New Zealand
Level 8 - NCEA Level 3

# Optimisation with Trig Functions (sin/cos/tan)

## Interactive practice questions

We want to find the maximum value of the area of the rectangle shown below, where the measurements are in centimetres and $0\le\theta\le\frac{\pi}{2}$0θπ2.

a

Order the following steps you would take to find the maximum value of the area.

 $A$A Find the derivative of the area with respect to the angle. $B$B Test each possible maximum value. $C$C Find any possible values of the angle which might maximise the area. $D$D Find an expression for the area in terms of the angle. $E$E Calculate the maximum value of the area.
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b

Find an expression for $x$x in terms of $\theta$θ.

c

Find an expression for $y$y in terms of $\theta$θ.

d

Hence find an expression for the area, $A$A, in terms of $\theta$θ.

e

By differentiating $A$A with respect to $\theta$θ, and then using a CAS calculator, find the possible values of $\theta$θ which could maximise the area.

f

What can we conclude about $A$A when $\theta=\frac{\pi}{4}$θ=π4?

Since $A'$A is positive when $\theta<\frac{\pi}{4}$θ<π4 and positive when $\theta>\frac{\pi}{4}$θ>π4, $\theta=\frac{\pi}{4}$θ=π4 gives the minimum value of $A$A.

A

Since $A'$A is negative when $\theta<\frac{\pi}{4}$θ<π4 and negative when $\theta>\frac{\pi}{4}$θ>π4, $\theta=\frac{\pi}{4}$θ=π4 gives the maximum value of $A$A.

B

Since $A'$A is negative when $\theta<\frac{\pi}{4}$θ<π4 and positive when $\theta>\frac{\pi}{4}$θ>π4, $\theta=\frac{\pi}{4}$θ=π4 gives the minimum value of $A$A.

C

Since $A'$A is positive when $\theta<\frac{\pi}{4}$θ<π4 and negative when $\theta>\frac{\pi}{4}$θ>π4, $\theta=\frac{\pi}{4}$θ=π4 gives the maximum value of $A$A.

D

Since $A'$A is positive when $\theta<\frac{\pi}{4}$θ<π4 and positive when $\theta>\frac{\pi}{4}$θ>π4, $\theta=\frac{\pi}{4}$θ=π4 gives the minimum value of $A$A.

A

Since $A'$A is negative when $\theta<\frac{\pi}{4}$θ<π4 and negative when $\theta>\frac{\pi}{4}$θ>π4, $\theta=\frac{\pi}{4}$θ=π4 gives the maximum value of $A$A.

B

Since $A'$A is negative when $\theta<\frac{\pi}{4}$θ<π4 and positive when $\theta>\frac{\pi}{4}$θ>π4, $\theta=\frac{\pi}{4}$θ=π4 gives the minimum value of $A$A.

C

Since $A'$A is positive when $\theta<\frac{\pi}{4}$θ<π4 and negative when $\theta>\frac{\pi}{4}$θ>π4, $\theta=\frac{\pi}{4}$θ=π4 gives the maximum value of $A$A.

D
g

Hence find the maximum value of the area of the rectangle.

Easy
Approx 6 minutes

Lisa finds a tiny burn in her new rectangular scarf, which measures $60$60 cm by $420$420 cm.

She decides that the best solution is to cut off one corner of the scarf through the burn.

The burn is located $6$6 cm from one edge of the scarf and $14$14 cm from the end of the scarf.

Two corridors meet at a right angle as shown in the diagram. One has a width of $4$4 metres, and the other has a width of $5$5 metres.

The angle $\theta$θ is the angle between the wall and the diagonal of the corridor.

A trough for holding water is formed by taking a piece of sheet metal $210$210 cm wide and folding the $70$70 cm on either end up as shown below.

We want to find the angle $\theta$θ which will maximise the amount of water that the trough can hold.

### Outcomes

#### M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

#### 91578

Apply differentiation methods in solving problems