Question

At a local grocery store the demand for ground beef is approximately 50 pounds per week when the price per pound is $4, but is only 40 pounds per week when the price rises to $5.50 per pound. Assuming a linear relationship between the demand x and the price per pound p, express the price as a function of demand. Use this model to predict the demand if the price rises to $5.80 per pound.

Answer 1

To find the price as a function of demand for ground beef, we calculate the slope with the given data, determine the y-intercept, and then use the slope-intercept form. The developed linear demand function allows us to predict demand of approximately 38 pounds of ground beef per week for a price increase of $5.80 per pound.

Step-by-step explanation:

To express the price as a function of demand for ground beef, we can use the given two points: (50, $4) and (40, $5.50). First, we calculate the slope (m) of the price-demand line using the slope formula, m = (y2 - y1) / (x2 - x1). In this context, it becomes m = (5.50 - 4) / (40 - 50) = 1.50 / (-10) = -0.15.

Next, we use the slope-intercept form of the linear equation, which is p = mx + b, where m is the slope and b is the y-intercept. We substitute one point into the equation to solve for b. Using the point (50, $4): 4 = -0.15(50) + b, we find b = 4 + 7.5 = $11.50. Now we have the price-demand function, which is p = -0.15x + 11.50.

Finally, to predict the demand if the price rises to $5.80 per pound, we solve for x in the price-demand function: 5.80 = -0.15x + 11.50. This gives us x = (11.50 - 5.80) / -0.15. Calculating x, we get x = 38 pounds. Therefore, the demand would be approximately 38 pounds of ground beef per week at the price of $5.80 per pound.