The length of � meters of a certain metal rod at temperature 0℃ is given by �=1+0.00005�+0.0000004� 2 .Determine the rate of change of length,in��℃ when the temperature is:(a)100℃ and (b)400℃

Answer 1

To find the rate of change of length, we need to take the derivative of the length function with respect to temperature.

Given:
ℓ = 1 + 0.00005t + 0.0000004t^2

Taking the derivative with respect to temperature t:
dℓ/dt = 0 + 0.00005 + 2(0.0000004)t

Simplifying:
dℓ/dt = 0.00005 + 0.0000008t

a) When the temperature is 100℃:
dℓ/dt = 0.00005 + 0.0000008(100)
= 0.00005 + 0.00008
= 0.00013

Therefore, the rate of change of length when the temperature is 100℃ is 0.00013 inches/℃.

b) When the temperature is 400℃:
dℓ/dt = 0.00005 + 0.0000008(400)
= 0.00005 + 0.00032
= 0.00037

Therefore, the rate of change of length when the temperature is 400℃ is 0.00037 inches/℃.