Question

How to solve extended response questions using calculus?

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

Answer 1

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`