RI Sec 3 1998 EOY P2 Q16
A solid cuboid has a base of dimensions (8 - x) cm by (x + 4) cm and height 1 cm.
(a)
(i) Show that the volume of the cuboid, V, in cm³, is given by V= 32 + 4x - x².
Hence, show that V = 36 - (x - 2)²
(ii) Find the dimensions of the cuboid if the cuboid has the maximum possible volume.
(iii) Find the dimensions of the cuboid if the cuboid has a volume of 20 cm³.
(b)
(i) Find an expression, in terms of x, for the total surface area of the cuboid.
(ii) If the total surface area is 40 cm², find the volume of the cuboid.
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Answer:
(a)
(i) V = 1 * (8 - x) * (x + 4)
= (8 - x)(x + 4)
= 8x - x² + 32 - 4x
= 32 + 4x - x² (shown)
V= 32 - (x² - 4x) => complete the square
V = 32 - (x² - 4x + (-2)² - (-2)²)
V = 32 - (x -2)² + 4
V = 36 - (x - 2)² (shown)
(ii) Maximum possible value of V = 36
This happens when (x - 2)² = 0, or when x = 2
Hence, the dimensions are
(8 - 2) cm by (2 + 4) cm by 1 cm
6 cm by 6 cm by 1 cm
(iii) thus, 36 - (x - 2)² = 20
(x - 2)² = 16
x - 2 = 4 or x - 2 = -4
x = 6 or x = -2
When x = 6,
dimensions = (8 - 6) cm by (6 + 4) cm by 1 cm
= 2 cm by 10 cm by 1 cm
When x = -2,
dimensions = (8 - (-2) ) cm by (-2 + 4) cm by 1 cm
= 10 cm by 2 cm by 1 cm
(b)
(i) Total surface area = 1 * (8 - x) * 2 + 1 * (x + 4) * 2 + (8 - x)(x + 4) * 2
= 16 - 2x + 2x + 8 + 2 (32 + 4x - x²)
= 88 + 8x - 2x²
(ii) 88 + 8x - 2x² = 40
2x² - 8x - 88 + 40 = 0
x² - 4x - 24 = 0
Using formula, x = (-b ± √(b² - 4ac) ) / 2a,
x ≈ 7.29 or -3.29
When x = 7.29,
volume of cuboid = 36 - (7.29 - 2)² = 8.0159 cm³
When x = -3.29,
volume of cuboid = 36 - (-3.29 - 2)² = 8.0159 cm³
Hence, volume of cuboid = 8.0159 cm³
