sectional area (the area of the face perpendicular to the length) can be calculated as the area of a trapezoid with bases 2 cm and 5 cm and height 4 cm? Wait, no, maybe the cross - section is a triangle? Wait, no, the side is 4 cm, 5 cm, 2 cm. Wait, maybe it's a trapezoidal prism where the two parallel sides are 2 and 5, the height of the trapezoid is 4, and the length is 6.

Wait, let's recalculate the area of the trapezoid: \( A=\frac{(a + b)h}{2}\), where \( a = 2\), \( b = 5\), \( h = 4\). So \( A=\frac{(2 + 5)\times4}{2}=\frac{7\times4}{2}=14\). Then volume \( V=A\times6 = 84\). Wait, but let's check another approach. Suppose the base is a triangle? No, the sides are 2, 4, 5? No, that doesn't make sense. Wait, maybe the prism is a trapezoidal prism, and the calculation is correct.

Wait, another way: maybe the front face is a trapezoid with bases 2 and 5, height 4, and the length is 6. So area of trapezoid is \( (2 + 5)/2\times4=14\), volume is \( 14\times6 = 84\).

Answer:

\( 84 \)