Final answer:
To determine the interval with the greatest average rate of change for the cost to produce tires, we calculate the average rate of change for each interval and compare the values. The interval between 46 and 48 tires has the highest average rate of change of 69.
Step-by-step explanation:
To determine which interval has the greatest average rate of change for the cost to produce tires, we need to compute the average rate of change for each interval and compare the values. The average rate of change can be calculated by finding the difference in cost between the endpoints of the interval and dividing it by the difference in the number of tires produced.
Let's calculate the average rate of change for each interval:
a) Between 2 and 4 tires:
Average rate of change = (C(4) - C(2))/(4 - 2) = (0.45(4)^2 + 12(4) + 450 - (0.45(2)^2 + 12(2) + 450))/(4 - 2)
= (31.2 - 11.7)/2 = 19.5/2 = 9.75
b) Between 46 and 48 tires:
Average rate of change = (C(48) - C(46))/(48 - 46) = (0.45(48)^2 + 12(48) + 450 - (0.45(46)^2 + 12(46) + 450))/(48 - 46)
= (3552 - 3414)/2 = 138/2 = 69
c) Between 12 and 14 tires:
Average rate of change = (C(14) - C(12))/(14 - 12) = (0.45(14)^2 + 12(14) + 450 - (0.45(12)^2 + 12(12) + 450))/(14 - 12)
= (109.2 - 64.8)/2 = 44.4/2 = 22.2
d) Between 22 and 24 tires:
Average rate of change = (C(24) - C(22))/(24 - 22) = (0.45(24)^2 + 12(24) + 450 - (0.45(22)^2 + 12(22) + 450))/(24 - 22)
= (392.4 - 352.8)/2 = 39.6/2 = 19.8
Based on our calculations, the interval between 46 and 48 tires has the greatest average rate of change for the cost to produce tires (69).