Question

Rip van Winkle has fallen asleep for a very long time.
The length of his beard (in millimeters) as a function of time (in weeks) is graphed.
A first quadrant coordinate plane. The horizontal axis is from zero to thirty-five with a scale of five and is titled Time in weeks. The vertical axis is from zero to ninety with a scale of ten and is titled Beard length in millimeters. The graph of the line is y equals two point five x plus ten.
\[\small{5}\]
\[\small{10}\]
\[\small{15}\]
\[\small{20}\]
\[\small{25}\]
\[\small{30}\]
\[\small{10}\]
\[\small{20}\]
\[\small{30}\]
\[\small{40}\]
\[\small{50}\]
\[\small{60}\]
\[\small{70}\]
\[\small{80}\]
How long did it take each time Rip van Winkle's beard grew by
\[5\] millimeters?
Choose 1 answer:
Choose 1 answer:
(Choice A)
\[7.5\] weeks
A
\[7.5\] weeks
(Choice B)
\[5\] weeks
B
\[5\] weeks
(Choice C)
\[2.5\] weeks
C
\[2.5\] weeks
(Choice D)
\[2\] weeks
D
\[2\] weeks

Answer 1

To determine how long it takes for Rip van Winkle's beard to grow by 5 millimeters, we need to analyze the given equation of the line, which represents the growth of the beard:

\[
y = 2.5x + 10
\]

Here, \(y\) represents the beard length in millimeters, and \(x\) represents time in weeks. The slope \(2.5\) indicates the rate of growth of the beard in millimeters per week.

To find out how long it takes for the beard to grow by \(5\) millimeters, we can first determine the increase in \(y\) for an increase in \(x\):

1. The beard length increases by \(5\) millimeters; therefore we set up the equation:

\[
\Delta y = 5
\]

Using the slope \(2.5\), we can express this as:

\[
\Delta y = m \cdot \Delta x
\]

Where \(m\) (the slope) is \(2.5\), and \(\Delta x\) is the change in time we want to find.

Setting up the equation:

\[
5 = 2.5 \cdot \Delta x
\]

Now we can solve for \(\Delta x\):

\[
\Delta x = \frac{5}{2.5} = 2
\]

Therefore, Rip van Winkle's beard grows by \(5\) millimeters in \(2\) weeks.

The correct answer is:

(Choice D)
\[\small{2}\] weeks