Question

You start out driving from your house to an old friend’s house in a different town, which is 280 miles away. After an hour and a half, you have gone 105 miles.

a) Assuming your speed is constant, find a linear function , which gives the distance from your house (in miles) as a function of time (in hours).

b) Find and interpret D(1).

c) Solve D(t)=280

d) Solve D(t)^3>=140 and interpret the result.

e) Considering the context of this function, what is the domain?

f) What is the range?

Answer 1

Part A:

Given that after an hour and a half, you have gone 105 miles and that your speed is constant, this means that the slope which represents the speed with which you are driving is given by

$\frac{105}{1.5} =70 \ miles \ per \ hour$

At 0 hours, he has travelled 0 miles.

The equation of a straight line is given by y = mx + c, where m represent the slope and c is the y-intercept (the value of y when x is 0).

Therefore, a linear function , which gives the distance from the house (in miles) as a function of time (in hours) is given by D = 70t.



Part B:

D(1) is the value of D when t = 1.

D(1) = 70(1) = 70

This means that the distance traveled after 1 hour is 70 miles.



Part C:

D(t) = 280 means the time it takes you to travel 280 miles.

70t = 280

t = 280 / 70 = 4

t = 4.



Part D:

$D(t)^3 \geq 140 \\ \\ \Rightarrow 70t^3 \geq 140 \\ \\ \Rightarrow t^3 \geq \frac{140}{70} =2 \\ \\ \Rightarrow t \geq \sqrt[3]{2}$



Part E:

The domain is a set of the possible values of the independent variable, from the context of the function, the independent variable is the time it took him for the journey.

Therefore, the domain of the function is [0, 4].



Part F:

The range is the set of the possible values of y, y in the context of the question is the distance travelled at each particular time.

Therefore, the range of the function is [0, 280].

Answer 2

a) D(t)=70t. To find this, consider that we traveled 105 miles in the first 1.5 hours. Since the speed is constant, we simply divide the 105 miles by the 1.5 hours. 105/1.5=70, which tells us we are going 70 mph. To express this as a function, we write the speed of travel (70) times the number of hours (t). Thus D(t)=70t. b) D(1)=70. This means that we traveled 70 miles after one hour. To find this, plug in 1 hour for t and simplify: 70(1)=70. c) t=4. To solve this, set 280=70t and solve for t. 280/70=4. d) t=.0741784872. To solve, take the cube root of both sides. This gives us D(t)=5.192494102. This can be written 70t=5.192494102. Dividing both sides by 70 we get that t=0741784872. e) Domain is t-interval [0,4]. To find this, consider that the trip is 280 miles and we are traveling at 70 mph. Dividing 280 by 70, we see that we will make the trip in 4 hours. Thus our trip starts at t=0 and ends at t-4. f) Range is the D-interval [0,280]. To find this, consider that the domain is [0,4]. d(0)=0 and D(4)=280. Thus the range is [0,280].