Rates of change Q2 (1 Viewer)

pikachu975 · Feb 12, 2017

In a rectangle, the length is always twice the breadth. If the breadth increases at the constant rate of 1/2 cm/min (the figure remaining a rectangle) find the rate of change of (i) the length (ii) the area (iii) the perimeter at the instant when the breadth is 8 cm.

Thanks for any help!

InteGrand · Feb 12, 2017

pikachu975 said:
In a rectangle, the length is always twice the breadth. If the breadth increases at the constant rate of 1/2 cm/min (the figure remaining a rectangle) find the rate of change of (i) the length (ii) the area (iii) the perimeter at the instant when the breadth is 8 cm.

Thanks for any help!

$\noindent Let the length be $2x$ and the breadth $x$, where $x=x(t)$ is a function of time $t$ with $\dot{x}(t) \equiv 0.5$.$

$\noindent i) Just double the breadth's constant rate of growth, i.e. 1 cm/min.$

$\noindent ii) We have area being $A = 2x\cdot x = 2x^{2}$, so $\dot{A} = 4x\times \dot{x} = 2x = 16$ when $x = 8$ (i.e. Answer is 16 square centimetres per minute).$

$\noindent iii) The perimeter is $P = 6x$, so the answer is $6\times 0.5 = 3$, i.e. 6 cm/min.$

InteGrand · Feb 12, 2017

(Dots refer to time derivatives.)

calamebe · Feb 12, 2017

i) L = 2B, so dL/dt = 2dB/dt = 1 cm/min

ii) A = LB = 2B^2, so dA/dt = 4B*dB/dt = 2B cm/min

iii) P = 2L + 2B = 6B, dP/dt = 6*dB/dt = 3 cm/min, which is independent of B, and so when B=8 the perimeter changes at 3 cm/min.

EDIT: Nevermind look above

pikachu975 · Feb 12, 2017

InteGrand said:
$\noindent Let the length be $2x$ and the breadth $x$, where $x=x(t)$ is a function of time $t$ with $\dot{x}(t) \equiv 0.5$.$

$\noindent i) Just double the breadth's constant rate of growth, i.e. 1 cm/min.$

$\noindent ii) We have area being $A = 2x\cdot x = 2x^{2}$, so $\dot{A} = 4x\times \dot{x} = 2x = 16$ when $x = 8$ (i.e. Answer is 16 square centimetres per minute).$

$\noindent iii) The perimeter is $P = 6x$, so the answer is $6\times 0.5 = 3$, i.e. 6 cm/min.$

Oh wait so does the breadth = 8 cm apply for ii as well? I thought it was only for iii

InteGrand · Feb 12, 2017

pikachu975 said:
Oh wait so does the breadth = 8 cm apply for ii as well? I thought it was only for iii

Whole question was for when breadth is 8 cm. (I.e. Do (i), (ii) and (iii) at the moment when breadth is 8 cm.)

Rates of change Q2 (1 Viewer)

pikachu975

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InteGrand

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calamebe

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