4.7 Related Rates - Introduction to Calculus

Related Rates

Implicit differentiation also connects quantities that change with time. The ladder example uses the fixed-length relation

x^2+y^2=10^2

Here

x

is the foot's distance from the wall and

y

is the top's height. Both depend on time. Differentiate with respect to the same variable

t

2x\frac{dx}{dt}+2y\frac{dy}{dt}=0

A downward velocity makes

\dfrac{dy}{dt}<0

; a positive

\dfrac{dx}{dt}

means that the foot is moving away from the wall.

\displaystyle x^2+y^2=10^2

Height y in metres8Downward speed in metres per second0.5

\displaystyle \frac{dy}{dt}=-0.5\,\mathrm{m/s},\quad \frac{dx}{dt}=0.666667\,\mathrm{m/s}

\displaystyle \frac{d\theta}{dt}=-0.083333\,\mathrm{rad/s}

The angle decreases as the foot moves away from the wall. The control stays away from the vertical position where these formulas divide by zero.

QUESTION

For a fixed-length ladder,

\displaystyle x>0

\displaystyle y>0

, and

\displaystyle \dfrac{dy}{dt}<0

What is the sign of

\displaystyle \dfrac{dx}{dt}

A $\displaystyle 10\,\mathrm m$ ladder leans against a perpendicular wall

(a) The top slides down at $\displaystyle 0.5\,\mathrm{m/s}$ How fast is the foot moving when the top is $\displaystyle 8\,\mathrm m$ above the ground?

(b) How fast is the angle between the ladder and the ground changing at that moment?

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Related Rates

Implicit differentiation also connects quantities that change with time. The ladder example uses the fixed-length relation

x^2+y^2=10^2

Here

x

is the foot's distance from the wall and

y

is the top's height. Both depend on time. Differentiate with respect to the same variable

t

2x\frac{dx}{dt}+2y\frac{dy}{dt}=0

A downward velocity makes

\dfrac{dy}{dt}<0

; a positive

\dfrac{dx}{dt}

means that the foot is moving away from the wall.

\displaystyle x^2+y^2=10^2

Height y in metres8Downward speed in metres per second0.5

\displaystyle \frac{dy}{dt}=-0.5\,\mathrm{m/s},\quad \frac{dx}{dt}=0.666667\,\mathrm{m/s}

\displaystyle \frac{d\theta}{dt}=-0.083333\,\mathrm{rad/s}

The angle decreases as the foot moves away from the wall. The control stays away from the vertical position where these formulas divide by zero.

QUESTION

For a fixed-length ladder,

\displaystyle x>0

\displaystyle y>0

, and

\displaystyle \dfrac{dy}{dt}<0

What is the sign of

\displaystyle \dfrac{dx}{dt}

A $\displaystyle 10\,\mathrm m$ ladder leans against a perpendicular wall

(a) The top slides down at $\displaystyle 0.5\,\mathrm{m/s}$ How fast is the foot moving when the top is $\displaystyle 8\,\mathrm m$ above the ground?

(b) How fast is the angle between the ladder and the ground changing at that moment?

Related Rates

Related Rates

Connect the ladder's changing quantities

KNOWLEDGE QUIZ

Follow a sliding ladder

Related Rates

Related Rates

Connect the ladder's changing quantities

KNOWLEDGE QUIZ

Follow a sliding ladder