Question

math | graded assignment | unit test, part 2 | unit 4
(score for question 3: ___ of 5 points)

1. a group of workers is harvesting berries at a constant rate. an equation that represents the number of baskets of berries the workers picked over a period of time, in hours, is ( y = 5x + 10 ).

(a) what are the slope and the ( y )-intercept for the equation that represents the number of baskets of berries the workers picked over a period of time?
(b) what are the rate of change and the initial amount? explain the meaning of the rate of change and the initial amount for the situation.
(c) use graph paper to draw a graph representing the situation. label the axes appropriately.
answer:

Explanation:

Step1: Recall the slope - intercept form

The equation of a line in slope - intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.
For the equation \(y = 5x+10\), by comparing with \(y = mx + b\).

Step2: Identify slope and \(y\) - intercept

We have \(m = 5\) (slope) and \(b = 10\) (\(y\) - intercept).

Step3: Relate to rate of change and initial amount

In the context of the problem, the rate of change (slope) represents the number of baskets of berries picked per hour. So the rate of change is \(5\) baskets per hour.
The \(y\) - intercept (\(b = 10\)) represents the initial amount of baskets (the number of baskets before any time \(x\) (in hours) has passed). So the initial amount is \(10\) baskets.

Answer:

a) The slope is \(5\) and the \(y\) - intercept is \(10\).
b) The rate of change is \(5\) baskets per hour (it means the workers pick \(5\) baskets of berries each hour) and the initial amount is \(10\) baskets (the number of baskets at the start, when \(x = 0\) hours).
c) On the \(x\) - axis, label "Time (hours)" and on the \(y\) - axis, label "Number of baskets". Plot the \(y\) - intercept at the point \((0,10)\). Then, using the slope of \(5\) (for every \(1\) unit increase in \(x\), \(y\) increases by \(5\)), plot other points such as \((1,15)\), \((2,20)\) etc. and draw a straight line through the points.