Question

question
the temperature of an object, in degrees f, can be modeled by the function ( d(t) ), where ( t ) is measured in seconds. the value of ( d(6) = -5 ). selected values of ( d(t) ) are shown in the table below. use a linear approximation when ( t = 6 ) to estimate the temperature of the object at time ( t = 6.2 ). use proper units.

( t )	0	3	6	8	11

answer
attempt 1 out of 5

degrees f

Explanation:

Step1: Recall Linear Approximation Formula

The linear approximation of a function \( D(t) \) at \( t = a \) is given by \( L(t) = D(a) + D'(a)(t - a) \). Here, \( a = 6 \), \( t = 6.2 \), \( D(6) = 19 \) (from the table), and \( D'(6) = -5 \).

Step2: Substitute Values into Formula

Substitute \( a = 6 \), \( t = 6.2 \), \( D(6) = 19 \), and \( D'(6) = -5 \) into the linear approximation formula:
\( L(6.2) = D(6) + D'(6)(6.2 - 6) \)
\( L(6.2) = 19 + (-5)(0.2) \)

Step3: Calculate the Result

First, calculate \( (-5)(0.2) = -1 \). Then, \( 19 - 1 = 18 \).

Answer:

18 degrees F