C2 help

I'm really struggling with the following C2 level question and so any help would be really appreciated.

A rectangular sheet of metal measures 50 cm by 40 cm. Squares of side x cm are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray.

(a) Show that the volume, V cm^3 of the tray is given by V = 4x(x^2 - 45x + 500).

Okay, so I got this part of the question.

(b) State the range of possible values of x.

The back of the textbook gives this answer as x is greater than or equal to 0 and less than or equal to 20, but I'm not sure why this is the case.

(c) Find the value of x for which V is a maximum.

(d) Hence find the minimum value of V.

(e) Justify that the value of V you have found in part (c) is a maximum.

Thank you.

Cathy

Reply 1

nota bene

For b) by logical reasoning you should get to the answer...
x is not less than 0 since we can't have a negative length of the side of the squares - right?
the rectangle is 40 cm, and if you are to cut out two squares you can just get the sidelength 20cm.

Draw a diagram and it should be very clear :smile:

edit:For c) maybe differentiate? and put the derivative =0 solve the quadratic and take the 'right' value of x. To justify this (e) I assumed they want you to look at the second derivative.

Reply 2

CathyLou

Okay. Thanks very much for your help. :smile:

Cathy

Reply 3

CathyLou

nota bene

edit:For c) maybe differentiate? and put the derivative =0 solve the quadratic and take the 'right' value of x. To justify this (e) I assumed they want you to look at the second derivative.

For (c) I got that f'(x) = 500 - 90x + 3x^2 but I don't know how to find the solutions as I'm sure that it won't factorise.

Could someone please help?

Thank you.

Cathy

Reply 4

nota bene

CathyLou

For (c) I got that f'(x) = 500 - 90x + 3x^2 but I don't know how to find the solutions as I'm sure that it won't factorise.

Could someone please help?

Thank you.

Cathy

I might have given you bad advice...haven't tried it myself, but if it doesn't facvtorise...then use the quadratic formula
latex]
edit: and I don't agree with your differentiation... I get

12x^2-360x+2000

Reply 5

steve2005

CathyLou

For (c) I got that f'(x) = 500 - 90x + 3x^2 but I don't know how to find the solutions as I'm sure that it won't factorise.

Could someone please help?

Thank you.

Cathy

Complete the square or use the formula. I like completing the square.