NZ Level 8 (NZC) Level 3 (NCEA) [In development]
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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.
Step 1 $\editable{}$
Step 2 $\editable{}$
Step 3 $\editable{}$
Step 4 $\editable{}$
Step 5 $\editable{}$
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
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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

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