q 0 125 250 375 500 625 750
C'(q) 27 35 41 45 43 46 51
Give fixed cost = 8800 and cost of producing 250 units is 17425, how much would the total cost increase if production were increased one unit, to 251 units?
---
A cup of coffee at 90 degrees celsius is put into a 30 degree celsius room when t =0 . The coffee's temperature, f (t) , is changing at a rate given by f '(t)=-8(0.8) t degrees celsius per minute, where t is in minutes.
Estimate the coffee's temperature when t =10
To find the increase in total cost when production is increased by one unit, we need to calculate the marginal cost (C'(q)) at the production level of 250 units and then subtract it from the marginal cost at the production level of 251 units.
Given the data:
Production levels (q): 0, 125, 250, 375, 500, 625, 750
Marginal costs (C'(q)): 27, 35, 41, 45, 43, 46, 51
We can assume that the marginal cost is constant between the given production levels.
To find the marginal cost at the production level of 250 units:
Looking at the production level of 250 units, we see that the marginal cost is 41. This indicates that the cost of producing the 250th unit is 41 units of cost.
To find the marginal cost at the production level of 251 units:
Since the marginal cost is assumed to be constant between the given production levels, we can assume that the marginal cost at the production level of 250 units is also 41 units of cost.
Therefore, the increase in total cost when production is increased by one unit, to 251 units, would be equal to the marginal cost at the production level of 251 units, which is also 41 units of cost.
Hence, the total cost would increase by 41 units of cost.
---
To estimate the coffee's temperature when t = 10, we can use the given rate of change of the temperature f '(t) = -8(0.8) t.
Given that the coffee's temperature is at 90 degrees Celsius at t = 0, we can use the rate of change to estimate its temperature at t = 10.
Using the rate of change formula, we substitute t = 10:
f '(t) = -8(0.8) t
f '(10) = -8(0.8)(10)
Evaluating the expression, we get:
f '(10) = -64
This indicates that the temperature of the coffee is decreasing at a rate of 64 degrees Celsius per minute at t = 10.
To estimate the coffee's temperature at t = 10, we subtract the rate of change from the initial temperature:
Estimated temperature at t = 10 = Initial temperature - Rate of change
Estimated temperature at t = 10 = 90 - 64
Calculating this expression, we find:
Estimated temperature at t = 10 = 26 degrees Celsius
Hence, we estimate that the coffee's temperature at t = 10 is 26 degrees Celsius.
To calculate how much the total cost would increase if production were increased one unit, we can use the information provided in the problem.
Given:
q = 250 units
C(q) = 17425 (cost of producing 250 units)
We can calculate the cost per unit, C'(q), for 250 units by finding the slope of the total cost function at that point:
C'(q) = 41 (for q = 250 units)
To find the increase in cost when production is increased by one unit (from 250 to 251 units), we need to calculate the difference in total costs:
Increase in total cost = C'(q) * 1
Increase in total cost = 41 * 1
Increase in total cost = 41
Therefore, the total cost would increase by 41 units if production were increased one unit to 251 units.
---
To estimate the temperature of the coffee when t = 10 minutes, we can use the given rate of change of temperature, f '(t), and the initial temperature of the coffee.
Given:
f '(t) = -8(0.8)t (rate of change of temperature)
t = 0 (initial time)
f(t) = 90 degrees celsius (initial temperature)
To find the coffee's temperature at t = 10 minutes, we need to integrate the rate of change of temperature with respect to time:
f(t) = ∫[-8(0.8)t] dt
Integrating, we get:
f(t) = -4(0.8)t^2 + C
To find the value of C, we can use the initial temperature provided:
90 = -4(0.8)(0)^2 + C
90 = C
Substituting C = 90 into the equation:
f(t) = -4(0.8)t^2 + 90
Now, we can find the temperature at t = 10 minutes:
f(10) = -4(0.8)(10)^2 + 90
Calculating, we get:
f(10) = -4(0.8)(100) + 90
f(10) = -320 + 90
f(10) = -230
Therefore, the coffee's estimated temperature when t = 10 minutes is -230 degrees celsius.