A price p (in dollars) and demand x for a product are related by 2x^2 - 3xp + 50p^2 = 20800?
p and x are functions of t (time), dp/dt and dx/gt their rates of change over time. Given that p and x are related by 2x² - 3xp + 50p² = 20800, when p = 20, x is defined by 2x² - 60x + 20000 = 20800, (x > 0) x² - 30x - 400 = 0, (x > 0) (x - 40)(x + 10) = 0, (x > 0) hence x = 40 Differentiating each side of the equation with respect to t: 4x(dx/dt) - 3p(dx/dt) - 3x(dp/dt) + 100p(dp/dt) = 0 Therefore: dx/dt = (3x - 100p)(dp/dt)/(4x - 3p) At p = 20, x = 40 and dp/dt = 2 Hence: dx/dt = (3*40 - 100*20)*2/(4*40 - 3*20) . . . .= (120 - 200)*2/(160 - 60) . . . . = -0.4 Rate of change of demand = - 0.4 unit per month At the initial price of 20 dollars per unit, if the price increases at the rate of 2 dollars/month then the demand will decrease at the rate of 0.4 unit per month.
A manufacturer can sell a certain product for $80 per unit. Total cost consists of a fixed overhead of $4,500 plus production cost of $50 per unit. How many units must the manufacturer sell to realize a profit of $900?
A manufacturer can sell a certain product for $80 per unit. Total cost consists of a fixed overhead of $4,500 plus production cost of $50 per unit. How many units must the manufacturer sell to realize a profit of $900?This seems quite straightforward. He makes $30 on each item out of which has to come the overhead and the profit that total $5400. Divide this by $30. I get 180. What do you get?
Optimization: A fertilizer producer finds that it can sell its product at a price of....?
A fertilizer producer finds that it can sell its product at a price of p=300 - 0.1x dollars per unit when it produces x units of fertilizer. The total production cost (in dollars) for x units is C(x) = 15000 + 125x + 0.025x^2. If the production capacity of the firm is at most 1000 units of fertilizer in a specified time, how many units must be manufactured and sold in that time to maximize the profit?
When the price of a good changes to Rs 11 per unit the consumer demand falls from 11 units to 7 units. The price elasticity of demand is -1. What was the price before change? Use expenditure approach of price elasticity of demand to answer?
Under the TE approach, if elasticity of demand = -1, then the total expenditure incurred by the household remains the same as before the price change. In other words, consumer's total expenditure on the product remains constant inspite of a price change when Ed =1.Let the old price of the good be pNew price is p1 = Rs. 11Initial quantity demanded be q = 11 unitsNew quantity demanded be q1 = 7 unitsNow, the TE incurred by the household after the price change = p1 x q1 = 11 x 7 = Rs. 77.Since demand is unitary elastic, TE remains constant. So at the initial price too, TE was the same.So TE before the price change = p x q = p x 11 = Rs. 77. So p = Rs. 7Hence p = Rs. 7.Verification :Old price : Rs. 7 x 11 units = Rs. 77 TENew price : Rs. 11 x 7 units = Rs. 77 TEFeel free to comment for any clarification.
What is the difference between unit sales, unit sales price and the cost of unit price?
Units Sales - if you were a Tesla National Distributor and you sold the cars in bulk to dealers - that will be classified as Unit Sales.Although the average Teslar retails for £150k in the UK, you may be selling units of 10 for £1m - that will be your Unit Sales.Unit Sales Price - using the example above. This will be the price you sold each Teslar for. In this case £100k (Remember you sold 10 for £1m)Cost of Unit Price - This is what it costs your company to acquire each of the cars. If Elon Musk is feeling generous and allows you a substantial discount by selling the car to you at £75k - that will be your cost of each unit.Generally, this tends to be just the direct cost of the item. (What it cost you to make/acquire - direct/variable cost NOT Fixed costs)Hope this helpsBoomy Tokan
What is the formula to determine variable cost per unit?
As a formula:VC per unit = TVC / units producedTC = TFC + TVCTC = Total costsTFC = Total Fixed costs (do not change with the level of production)TVC = Total Variable Costs (Total VC changes when the production level changes; costs per unit do not change)General ExplanationIn general the total of the variable cost per unit increases resp. decreases when the number of production units increases or decreases.This is the opposite of so-called constant, fixed or sunk costs that do NOT change when the number of production units change.Particular explanationTo give you an example:When you lease a plant (production site) the monthly lease costs do not change, whether you would produce 10,000 units or 0 units. You would still need to pay the rent.However, when you produce some item you need the material for that item.Let’s say that we have a production site for steel rivets. The level of steel varies with the number of rivets you are going to produce. No rivets, no steel, many rivets much steel.Of course, in the real world the variable costs per unit can change, for example because of a volume discount when you would buy more steel, or due to price fluctuations on the steel market.Finally , if you have the total variable costs AND you know the number of items produced, you can calculate the variable costs per unit: TVC / number of units.There are some different flavours to this formula which have in common that they add a certain level of sophistication but at the end of the day it all boils down to what I described here before.Good luck!
Consumer And Producer Surplus 10 POINTS!?
First, find the equilibrium quantity x. How? Equate the two equations, then solve for x: –0.01x^2 – 0.1x + 191 = 0.02x^2 + 0.3x + 16 0.02x^2 + 0.01x^2 + 0.3x + 0.1x + 16 - 191 = 0. 0.03x^2 + 0.4x - 175 = 0 Using the quadratic formula: x =70 the equilibrium quantity is 7,000 units Now, solve for the equilibrium price by substituting the value of x to the demand equation: p = –0.01x^2 – 0.1x + 191 p = –0.01(70)^2 – 0.1(70) + 191 p = $135 Substituting the value of x to the supply equation should result in the same equilibrium price. p = 0.02x^2 + 0.3x + 16 p = 0.02(70^2 + 0.3(70) + 16 p = $135 If the market price is set at the equilibrium price, find the consumer's surplus and the producers' surplus. Substituting equilibrium price to the demand equation would give us the quantity that consumers are willing to buy at that price: 135 = –0.01x^2 – 0.1x + 191 0.01x^2 + 0.1x - 191 + 135 = 0 0.01x^2 + 0.1x - 56 = 0 Using the quadratic formula: x = 70 Meaning: Consumer's surplus at the equilibrium price is 70 - 70 = 0. This is to be expected, since consumers won't demand more of the product at a price higher than the equilibrium price. Substituting equilibrium price to the supply equation would give us the quantity that consumers are willing to sell at that price: 135 = 0.02x^2 + 0.3x + 16 0.02x^2 + 0.3x + 16 - 135 = 0 0.02x^2 + 0.3x - 119 = 0 Using the quadratic formula: x = 70 Meaning: Producer's surplus at the equilibrium price is 70 - 70 = 0. This is to be expected, since producers won't supply more of the product at a price lower than the equilibrium price.
The equation p = -x^2 + 8x +5 gives the price p, in dollars, for a product when x million units are produced.?
-x² + 8x + 5 = 0 x² - 8x - 5 = 0 quadratic equation: x = [ -b ± √(b² - 4ac) ] / 2a x = [ -(-8) ± √((-8)² - 4(1)(-5)) ] / 2(1) x = [ 8 ± √(64 + 20) ] / 2 x = [ 8 ± √4√21 ] / 2 x = 4 ± √21 the solutions are x = (4 + √21) or (4 - √21) the positive solution is (4 + √21) which is 8.58257... which is 8.58 to two dp this means they must produce at least 8.58 million units even if the price is zero!
Why is marginal revenue not equal to price in a monopoly?
MR= change in revenue/change in quantitiy sold.Lets say I am selling Burgers. I can sell only 1 burger at $100 dollars but 2 at $80.Change in revenue = 160 - 100 = $60Change in Quantity = 2 - 1 = 1MR = $60/1 = $60 which is less than the price $80.Logic: The reason why MR is less than Price in Monopoly is that I cannot sell additional quantity unless I reduce the price of my product.And the price reduced is not just for the additional quantity but for all the products I am selling.In this case, when I reduce the price to $80, the additional burger I am selling costs me $20 on the first burger. And thus MR reduced to net $60.
Business math help cost revenue and profit problem?
Profit = Price - Costs P = (1256 - (2/3)x)x - 34000 - ((1/3)x + 222)x P = 1256x - (2/3)x^2 - 34000 - (1/3)x^2 - 222x P = 1034x - x^2 - 34000 Break even when P=0 x^2 - 1034x + 34000 = 0 x = 517 +/- 483 so x = $34 or x = $1000 are break even points Max R when dR/dx = 0 and d^2R/dx^2 < 0 R = 1256x - (2/3)x^2 dR/dx = 1256 - (4/3)x dR/dx = 0 when x = $942 (Price for max revenue) max R = $591,576 check for max: d^2R/dx^2 = -4/3 which is < 0 so this is a maxima. Profit function: P = 1034x - x^2 - 34000 max P when dP/dx = 0 and d^2P/dx^2 < 0 dP/dx = 1034 - 2x dP/dx = 0 when x = $517 (Price for max profit) P = $233,289 check max: d^2P/dx^2 = -2 which is < 0 so it is a maxima