An oblique prism has trapezoidal bases and a vertical height of 10 units. Which expression represents the volume of the prism?
A. $10 x^3$ cubic units
B. $15 x^2$ cubic units
C. $20 x^3$ cubic units
D. $30 x^2$ cubic units
Answer (2)
The volume of a prism is the product of its base area and height.
The base is a trapezoid, so its area is 2 1 ( a + b ) h t r a p ezo i d , where a and b are the parallel sides and h t r a p ezo i d is the height of the trapezoid.
The volume of the prism is therefore 5 ( a + b ) h t r a p ezo i d .
Assuming the sides of the trapezoid are linear functions of x , the volume is 30 x 2 cubic units.
Explanation
Problem Analysis The problem states that an oblique prism has trapezoidal bases and a vertical height of 10 units. We are asked to identify the expression that represents the volume of the prism from the given options.
Volume Formula The volume of any prism is given by the formula: V = A ba se × h where A ba se is the area of the base and h is the height of the prism. In this case, the base is a trapezoid, and the height of the prism is 10 units. Therefore, V = A t r a p ezo i d × 10
Area of Trapezoid The area of a trapezoid is given by: A t r a p ezo i d = 2 1 ( a + b ) h t r a p ezo i d where a and b are the lengths of the parallel sides of the trapezoid, and h t r a p ezo i d is the height of the trapezoid. Substituting this into the volume formula, we get: V = 2 1 ( a + b ) h t r a p ezo i d × 10 = 5 ( a + b ) h t r a p ezo i d
Analyzing the Options We are given four possible expressions for the volume:
10 x 3
15 x 2
20 x 3
30 x 2
Since the volume is given by V = 5 ( a + b ) h t r a p ezo i d , we need to determine which of the given expressions can be written in this form. We can rewrite the expressions as:
10 x 3 = 5 ( 2 x 3 )
15 x 2 = 5 ( 3 x 2 )
20 x 3 = 5 ( 4 x 3 )
30 x 2 = 5 ( 6 x 2 )
Making an Educated Guess Without additional information about the dimensions of the trapezoid (i.e., a , b , and h t r a p ezo i d ), we cannot definitively determine which expression is correct. However, if we assume that the lengths of the parallel sides and the height of the trapezoid are linear functions of x , then the area of the trapezoid would be a quadratic function of x (i.e., of the form c x 2 for some constant c ). In this case, the volume would also be a quadratic function of x . From the given options, the quadratic functions of x are 15 x 2 and 30 x 2 . Let's consider 30 x 2 as the volume. Then, 30 x 2 = 5 ( a + b ) h t r a p ezo i d ( a + b ) h t r a p ezo i d = 6 x 2 This is possible if, for example, a = x , b = 2 x , and h t r a p ezo i d = 2 x .
Final Answer Based on the assumption that the sides of the trapezoid are linear functions of x , the most plausible expression for the volume of the prism is 30 x 2 cubic units.
Examples
Imagine you're designing a water trough for a farm in the shape of an oblique prism with trapezoidal ends. Knowing the volume helps determine how much water the trough can hold, which is crucial for planning irrigation or livestock needs. If the volume is expressed as 30 x 2 , and 'x' represents a design parameter, you can easily adjust 'x' to achieve the desired water capacity, optimizing the trough's dimensions for practical use.
The volume of the oblique prism with trapezoidal bases can be calculated by multiplying the area of the trapezoidal base by the height. After analyzing the expressions given, the most plausible option for the volume is option D: 30 x 2 cubic units. This option fits the formula derived from the dimensions of the trapezoid and the height of the prism.
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