Function as a mathematical model problem. please help!?

Question

the demand for a certain item is given by the equation x=-10p+500 (0<=p<=50), where x untis of the item will be bought if its price is p pesos

a.) find the mathematical model expressing revenue R as afunction of x

b.) what is the revenue if 10 units were sold

c.) what quantity of x maximizes the revenue

d.) mathematical model expressing R as a function of p

e.) revenue if the price is 30 pesos

f.) what price should be charged to maximize revenue

g.) what is the maximum revenue

please show complete solution. thank you very much for answering!!!!

Answer 1

When you say &quot;where x untis of the item will be bought if its price is p pesos&quot; is the price per item or of the whole lot of x items?

Answer 2

x = -10p + 500 &lt;===&gt; p = -0.1x + 50

a) R = xp = -0.1x^2 + 50x

b) R(10)  = -0.1*100 + 50*10 = 510 pesos

c) R(x) is max when R&#039;(x) = dR/dx = 0

R&#039;(x) = -0.2x + 50. R&#039;(x) = 0 when x = 250.

Answer: the revenue is maximum when 250 units are sold.

d) R = xp = (-10p + 500)p = -10p^2 + 500p

e) R(30) = -10*900 + 500*30 = 6000 pesos

f) R(p) is max when R&#039;(p) = dR/dp = 0

R&#039;(p) = -20p + 500. R&#039;(p) = 0 when p = 25 pesos

g) The revenue is maximum when 250 units are sold at 25 pesos

R (max) = 25 * 250 = 6250 pesos