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Marek used interlocking block toys to construct a model of a hotel. There are 5x² – 2 levels in this hotel. Each level is 2x² + 1 inches high. Which expression shows the total height of this toy hotel?
10x⁴ – 4x² + 3 inches
10x⁴ + x² – 2 inches
7x² – 1 inches
10x⁴ – 2 inches
To find the total height of the hotel, we need to multiply the number of levels by the height of each level.
1. We have 5x² – 2 levels.
2. The height of each level is 2x² + 1 inches.
So, the total height ( H ) can be calculated as follows:
[ H = (text{number of levels}) times (text{height of each level}) ]
[ H = (5x² – 2)(2x² + 1) ]
Now, we can use the distributive property to expand this:
[ H = 5x² cdot 2x² + 5x² cdot 1 – 2 cdot 2x² – 2 cdot 1 ]
So, calculating each term gives us:
– ( 5x² cdot 2x² = 10x^4 )
– ( 5x² cdot 1 = 5x² )
– ( -2 cdot 2x² = -4x² )
– ( -2 cdot 1 = -2 )
Now, combine the like terms:
[ H = 10x^4 + 5x² – 4x² – 2 ]
That simplifies to:
[ H = 10x^4 + (5x² – 4