Question

As you travel from Detroit in a certain direction, the outside temperature, T (in degrees), depends on your distance, d (in miles), from Detroit. For a particular 100-mile trip, the temperature was given by the function T(d ) = d + 30 degrees. a. What was the total change in temperature from the beginning of the trip to the end of the trip? b. What was the average rate of change of the temperature, with respect to distance, from the beginning of the trip to the end of the trip? c. At what rate, was the temperature changing, with respect to distance, at d = 4 miles? d. At what rate was the rate of change of the temperature changing, with respect to distance, at d = 4 miles?

Answer 1

a)
`\Delta T= 100^(\circ)C`

b)
`\bigtriangledown T=1^(\circ)C.mile^(-1)`

c)
`\bigtriangledown T_4=1^(\circ)C.mile^(-1)`

d)
`\bigtriangledown T_4=1^(\circ)C.mile^(-1)`

Step-by-step explanation:

Given is the data of variation of temperature with respect to the distance traveled:

Temperature T as a function of distance d:

`T=(d+30) ^(\circ)C`...................................(1)

(a)

Total change in temperature from the start till the end of the journey:

`\Delta T= T_f-T_i`..............................(2)

where:

`T_f`= final temperature

`T_i`= initial temperature

∵In the start of the journey d = 0 miles & at the end of the journey d = 100 miles.

So, correspondingly we have the eq. (2) & (1) as:

`\Delta T= (100+30)-(0+30)`

`\Delta T= 100^(\circ)C`

(b)

Now, the average rate of change of the temperature, with respect to distance, from the beginning of the trip to the end of the trip be calculated as:

`\bigtriangledown T=(\Delta T)/(\Delta d)`......................(3)

where:

`\Delta d`= change in distance

`\bigtriangledown T=`change in temperature with respect to distance

putting the respective values in eq. (3)

`\bigtriangledown T=(100)/(100)`

`\bigtriangledown T=1^(\circ)C.mile^(-1)`

(c)

comparing the given function of the temperature with the general equation of a straight line:

`y=m.x+c`

We find that we have the slope of the equation as 1 throughout the journey and therefore the rate of change in temperature with respect to distance remains constant.

`\bigtriangledown T_4=1^(\circ)C.mile^(-1)`

(d)

comparing the given function of the temperature with the general equation of a straight line:

`y=m.x+c`

We find that we have the slope of the equation as 1 throughout the journey and therefore the rate of change in temperature with respect to distance remains constant.

`\bigtriangledown T_4=1^(\circ)C.mile^(-1)`