Question

you want to buy either a wood pellet stove or an electric furnace. the pellet stove costs $2160 and produces heat at a cost of $15.15 per 1 million btu (british thermal units). the electric furnace costs $1250 and produces heat at a cost of $33.15 per 1 million btu.
(a) write a function for the total cost y (in dollars) of buying the pellet stove and producing x million btu of heat
(b) write a function for the total cost y (in dollars) of buying the electric furnace and producing x million btu of heat
(c) use a graphing utility to graph the system of equations formed by the two cost functions.

Explanation:

Step1: Define variables for pellet stove

Let \( x \) be the number of million Btu of heat, and \( y \) be the total cost. The pellet stove has a fixed cost of $2160 and a variable cost of $15.15 per million Btu. So the cost function is \( y = 15.15x + 2160 \).

Step2: Define variables for electric furnace

For the electric furnace, fixed cost is $1250 and variable cost is $33.15 per million Btu. So the cost function is \( y = 33.15x + 1250 \).

Step3: Graph the system

To graph \( y = 15.15x + 2160 \) and \( y = 33.15x + 1250 \), we can find the y - intercepts and slopes. For \( y = 15.15x + 2160 \), y - intercept is 2160, slope is 15.15. For \( y = 33.15x + 1250 \), y - intercept is 1250, slope is 33.15. Using a graphing utility, we plot these two linear functions. The first line starts at (0, 2160) and rises with a slope of 15.15, the second line starts at (0, 1250) and rises with a slope of 33.15.

Answer:

(a) Pellet stove: \( y = 15.15x + 2160 \); (b) Electric furnace: \( y = 33.15x + 1250 \); (c) Graph the two linear functions \( y = 15.15x + 2160 \) and \( y = 33.15x + 1250 \)