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 slope - intercept form of a linear equation 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 can directly read off that \(m = 5\) (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. Since the slope \(m = 5\), the workers pick 5 baskets of berries per hour.
The \(y\) - intercept (\(b = 10\)) represents the initial amount. When \(x = 0\) (at the start, \(0\) hours of work), the number of baskets of berries is \(y=10\). So the initial amount of baskets is 10.

Answer:

a. Slope \(m = 5\), \(y\) - intercept \(b = 10\).
b. Rate of change is 5 (baskets per hour), initial amount is 10 (baskets at time \(x = 0\)). The rate of change means the workers harvest 5 baskets each hour, and the initial amount is the number of baskets they had before they started the harvesting (at \(x = 0\) hours).
c. On the \(x\) - axis, label "Time (hours)". On the \(y\) - axis, label "Number of baskets of berries". 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 another point. For example, when \(x = 1\), \(y=5\times1 + 10=15\) (point \((1,15)\)). Draw a straight line through these points.