Answer:
(a) h = 160 - 5·x
(b) V = 4·x² × h = 4·x² × (160 - 5·x) = 640·x² - 20·x³ = 20·x²·(32 - x)
∴ V = 20·x²·(32 - x)
Step-by-step explanation:
(a) The given dimensions of the cuboid (rectangular prism) are;
'4·x' meters by 'x' meters by 'h' meters
Let the 4·x meters represent the length, 'l', of the cuboid, let the x meters represent the width, 'w', of the cuboid, and let the h meters represent the height 'h' of the cuboid
Therefore, we are given that the cuboid is a wire cage, with the total length of the sides (edges) of the cuboid equal to 640 meters
Therefore, the sum of the edges are;
Top(4·x + x + 4·x + x) + Side(h + h + h + h) + Bottom(4·x + x + 4·x + x) = 640
20·x + 4·h = 640
∴ h = (640 - 20·x)/4 = 160 - 5·x
h = 160 - 5·x
(b) The volume of a cuboid, V, is given as follows;
V = The area of the base of the cuboid × The height of the cuboid
The area of the base of the cuboid = l × w = 4·x × x = 4·x²
The height of the cuboid = h = 160 - 5·x
∴ V = 4·x² × (160 - 5·x) = 640·x² - 20·x³
V = 640·x² - 20·x³ = 20·x²·(32 - x)
∴ V = 20·x²·(32 - x) QED.
