v1 Algebra, STD2 A4 2021 HSC 35

A toy store releases a limited edition LEGO set for $20 each. At this price, 3000 LEGO sets are sold each week and the revenue is `3000 xx 20=$60\ 000`.

The toy store considers increasing the price. For every dollar price increase, 15 fewer LEGO sets will be sold.

If the toy store charges `(20+x)` dollars for each LEGO set, a quadratic model for the revenue raised, `R`, from selling them is

`R=-15x^2+2700x+60\ 000`

What price should be charged per LEGO set to maximise the revenue? (2 marks)

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How many LEGO sets are sold when the revenue is maximised? (2 marks)

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Find the value of the intercept of the parabola with the vertical axis. (1 mark)

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Show Answers Only

a. `$110`

b. `1650`

c. `$60\ 000`

Show Worked Solution

a. `text{Highest revenue}\ (R_text{max})\ text(occurs halfway between)\ \ x= -20 and x=200.`

`text{Midpoint}\ =(-20 + 200)/2 = 90`

`:.\ text(Price of LEGO set for)\ R_text(max)`

`=90 + 20`

`=$110`

b. `text{LEGO sets sold when}\ R_{max}`

`=3000-(90 xx 15)`

`=1650`

c. `ytext(-intercept → find)\ R\ text(when)\ \ x=0:`

`R`	`= -15(0)^2 + 2700(0) + 60\ 000`
	`=$60\ 000`