How to solve extended response questions using calculus?

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Can someone please explain to me how to do question 22f and 24 a? Thanks!

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1 Answer
Ryan
Oct 5, 2017

22f: Maximum strength is #74005# whatever units.
24a: #V(x) = 2x*x*(90-3x)# ... needs explanation.

Explanation:

Ah, classic Units 1&2 Methods question (though, if I remember correctly these were both questions I could not be bothered doing before the test on this topic).

Assuming that you have worked through parts a to e on question 22:

#2x + 2y = 120#
#:. y=60-x#.
#:. S = 5x^2(60-x)#
# :. S = 300x^2-5x^3, x in (0,60)#

#:. (dS)/(dx) = 600x-15x^2#
#:.(dS)/(dx) = -15x(x-40)#

Obviously the maximum value is at #x=40#.

But this is an absolute maximum question across a given domain, i.e., now, according to part f:

#S:(0,19] rarr RR, S(x) = 300x^2-5x^3#

You can see that because of the shape of a negative cubic, the graph increases as #x# approaches 19.

#S(19) = 74005#, which your calculator will very nicely tell you.

Onwards to question 24:

A rectangular prism has 12 sides, 4 length plus 4 height plus 4 breadth, i.e.:

#4l + 4b + 4h = 360#

Since #l = 2b#

#4h = 360 - 12b#

Rename #b# as #x#, then:
#h = 90 - 3x#

Since #V=l*b*h#

#V = 2x*x*(90-3x) = 180x^2 - 6x^3#

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