The marginal cost function is MC(q)=300-10q, where 'q' is the number of tons of coffee produced. Fixed costs are $1000. The marginal revenue function is MR(q)=500-20q. The value of the profit function P(q) when q=25 is closest to? And the Average Cost when q=20 is closest to? And when the change in revenue if sales increase from 5 to 20 tons is closest to?
To find the value of the profit function P(q) when q=25, we need to subtract the total cost from the total revenue at that quantity level.
First, let's calculate the total cost (TC) when q=25. The total cost includes both fixed costs and variable costs (which in this case is the marginal cost). We can find the fixed cost from the given information, which is $1000:
Fixed Costs (FC) = $1000
Variable Costs (VC) = Marginal Cost (MC) * Quantity (q)
= (300 - 10q) * q
Total Cost (TC) = FC + VC
= $1000 + (300 - 10q) * q
Substituting q=25 into the equation, we have:
TC = $1000 + (300 - 10 * 25) * 25
= $1000 + 0 * 25
= $1000
Next, let's calculate the total revenue (TR) when q=25. The total revenue at a quantity level is given by the marginal revenue multiplied by the quantity:
Total Revenue (TR) = Marginal Revenue (MR) * Quantity (q)
= (500 - 20q) * q
Substituting q=25 into the equation, we have:
TR = (500 - 20 * 25) * 25
= (500 - 500) * 25
= $0
Finally, to find the profit (P), we subtract the total cost from the total revenue:
P = TR - TC
= $0 - $1000
= -$1000
Therefore, when q=25, the value of the profit function P(q) is approximately -$1000.
To find the average cost when q=20, we divide the total cost by the quantity:
Average Cost (AC) = Total Cost (TC) / Quantity (q)
= (FC + VC) / q
= ($1000 + (300 - 10q) * q) / q
Substituting q=20 into the equation, we have:
AC = ($1000 + (300 - 10 * 20) * 20) / 20
= ($1000 + 0 * 20) / 20
= $1000 / 20
= $50
Therefore, when q=20, the average cost is approximately $50.
To find the change in revenue if sales increase from 5 to 20 tons, we need to subtract the total revenue at 5 tons from the total revenue at 20 tons:
Change in Revenue = Total Revenue (TR) at 20 tons - Total Revenue (TR) at 5 tons
Total Revenue (TR) = (500 - 20q) * q
Substituting q=20 into the equation, we have:
TR at 20 tons = (500 - 20 * 20) * 20
= (500 - 400) * 20
= $100 * 20
= $2000
Substituting q=5 into the equation, we have:
TR at 5 tons = (500 - 20 * 5) * 5
= (500 - 100) * 5
= $400 * 5
= $2000
Now, let's calculate the change in revenue:
Change in Revenue = TR at 20 tons - TR at 5 tons
= $2000 - $2000
= $0
Therefore, the change in revenue if sales increase from 5 to 20 tons is $0.
To find the profit function, we need to subtract the total cost function from the total revenue function:
Total cost (TC) = Fixed costs + Variable costs
Variable costs = MC(q) * q
TC(q) = 1000 + (300 - 10q) * q
TC(q) = 1000 + 300q - 10q^2
Total revenue (TR) = MR(q) * q
TR(q) = (500 - 20q) * q
TR(q) = 500q - 20q^2
Profit function (P) = TR(q) - TC(q)
P(q) = (500q - 20q^2) - (1000 + 300q - 10q^2)
P(q) = 500q - 20q^2 - 1000 - 300q + 10q^2
P(q) = -10q^2 + 200q - 1000
To find the profit when q = 25, we substitute q = 25 into the P(q) equation:
P(25) = -10(25)^2 + 200(25) - 1000
P(25) = -10(625) + 5000 - 1000
P(25) = -6250 + 4000 - 1000
P(25) = -3250
Therefore, the closest value of the profit function P(q) when q = 25 is -3250.
To find the average cost when q = 20, we divide the total cost (TC) by the quantity (q):
Average cost (AC) = TC(q) / q
AC(q) = (1000 + 300q - 10q^2) / q
AC(20) = (1000 + 300(20) - 10(20)^2) / 20
AC(20) = (1000 + 6000 - 10(400)) / 20
AC(20) = (7000 - 4000) / 20
AC(20) = 3000 / 20
AC(20) = 150
Therefore, the closest value of the average cost when q = 20 is 150.
To find the change in revenue if sales increase from 5 to 20 tons, we calculate the difference in total revenue (TR):
TR(20) - TR(5) = [(500(20) - 20(20)^2) - (500(5) - 20(5)^2)]
TR(20) - TR(5) = (10000 - 20(400)) - (2500 - 20(25))
TR(20) - TR(5) = (10000 - 8000) - (2500 - 500)
TR(20) - TR(5) = 2000 - 2000
TR(20) - TR(5) = 0
Therefore, the closest value of the change in revenue if sales increase from 5 to 20 tons is 0.