NCEA Calculus 2013 TWO(e - Solution & Explanation

Question

A copper sheet of width 24 cm is folded, as shown, to make spouting.

Cross-section:

Loading diagram...

Find angle

\theta

which gives the maximum cross-sectional area.

*You do not need to prove that you have found a maximum.*

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

Official Answer

A(\theta)=64\sin\theta+64\sin\theta\cos\theta

OR

A(\theta)=64\sin\theta+32\sin2\theta

A'(\theta)=64\cos\theta+64\cos^2\theta-64\sin^2\theta

OR

A'(\theta)=64\cos\theta+64\cos2\theta

\quad\quad=64\cos\theta+64\cos^2\theta-64(1-\cos^2\theta)

\quad\quad=64(2\cos^2\theta+\cos\theta-1)

Minimum when

A'(\theta)=0

2\cos^2\theta+\cos\theta-1=0

(2\cos\theta-1)(\cos\theta+1)=0

Or

\cos\theta=\frac{1}{2}

or

\cos\theta=-1

(NO)

\theta=60^{\circ}

or

\theta=\frac{\pi}{3}

Grading Criteria

Achievement (u)

-

Merit (r)

Correct derivative.

Excellence T1

Correct solution with correct derivatives.

Excellence T2

-

2013 TWO(e)

Achievement (u)

Merit (r)

Excellence T1

Excellence T2