A cube and a cuboid have the same volume. The dimensions of the cuboid are in the ratio 1:2: 4. If the difference between the cost of painting the cuboid and cube (whole surface area) at the rate of Rs 5 per m2 is Rs 80. Find their volumes.
Answers
Given :-
A cube and a cuboid have the same volume. The dimensions of the cuboid are in the ratio 1:2: 4. If the difference between the cost of painting the cuboid and cube (whole surface area) at the rate of Rs 5 per m2 is Rs 80.
To Find :-
Volume
Solution :-
Let the ratio be x, 2x and 4x
Volume = x × 2x × 4x
Volume = 8x³
Surface Area of cuboid = 2(lb + bh + lh)
Surface Area of cuboid = 2(x × 2x + 2x × 4x + x × 4x)
Surface Area of cuboid = 2(2x² + 8x² + 4x²)
Surface Area of cuboid = 2(14x²)
Surface Area of cuboid = 28x²
Volume of cuboid = Volume of cube
Volume of cube = 8x³
Edge = ∛(8x)³
Edge = 2x
Surface Area of cube = 6a²
Surface Area of cube = 6(2x)²
Surface Area of cube = 6(4x²)
Surface Area of cube = 24x²
_________________________________________________________Cost of painting the cuboid = 5 × Surface Area
Cost of painting the cuboid = 5 × 28x²
Cost of painting the cuboid = 140x²
Cost of painting the cube = 5 × Surface area
Cost of painting the cube = 5 × 24x²
Cost of painting the cube = 120x²
Difference = 80
140x² - 120x² = 80
20x² = 80
x² = 80/20
x² = 4
x = √4
x = 2
Volume of cube = 8x³
Volume = 8 × 2 × 2 × 2
Volume = 64 m²
Volume of cuboid = Volume of cube
Volume of cuboid = 64 m²
Answer:
Let each side of cube be ′a′ m.
Ratio of sides of cuboid is 1:2:4.
Let the sides be ′x′ m
Volume of cube= volume of cuboid
∴a3=x×2x×4x=8x3a3=8x3⇒a=2xA/Q[6a2−2(lb+lh+bh)]×5=80{6×(2x)2+2[2x2+8x2+4x2]}×5=80⇒[−24x2+2(14x2)]×5=80⇒(−24x2+28x2)×5=80⇒4x2×5=80⇒20x2=80⇒x2=4⇒x=2∴a=2×2=4
∴ Volume of cube =(4)3=64m
Step-by-step explanation: