NCEA Calculus 2017 TWO(e - Solution & Explanation

Question

A rectangle is inscribed in a semi-circle of radius

r

, as shown below.

Loading diagram...

Show that the maximum possible area of such a rectangle occurs when

x = \dfrac{r}{\sqrt{2}}

.

*You do not need to prove that your solution gives the maximum area.*

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

Official Answer

Area

=2x\sqrt{r^2-x^2}

A(x)=2x(r^2-x^2)^{\frac{1}{2}}

A'(x)=2(r^2-x^2)^{\frac{1}{2}}+2x\cdot\frac{1}{2}(r^2-x^2)^{-\frac{1}{2}}\cdot(-2x)

=2\sqrt{r^2-x^2}-\dfrac{2x^2}{\sqrt{r^2-x^2}}

A'(x)=0\Rightarrow\sqrt{r^2-x^2}=\dfrac{x^2}{\sqrt{r^2-x^2}}

r^2-x^2=x^2

2x^2=r^2

x^2=\dfrac{r^2}{2}

x=\dfrac{r}{\sqrt{2}}

Grading Criteria

Achievement (u)

-

Merit (r)

Correct derivative.

Excellence T1

Correct solution presented in a correct mathematical manner.

Excellence T2

-

Video Explanation

NCEA Level 3 Calculus Differentiation 2017 NZQA Exam - Worked Answers by infinityplusone(starts at 41:25)

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2017 TWO(e)

Achievement (u)

Merit (r)

Excellence T1

Excellence T2