Expressions Help

eddie

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A sweet manufacturer found that the sales figures for a certain item depended on its selling price. The company’s market research department advised that the maximum number of items that could be sold weekly was 20000 and that the number decreased by 100 for every 1p increase in its price. The total production cost consists of a set-up cost of £200 plus 50p for every item manufactured.

If the price increase one week is p pence, find expressions in terms of p for:

a) Show that the profit £P is given by
P = 250p – 10200 – p2
 
A sweet manufacturer found that the sales figures for a certain item depended on its selling price. The company’s market research department advised that the maximum number of items that could be sold weekly was 20000
At what price? In order to get the answer below, the number sold must be a linear function of the price. This "2000 maximum" must be at the lowest price. What is that price?

and that the number decreased by 100 for every 1p increase in its price. The total production cost consists of a set-up cost of £200 plus 50p for every item manufactured.

If the price increase one week is p pence, find expressions in terms of p for:

a) Show that the profit £P is given by
P = 250p – 10200 – p2
Taking "\(\displaystyle p_0\)" to be the price at which 20000 items can be sold, and "x" to be the number of p less than that, the price would be \(\displaystyle p_0- x\) and the number sold at that price would be \(\displaystyle 200000- 100x\) so the net profit, price of each times the number sold, would be \(\displaystyle (p_0- x)(20000- 100x)\).
The cost is then \(\displaystyle 200+ 50(p_0- x)\). The actual profit is the net profit minus the cost, \(\displaystyle (p_0- x)(20000000- 10x)(200+ 50(p_0- x))\).
 
A sweet manufacturer found that the sales figures for a certain item depended on its selling price. The company’s market research department advised that the maximum number of items that could be sold weekly was 20000 and that the number decreased by 100 for every 1p increase in its price. The total production cost consists of a set-up cost of £200 plus 50p for every item manufactured.

If the price increase one week is p pence, find expressions in terms of p for:

a) Show that the profit £P is given by
P = 250p – 10200 – p2

Something seems strange here, at least IMO. Suppose you initially sell this sweet for s0 and make m0 of the sweets for a initial profit of P0. Now increase the price of the sweet zero pence. Our profit is now actually a loss of £10200?
 
Last edited:
At what price? In order to get the answer below, the number sold must be a linear function of the price. This "2000 maximum" must be at the lowest price. What is that price?


Taking "\(\displaystyle p_0\)" to be the price at which 20000 items can be sold, and "x" to be the number of p less than that, the price would be \(\displaystyle p_0- x\) and the number sold at that price would be \(\displaystyle 200000- 100x\) so the net profit, price of each times the number sold, would be \(\displaystyle (p_0- x)(20000- 100x)\).
The cost is then \(\displaystyle 200+ 50(p_0- x)\). The actual profit is the net profit minus the cost, \(\displaystyle (p_0- x)(20000000- 10x)(200+ 50(p_0- x))\).


In the question there is no figure for my guess is that the price is actually p
 
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