Part (a)
Answer: x^3 + 12x^2 + 47x + 60
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Work Shown:
Let's say we have
- L = length = x+5
- W = width = x+4
- H = height = x+3
To find the volume of the block, we multiply the length width and height.
So,
volume = (length)*(width)*(height)
V = L*W*H
V = LW*(x+3) .... replace H with (x+3)
V = LW*x + LW*3 .... distribute
V = Lx*(W) + 3L*(W)
V = Lx(x+4) + 3L(x+4) .... replace W with x+4
V = Lx^2 + 4Lx + 3Lx + 12L .... distribute
V = Lx^2 + 7Lx + 12L
V = x^2*(L) + 7x*(L) + 12*(L)
V = x^2*(x+5) + 7x*(x+5) + 12*(x+5) .... replace L with x+5
V = x^3 + 5x^2 + 7x^2 + 35x + 12x + 60 .... distribute
V = x^3 + 12x^2 + 47x + 60
x^3 + 12x^2 + 47x + 60 is the polynomial in standard form that represents the volume of the block.
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Part (b)
Answer: 336 cubic feet
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Work Shown:
We can plug x = 3 into the polynomial we found
V = x^3 + 12x^2 + 47x + 60
V = (3)^3 + 12(3)^2 + 47(3) + 60
V = 27 + 12(9) + 47(3) + 60
V = 27 + 108 + 141 + 60
V = 336
Or we could find the volume like this
V = L*W*H
V = (x+5)*(x+4)*(x+3)
V = (3+5)*(3+4)*(3+3)
V = (8)*(7)*(6)
V = 336
Either way, we get the same volume of 336 cubic feet.
