Optimisation AS Level!

Could someone explain to me how to do these two optimisation questions.

1)
An open toppped box has height h cm and square base of side xcm.

The box has capacity Vcm^3. The area of its external surface consisting of its horizontal base and four vertical faces is Acm^2

a) Find expressions for V and A in terms of x and h.

b) It is given that A = 3000

i) Show that V=750x - 1/4x^3

ii) Find the positive value of x for which DV/dx = 0 giving your answer in surd form.

2)
Small trays are to be made from rectangular pieces of card. Each piece of card is 8cm by 5cm and the tray is formed by removing squaress of side xcm from each corner and folding the remaining card along the dashed lines.

a) Explain why 0 < x < 2.5

b) Show that the volume Vcm^3, of a tray is given by

V = 4x^3 - 26x^2 + 40x

c) Find the value of x for which dV/dx = 0

d) Calculate the greatest possible volume of a tray

Reply 1

mikesgt2

1)
a) You should be able to do these.
b)
(i) Substitute 3000 into the equation you found for A in part a) in order to find h in terms of x. Then substitute this expression for h into your equation for V. Rearrange to give the required result.
(ii) Differentiate V wrt x, set to zero, solve quadratic.

Reply 2

GH

13

mikesgt2

1)
a) You should be able to do these.
b)
(i) Substitute 3000 into the equation you found for A in part a) in order to find h in terms of x. Then substitute this expression for h into your equation for V. Rearrange to give the required result.
(ii) Differentiate V wrt x, set to zero, solve quadratic.