Calculus of Trigonometric Functions

NZ Level 8 (NZC) Level 3 (NCEA) [In development]

Optimisation with Trig Functions (sin/cos/tan)

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}$0≤`θ`≤π2.

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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