Question

the scatter plot shows the average monthly temperature, x, and the monthly heating cost of a family, y, for 25 different months.
monthly heating cost (in dollars)
average monthly temperature (in °f)
use the equation of the line of best fit, $y = -1.24x + 97.40$, to answer the questions below.
give exact answers, not rounded approximations.
(a) what is the predicted heating cost for a month with an average temperature of 0 °f? $\square$
(b) for an increase of one degree fahrenheit, what is the predicted decrease in the monthly heating cost? $\square$
(c) what is the predicted heating cost for a month with an average temperature of 35 °f? $\square$

Explanation:

Part (a)

Step 1: Substitute \( x = 0 \) into the equation

The equation of the line of best fit is \( y=-1.24x + 97.40 \). We need to find the value of \( y \) when \( x = 0 \). Substitute \( x = 0 \) into the equation:
\( y=-1.24(0)+97.40 \)

Step 2: Simplify the expression

Simplify the right - hand side of the equation. Since \( - 1.24\times0 = 0 \), we have \( y=97.40 \).

Part (b)

Step 1: Analyze the slope of the line

The equation of a line in slope - intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. In the equation \( y=-1.24x + 97.40 \), the slope \( m=-1.24 \). The slope represents the rate of change of \( y \) with respect to \( x \). For a linear equation \( y=mx + b \), when \( x \) increases by 1 unit, \( y \) changes by \( m \) units. Since the slope is - 1.24, when the average monthly temperature (\( x \)) increases by 1 degree Fahrenheit, the monthly heating cost (\( y \)) decreases by \( 1.24 \) dollars.

Part (c)

Step 1: Substitute \( x = 35 \) into the equation

We substitute \( x = 35 \) into the equation \( y=-1.24x + 97.40 \). So we have:
\( y=-1.24\times35+97.40 \)

Step 2: Calculate the value of \( - 1.24\times35 \)

First, calculate \( -1.24\times35=-43.4 \)

Step 3: Calculate the value of \( y \)

Then, \( y=-43.4 + 97.40=54 \)

Answer:

s:
(a) The predicted heating cost for a month with an average temperature of \( 0^{\circ}\text{F} \) is \( \$97.40 \).
(b) For an increase of one degree Fahrenheit, the predicted decrease in the monthly heating cost is \( \$1.24 \).
(c) The predicted heating cost for a month with an average temperature of \( 35^{\circ}\text{F} \) is \( \$54 \).