Question

question 3 of 25
a company produces items in small batches. the machinery needs to warm up before the items can be produced. the scatterplot shows the time needed to produce batches of different numbers of items.
based on the line of best fit, which of these statements is true?
a. each item requires about 3 minutes of production time.
b. each item requires about 12 minutes of production time.
c. the equipment starts producing items after about 20 minutes of warming up.
d. the equipment starts producing items after about 25 minutes of warming up.

Explanation:

Step1: Find the equation of the line of best fit

The line of best fit passes through the points \((3,44)\) and \((5,60)\).
The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{60 - 44}{5 - 3}=\frac{16}{2}=8\).
Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(3,44)\) and \(m = 8\), we get \(y-44=8(x - 3)\).
Expanding, \(y-44=8x-24\), so \(y = 8x+20\).

Step2: Analyze each option

• Option A:

The slope of the line \(y = 8x + 20\) is \(8\), which means the production time per item is \(8\) minutes (not \(3\) minutes).

• Option B:

Since the slope is \(8\), each item requires about \(8\) minutes (not \(12\) minutes) of production time.

• Option C:

The \(y\) - intercept of the line \(y=8x + 20\) is \(20\). In the context of the problem, when \(x = 0\) (no items produced), \(y=20\). This represents the warm - up time. So the equipment starts producing items after about \(20\) minutes of warming up.

• Option D:

Since the \(y\) - intercept is \(20\) (not \(25\)), this option is incorrect.

Answer:

C. The equipment starts producing items after about 20 minutes of warming up.