A certain type of candle is packaged in boxes that measure 42 cm by 17 cm by 10 cm. The candle company that produced the above packaging has now designed shorter candles. A smaller box will be created by decreasing each dimension of the larger box by the same length. The volume of the smaller box will be at the most 2220 cm3 . What are the maximum dimensions of the smaller box? Solve this problem algebraically. Please use The Factor Theorem.
+128408 Let the decrease in each dimension = x
So....we have that
(42 - x) ( 17 - x) (10 - x) ≤ 2220 simplify
-x^3 + 69x^2 -1304x + 7140 ≤ 2220
-x^3 + 69x^2 -1304x + 4920 ≤ 0 mutiply through by -1 and reverse the inequality sign
x^3 - 69x^2 + 1304x - 4920 ≥ 0 (1)
Note that one possible root that makes the left side = 0 is wnen x = 5
So....using sunthetic divsion, we can find the remaining polynomial
5 [ 1 -69 1304 - 4920 ]
5 -320 4920
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1 -64 984 0
The remaining poynomial is x^2 - 64x + 984
Setting this to 0 and solving for x gives the other two roots of x ≈ 25.675 or x ≈ 38.3246
(1) will be true on these intervals
[5, 25.6754] and [ 38.3246 , inf )
So....when x = 5 the max dimensions of the box are 37, 12 and 5 cm
