Question

find the volume of the oblique rectangular prism below. round your answer to the nearest tenth if necessary.

Explanation:

Step1: Find the height of the triangular face

We have a right triangle with angle \(58^\circ\) and adjacent side \(6\). We can find the height (opposite side) using \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) or \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)? Wait, no, actually, the height of the parallelogram (the base face) can be found using the side \(6\) and angle \(58^\circ\). Wait, the base of the oblique prism: the base is a parallelogram with sides \(12\) and \(6\)? Wait, no, looking at the diagram, the dimensions are \(6\), \(12\), \(10\), and angle \(58^\circ\). Wait, the volume of an oblique prism is the area of the base times the height (the distance between the bases, which is the length of the edge perpendicular to the base). Wait, actually, for an oblique rectangular prism (wait, no, it's an oblique prism with a rectangular base? Wait, no, the base is a parallelogram? Wait, no, the diagram shows a right angle at the bottom left, and angle \(58^\circ\). Wait, maybe the base is a parallelogram with sides \(12\) and \(6\), and the height of the parallelogram is \(6\times\sin(58^\circ)\)? Wait, no, let's re-examine.

Wait, the volume of a prism is \(V = \text{Area of base} \times \text{height (the length of the edge perpendicular to the base)}\). Wait, in this case, the base is a parallelogram? Wait, no, the diagram has a right angle, and then angle \(58^\circ\). Wait, maybe the base is a rectangle? No, it's oblique. Wait, the given dimensions: \(6\), \(12\), \(10\), and angle \(58^\circ\). Let's see: the area of the base (the parallelogram) is base (12) times height. The height of the parallelogram can be found from the right triangle with side \(6\) and angle \(58^\circ\). Wait, the side \(6\) is adjacent to angle \(58^\circ\), and the height of the parallelogram is \(6\times\sin(58^\circ)\)? Wait, no, \(\sin(58^\circ)=\frac{\text{height}}{6}\), so height \(h = 6\times\sin(58^\circ)\). Wait, no, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), but here, if the angle is \(58^\circ\), and the adjacent side is \(6\), then the height (opposite side) is \(6\times\sin(58^\circ)\)? Wait, no, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), so \(\tan(58^\circ)=\frac{\text{height}}{6}\), so height \(h = 6\times\tan(58^\circ)\)? Wait, no, maybe I got the angle wrong. Wait, the right angle is at the bottom left, so the triangle is a right triangle with legs \(6\) (adjacent) and \(h\) (opposite), and angle \(58^\circ\) between the adjacent side and the hypotenuse? No, angle \(58^\circ\) is between the side of length \(6\) and the base of length \(12\)? Wait, maybe the base of the parallelogram is \(12\), and the height of the parallelogram is \(6\times\sin(58^\circ)\)? Wait, no, let's calculate:

Wait, the volume of the prism: the base is a parallelogram with area \(A = \text{base} \times \text{height of parallelogram}\). The base of the parallelogram is \(12\), and the height of the parallelogram is \(6\times\sin(58^\circ)\)? Wait, no, \(6\) is a side, and angle \(58^\circ\), so the height of the parallelogram (the distance between the two sides of length \(12\)) is \(6\times\sin(58^\circ)\). Then, the area of the base (parallelogram) is \(12 \times (6\times\sin(58^\circ))\)? No, that doesn't seem right. Wait, maybe the base is a rectangle? No, it's oblique. Wait, another approach: the volume of an oblique prism is equal to the area of the right - angled base (if we consider the projection) times the length of the edge. Wait, maybe the base is a parallelogram with sides \(10\) and \(12\), and the height of the parallelogram is \(6\times\sin(58^\circ)\)? No, let's look at the dimensions again. The diagram has \(6\), \(12\), \(10\), and angle \(58^\circ\). Wait, the length of the prism (the edge perpendicular to the base) is \(10\). Wait, maybe the base is a parallelogram with area \(A = 12\times(6\times\sin(58^\circ))\)? No, wait, the right triangle: the side \(6\) is one leg, angle \(58^\circ\), so the height of the parallelogram (the base) is \(6\times\sin(58^\circ)\)? Wait, \(\sin(58^\circ)\approx0.8480\), so \(6\times\sin(58^\circ)\approx6\times0.8480 = 5.088\). Then the area of the base (parallelogram) is base (\(12\)) times height (\(5.088\))? No, that doesn't seem right. Wait, maybe the base is a rectangle with length \(12\) and width \(6\), but it's oblique, so the height of the prism (the distance between the two rectangular bases) is \(10\times\sin(58^\circ)\)? No, I'm confused. Wait, let's recall: the volume of any prism is \(V=\text{Area of the base}\times\text{height}\), where "height" is the perpendicular distance between the two bases.

Looking at the diagram: the base is a parallelogram with sides \(12\) and \(6\), and the angle between them is \(58^\circ\)? No, the right angle is at the bottom left, so the triangle is a right triangle with legs \(6\) (adjacent) and \(h\) (opposite), and angle \(58^\circ\). Wait, the side \(6\) is adjacent to angle \(58^\circ\), so the height of the parallelogram (the base) is \(6\times\sin(58^\circ)\). Then the area of the base (parallelogram) is \(12\times(6\times\sin(58^\circ))\)? No, that would be if the base is a parallelogram with base \(12\) and height \(6\sin58^\circ\). Then the volume of the prism is the area of the base times the length of the edge perpendicular to the base, which is \(10\). Wait, that makes sense. So:

Step1: Calculate the height of the parallelogram base

The height \(h\) of the parallelogram (base) is found using the right triangle with side \(6\) and angle \(58^\circ\). Using \(\sin(58^\circ)=\frac{h}{6}\), so \(h = 6\times\sin(58^\circ)\).

\(\sin(58^\circ)\approx0.8480\), so \(h\approx6\times0.8480 = 5.088\).

Step2: Calculate the area of the base (parallelogram)

The area of a parallelogram is \(A=\text{base}\times\text{height}\). Here, the base of the parallelogram is \(12\), and the height is \(h\approx5.088\). So \(A = 12\times5.088\approx61.056\).

Step3: Calculate the volume of the prism

The volume of the prism is \(V = A\times\text{length of the edge perpendicular to the base}\). The edge perpendicular to the base is \(10\) (from the diagram: the side labeled \(10\) is perpendicular to the base? Wait, maybe I got the base wrong. Wait, maybe the base is a rectangle with length \(12\) and width \(10\), and the height of the prism (the distance between the two rectangles) is \(6\times\sin(58^\circ)\). Let's try that.

If the base is a rectangle with area \(12\times10 = 120\), and the height (distance between the bases) is \(6\times\sin(58^\circ)\approx5.088\), then volume \(V = 120\times5.088\approx610.56\). Wait, that seems more reasonable. Wait, let's re - examine the diagram: the dimensions are \(6\), \(12\), \(10\), and angle \(58^\circ\). The right angle is at the bottom left, so the triangle is a right triangle with legs \(6\) (adjacent) and \(h\) (opposite), angle \(58^\circ\). So \(h = 6\times\sin(58^\circ)\) (since \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)? No, wait, \(\sin(58^\circ)=\frac{h}{6}\) implies \(h = 6\times\sin(58^\circ)\), but actually, in a right triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), but here, if the angle is \(58^\circ\), and the hypotenuse is \(6\)? No, that can't be. Wait, maybe it's \(\cos(58^\circ)=\frac{\text{adjacent}}{\text{hypotenuse}}\), but no. Wait, I think I made a mistake. Let's use \(\sin(58^\circ)=\frac{\text{height}}{6}\), so height \(h = 6\times\sin(58^\circ)\approx6\times0.8480 = 5.088\). Then, the volume of the prism: the base is a rectangle with length \(12\) and width \(10\), and the height (the distance between the two rectangles) is \(h\approx5.088\). So \(V=12\times10\times5.088 = 120\times5.088\approx610.6\). Wait, that makes sense. Let's check again.

Wait, the formula for the volume of an oblique prism is the same as a right prism: \(V=\text{Area of base}\times\text{height}\), where height is the perpendicular distance between the bases. In this case, the base is a rectangle with length \(12\) and width \(10\), so area of base \(A = 12\times10=120\). The height (perpendicular distance between the bases) is the length of the side \(6\) projected along the direction perpendicular to the base, which is \(6\times\sin(58^\circ)\) (since the angle between the side \(6\) and the base is \(58^\circ\), so the perpendicular component is \(6\times\sin(58^\circ)\)).

So:

Step1: Calculate the height of the prism (perpendicular distance between bases)

\(h = 6\times\sin(58^\circ)\)

\(\sin(58^\circ)\approx0.8480\), so \(h\approx6\times0.8480 = 5.088\)

Step2: Calculate the area of the base (rectangle)

\(A=12\times10 = 120\)

Step3: Calculate the volume of the prism

\(V = A\times h=120\times5.088\approx610.6\)

Answer:

\(\approx 610.6\)