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

The linear relationship between demand (x) and price per pound (y) is expressed as y = -0.15x + 11.5. Predicting the demand when the price is $5.80 per pound yields approximately 38 pounds.

To express the price as a function of demand in a linear relationship, we can use the point-slope form of a linear equation:

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

In this case, let x represent the demand in pounds and y represent the price per pound. We have two points: (50, 4) and (40, 5.5).

Let's calculate the slope m:

`\[ m = (y_2 - y_1)/(x_2 - x_1) \]\[ m = (5.5 - 4)/(40 - 50) \]\[ m = (1.5)/(-10) = -0.15 \]`

Now, we can use the point-slope form with one of the points. Let's use (50, 4):

y - 4 = -0.15(x - 50)

Now, solve for y to express the price as a function of demand:

y = -0.15x + 11.5

So, the price per pound y as a function of demand x is y = -0.15x + 11.5.

Now, to predict the demand x if the price rises to $5.80 per pound, we can set y = 5.80 and solve for x:

`\[ 5.80 = -0.15x + 11.5 \]`

Subtract 11.5 from both sides:

`\[ -5.70 = -0.15x \]`

Divide both sides by -0.15:

`\[ x = (-5.70)/(-0.15) \]\[ x = 38 \]`

Therefore, the predicted demand, when the price rises to $5.80 per pound, is approximately 38 pounds.