Question

Graph the line that represents a proportional relationship between d and t with the property that an increase of 3 units in t corresponds to an increase of 8 units in d. What is the unit rate of change of d with respect to t? (That is, a change of l unit in t will correspond to a change of how many units in d?) The unit rate of change is Graph the relationship.

Answer 1

First, identify a relation between a change in time and a change in distance.

Let Δt be a change in time and Δd be a change in distance.

Since an increase of 3 units in t corresponds to an increase of 8 units in d, then:

`(\Delta t)/(\Delta d)=(3)/(8)`

To find a line with that characteristics, set the initial values for time and distance equal to 0, so that Δt = t and Δd =d. Then:

`\begin{gathered} (t)/(d)=(3)/(8) \\ \Rightarrow d=(8)/(3)t \end{gathered}`

Graph that line by finding two points on it. Notice that when t=0, then d=0 and when t=3, d=8. So, (0,0) and (3,8) belong to that line:

Since we know that Δd=(8/3)Δt, the unit rate of change will be found by plugging in Δt=1:

`\Delta d=8/3`

Since Δd is equal to 8/3 when Δt is equal to 1, then the unit rate of change is 8/3.

Therefore, the unit rate of change is 8/3. The graph is shown.