Algebraic Expressions and Identities: Problem-Solving with Rectangular Prism Geometry
The problem at hand involves dissecting a rectangular prism-like wooden block into smaller pieces and finding the algebraic expression for the area of the top surface of one of those smaller pieces.
Understanding the Dimensions
The rectangular prism has the following dimensions:
- Length: ((8x + 20)) cm
- Width: ((x - 1)) cm
It’s mentioned that this wooden block is cut into 4 equal pieces parallel to the shorter side. This means that the entire length ((8x + 20)) cm is divided into 4 parts.
Calculating the Length of Each Small Piece
Given that the division is parallel to the shorter side, the length of each smaller piece equals:
Now let’s simplify this expression:
- Divide both terms in the numerator by 4:
- ( \frac{8x}{4} = 2x )
- ( \frac{20}{4} = 5 )
Thus, the length of each small piece is (2x + 5) cm.
Finding the Area of the Top Surface of One Small Piece
The area of the top surface of one small piece is found by multiplying its length by its width:
Expanding the Expression
To determine the expression for the area, expand this product using the distributive property (also known as FOIL method for binomials):
-
Multiply:
- (2x \times x = 2x^2)
- (2x \times (-1) = -2x)
- (5 \times x = 5x)
- (5 \times (-1) = -5)
-
Combine all terms to get the final expression:
Conclusion
The algebraic expression for the area of the top surface of one of the smaller, equal pieces is (2x^2 + 3x - 5) square centimeters. The correct choice from the given options would be D) 2x² + 3x - 5.