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junior college 1 | H1 Maths
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Date Posted: 2 years ago
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a)Surface area of cube , A = total area of its six square faces = 6 × x × x = 6x²
When the total surface area is 216 cm²,
6x² = 216
x² = 36
x = √36 = 6 (only consider the positive square root since length is a positive value)
Volume of cube, V = x³ cm³ = 6³ cm³ = 216 cm³ (shown)
b) Rate at which volume is increasing → dV/dt
Given : Surface area is increasing at 16 cm²/s (a constant)
→ dA/dt = 16
Now A = 6x² , so dA/dx = 12x
Then, realise that dA/dt = dA/dx × dx/dt
16 = 12x dx/dt
dx/dt = 16/12x = 4/3x
Next, V = x³ so dV/dx = 3x²
dV/dt = dV/dx × dx/dt
= 3x² (4/3x)
= 4x
Lastly, Since x = 6 when the total surface area is 216 cm²,
dV/dt = 4(6) = 24
The volume is increasing at 24cm³/s
When the total surface area is 216 cm²,
6x² = 216
x² = 36
x = √36 = 6 (only consider the positive square root since length is a positive value)
Volume of cube, V = x³ cm³ = 6³ cm³ = 216 cm³ (shown)
b) Rate at which volume is increasing → dV/dt
Given : Surface area is increasing at 16 cm²/s (a constant)
→ dA/dt = 16
Now A = 6x² , so dA/dx = 12x
Then, realise that dA/dt = dA/dx × dx/dt
16 = 12x dx/dt
dx/dt = 16/12x = 4/3x
Next, V = x³ so dV/dx = 3x²
dV/dt = dV/dx × dx/dt
= 3x² (4/3x)
= 4x
Lastly, Since x = 6 when the total surface area is 216 cm²,
dV/dt = 4(6) = 24
The volume is increasing at 24cm³/s
Alternative working for b)
A = 6x², dA/dx = 12x
V = x³, dV/dx = 3x²
dA/dt = 16 (given)
dV/dt = dV/dx × dx/dA × dA/dt
= dV/dx ÷ dA/dx × dA/dt
= 3x² ÷ 12x × 16
= 4x
= 4(6) (since x = 6 when V and A = 216)
= 24
Second alternative : expressing V in terms of A
(not as easy to understand)
A = 6x²
V = x³
= (x²)³⁄²
= (6x² / 6)³⁄²
= (6x²)³⁄² / 6³⁄²
= A³⁄² / 6³⁄²
dV/dA = (3/2 A¹⁄²) / 6³⁄²
= 3(6x²)¹⁄² / 2(6³⁄²)
= 3(6¹⁄² x) / 2(6³⁄²)
= 3x / 2(6)
= 3x/12 = x/4
dV/dt = dV/dA × dA/dt
= x/4 × 16
= 4x
= 4(6)
= 24
A = 6x², dA/dx = 12x
V = x³, dV/dx = 3x²
dA/dt = 16 (given)
dV/dt = dV/dx × dx/dA × dA/dt
= dV/dx ÷ dA/dx × dA/dt
= 3x² ÷ 12x × 16
= 4x
= 4(6) (since x = 6 when V and A = 216)
= 24
Second alternative : expressing V in terms of A
(not as easy to understand)
A = 6x²
V = x³
= (x²)³⁄²
= (6x² / 6)³⁄²
= (6x²)³⁄² / 6³⁄²
= A³⁄² / 6³⁄²
dV/dA = (3/2 A¹⁄²) / 6³⁄²
= 3(6x²)¹⁄² / 2(6³⁄²)
= 3(6¹⁄² x) / 2(6³⁄²)
= 3x / 2(6)
= 3x/12 = x/4
dV/dt = dV/dA × dA/dt
= x/4 × 16
= 4x
= 4(6)
= 24
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