When the quantity of a good sold is x, the price for which each unit sells is p

Question

When the quantity of a good sold is x, the price for which each unit sells is p = 1200 - 3x. (a) For which range of values of the quantity of the good does this price relationship make sense? (b) Revenue is quantity times price. For what quantities of this good is the revenue 90.000? 110.000? 130,000? Suppose t hat the supply for raspberries relates price (p, measured in dollars per pound) to quantity (q, the number of pounds produced) according to p = 1 + 0.002q - 0.0000001q^2. (a) Find the value of q at the vertex of this quadratic, (b) Explain why this description of supply only makes sense for a quantity below this vertex value.

Explanation / Answer

15 (a). The range of values of the quantity for which the given price relationship makes sense is when p > 0 i.e. when 1200 -3x > 0 or when 1200 > 3x i.e when x < 1200/3 i.e. when x < 400.

(b) When the revenue is 90000, px = 90000 so that p = 90000/x. However, since p = 1200 – 3x, we have 90000/x = 1200 -3x or, 90000 = 1200x -3x2 or, 3x2 -1200x + 90000 = 0 or, x2-400x + 30000 = 0. On using the quadratic formula, we have x = [ -(-400) ± { (-400)2-4*1*30000}]/2*1 = [ 400 ± 160000- 120000]/2 = (400± 40000)/2 (400± 200)/2 = 600/2 i.e. 300 or, 200/2 = 100. Thus, the revenue is 90000 when the quantity sold is either 300 or,100. Similarly, when the revenue is 110000,we have 3x2-1200x+110000 = 0. Then, on using the quadratic formula, we have x = [ -(-1200) ± { (-1200)2-4*3*110000}]/2*3 =(1200 ± ( 1440000 – 1320000)/6 = (1200± 120000)/6 = (1200± 346.41)/6 = 1546.61/6 i.e. x = 257.77 = 258 ( on rounding off to the nearest whole number) or, x = 853.59/6 = 142.27 = 142 ( on rounding off to the nearest whole number). Thus, the revenue is110000 when the quantity sold is either 258 or,142. In a similar way, when the revenue is 130000,we have 3x2-1200x+130000 = 0. Then, on using the quadratic formula, we have x = [ -(-1200) ± { (-1200)2-4*3*130000}]/2*3 = (1200 ± ( 1440000 – 1560000)/6 . Since 1560000 is greater than 1440000, x is not a real number. Thus, the revenue will never be 130000.

16. (a) We have p = 1+0.002q – 0.0000001q2 or, p = -0.0000001(q2 – 20000q) = -0.0000001(q2 – 20000q + 100000000) + 10 = -0.0000001(q-10000)2 +10. This is the equation of a parabola opening downwards with vertex at the point (10000,10). Here q = 10000.

(b) When q = 10000, p = 1 + 0.002*10000 -0.0000001*(10000)2 = 1+ 20 – 10 =11 . If q > 10000, p will be less than 10 as the vetex is the highest point in a parabola opening downwards. This means that the price is highest q = 10000 and if q decreases below 10000 or increases beyond 10000, the price will fall. Therefore the supply at 10000 units and price of $11 per pound makes sense.

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