Paula has a large ball of yarn that she is going to use to knit a scarf for the winter. Every square inch on the scarf requires a certain number of yards of yarn from the ball.

Question

A linear model of this situation contains the values (20 , 244.4) and (25 , 185), where x represents the number of square inches knitted on the scarf, and y represents the number of yards remaining on the ball of yarn.

How many yards of yarn did Paula start with?
A.
482
B.
527
C.
11.88
D.
429.4

Answer 1

To find how many yards of yarn Paula started with, we can use the linear model given by the points (20, 244.4) and (25, 185). We can determine the equation of the line that models this relationship.

1. Calculate the slope (m) of the line using the two points (x₁, y₁) = (20, 244.4) and (x₂, y₂) = (25, 185):

   \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{185 - 244.4}{25 - 20} = \frac{-59.4}{5} = -11.88 \]

2. Using the point-slope form of the line equation, we can use one of the points to find the equation. Let's use point (20, 244.4):

   \[ y - y_1 = m(x - x_1) \]

   Substitute \(m = -11.88\), \(x_1 = 20\), and \(y_1 = 244.4\):

   \[ y - 244.4 = -11.88(x - 20) \]

   Distributing:

   \[ y - 244.4 = -11.88x + 237.6 \]

   Rearranging gives the equation:

   \[ y = -11.88x + 244.4 + 237.6 \]

   \[ y = -11.88x + 482 \]

So the equation of the line is \(y = -11.88x + 482\).

3. To find the initial amount of yarn, we need to evaluate the equation when \(x = 0\) (representing the condition before any yarn is used):

   \[ y = -11.88(0) + 482 = 482 \]

Thus, Paula started with 482 yards of yarn.

The correct answer is A. 482.