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.
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Form an equation for $V$V in terms of $x$x.
Give your answer in expanded form.
Solve for the possible value(s) of $x$x that could correspond to the box of largest volume.
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 | $\editable{}$ |
Find the maximum volume of the box.
When exam results were released online, the formula $M\left(t\right)=324+77t^2-t^4$M(t)=324+77t2−t4 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$0≤t≤8.5.
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.
Point $A$A with coordinates $\left(p,q\right)$(p,q) lies on the line $y=3x$y=3x.