NCEA Calculus 2014 TWO(e - Solution & Explanation

Question

A rectangle is drawn inside a right angled triangle, as shown in the diagram below.

Loading diagram...

Point B moves along the base of the triangle AC, beginning at point A, at a constant speed of 3 cm s

^{-1}

.

At what rate is the area of the rectangle changing when point B is 20 cm from point A?

*Show any derivatives that you need to find when solving this problem.*

Official Answer

Loading diagram...

\tan 30=\dfrac{h}{y}

h=y\tan 30

\cos 30=\dfrac{y+b}{50}

y+b=50\cos 30

b=50\cos 30-y

Area

=

base

\times

height

A=(50\cos 30-y)(y\tan 30)

=50y\sin 30-y^2\tan 30

=25y-\dfrac{y^2}{\sqrt{3}}

\dfrac{dA}{dy}=25-\dfrac{2y}{\sqrt{3}}

At

y=20

\dfrac{dA}{dy}=25-\dfrac{40}{\sqrt{3}}

\dfrac{dA}{dt}=\dfrac{dA}{dy}\times\dfrac{dy}{dt}

=\left(25-\dfrac{40}{\sqrt{3}}\right)\times 3

=5.72\;\text{cm}^2\text{s}^{-1}

Grading Criteria

Achievement (u)

Correct derivative for an incorrect but relevant expression for $A$ .

Merit (r)

A correct expression for $\dfrac{dA}{dy}$

Excellence T1

A correct solution. Units not Required.

Excellence T2

-

2014 TWO(e)

Achievement (u)

Merit (r)

Excellence T1

Excellence T2