help out with some questions please

Question

the price p (in dollars) and the quantity x sold of a certain product obey the demand equation x=-15/2p+300 for 0 < p less than or equal to 40

a how many units are sold if the price is 20$

b what price should the company charge to earn at least in $2000 in revenue

Answer 1

the price p (in dollars) and the quantity x sold of a certain product obey the demand equation x = - 15/2 p + 300 for 0 < p ≤ 40

a how many units are sold if the price is 20$

If p = 20, then

x = - 15/2 p + 300 = - (15/2) · 20 + 300 = - 150 + 300 = 150

150 units

b what price should the company charge to earn at least in $2000 in revenue

Revenue = (price of one unit) · (number of units) = x · p = 300 p - 7.5 p² = 2000

to find p we need to solve the equation

300 p - 7.5 p² = 2000, which can be rearranged as

7.5 p² - 300 p + 2000 = 0

p² - 40 p + 800 / 3 = 0

p = 20 ± √400 - 800/3 = 20 (1 ± √ 1/3)

To earn at least $2000 in revenue, the company should charge

20 (1 - √ 1/3) < p < 20 (1 + √ 1/3) or $8.46 ≤ p ≤ $31.53

Let us check our work

At p = 8.46, x = - 15/2 p + 300 = 236.55 ≈ 237 and the revenue xp = 2005.02 ≥ 2000

At p = 20, x = - 15/2 p + 300 = 150 and the revenue xp = 3000 ≥ 2000

At p = 31.53, x = - 15/2 p + 300 = 63.53 ≈ 64 and the revenue xp = 2017.92 ≥ 2000