A right triangle is formed by two positive

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

A right triangle is formed by two positive, time–dependent legs

x(t)

and

y(t)

whose lengths are measured in centimeters. At all times the hypotenuse has fixed length

d=15\text{ cm}

. (a) Starting from the geometric relationship among the three sides, obtain an expression that connects

\dfrac{dx}{dt}

and

\dfrac{dy}{dt}

at any time

t

. State all steps in your reasoning clearly. (b) Suppose

x

is increasing at a constant rate of

4\text{ cm/s}

. Without calculating a numerical value, determine the sign (positive, negative, or zero) of

\dfrac{dy}{dt}

and explain the physical significance of that sign in the context of the triangle. (c) Describe qualitatively how the magnitude of

\dfrac{dy}{dt}

changes as

x

increases from

3\text{ cm}

toward

15\text{ cm}

. Support your answer with reasoning based on the relationship you derived in part (a).