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6.08 Harder optimisation problems

Interactive practice questions

A box without cover is to be constructed from a rectangular cardboard that measures $90$90 cm by $42$42 cm by cutting out four identical square corners of the cardboard and folding up the sides.

Let $x$x be the height of the box, and $V$V the volume of the box.

a

Form an equation for $V$V in terms of $x$x.

Give your answer in expanded form.

b

Solve for the possible value(s) of $x$x that could correspond to the box of largest volume.

c

Complete the table to prove that when $x=9$x=9 cm the volume of the box is maximised.

$x$x $8$8 $9$9 $10$10
$\frac{dV}{dx}$dVdx $\editable{}$ $0$0 $\editable{}$
d

Find the maximum volume of the box.

Medium
10min

When exam results were released online, the formula $M\left(t\right)=324+77t^2-t^4$M(t)=324+77t2t4 was used to approximate the number of students, $M$M, logged onto the site at any time over the first $8.5$8.5 hours, $0\le t\le8.5$0t8.5.

Medium
13min

Charlie is fencing off a rectangular section of his backyard to use for a vegetable garden. He uses the existing back wall and has $24$24 metres of fencing to create the other three sides. He wishes to make the area for vegetables as large as possible.

Medium
6min

Point $A$A with coordinates $\left(p,q\right)$(p,q) lies on the line $y=3x$y=3x.

Medium
11min
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Outcomes

MA12-3

applies calculus techniques to model and solve problems

MA12-6

applies appropriate differentiation methods to solve problems

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