A company is planning to manufacture a new blender.After conducting extensive market surveys,the research department estimates a weekly demand of 600 blenders at a price of $50 per blender and a weekly demand of 800 blenders at a price of $40 per blender.Assuming the demand equation is linear,use the research department's estimates to find the revenue equation in terms of the demand x.

Question

A company is planning to manufacture a new blender.After conducting extensive market surveys,the research department estimates a weekly demand of 600 blenders at a price of $50 per blender and a weekly demand of 800 blenders at a price of $40 per blender.Assuming the demand equation is linear,use the research department's estimates to find the revenue equation in terms of the demand x.
A company is planning to manufacture a new blender.After conducting extensive market surveys,the research department estimates a weekly demand of 600 blenders at a price of $50 per blender and a weekly demand of 800 blenders at a price of $40 per blender.Assuming the demand equation is linear,use the research department's estimates to find the revenue equation in terms of the demand x.

Quizplus · Accepted Answer

To find the revenue equation in terms of demand x, we need to use the formula: revenue = price x quantity.

Let x be the weekly demand in units of blenders.

At a price of $50, the revenue equation is: 
R1 = 50x

At a price of $40, the revenue equation is: 
R2 = 40x

We can use the two data points to find the equation of a line that passes through them, which will give us the revenue equation in terms of demand x.

First, we need to find the slope of the line: 
slope = (R2 - R1) / (x2 - x1) 
where x1 = 600, x2 = 800, R1 = 50x1 = 30000, and R2 = 40x2 = 32000.

slope = (32000 - 30000) / (800 - 600) 
slope = 1000

Next, we need to find the y-intercept of the line: 
y-intercept = R1 - slope x x1 
y-intercept = 30000 - 1000 x 600 
y-intercept = 24000

Therefore, the revenue equation in terms of demand x is: 
R = 1000x + 24000

To maximize revenue, we need to find the quantity that corresponds to the highest point on the revenue curve. This occurs at the vertex of the parabolic revenue function, which is given by the formula: 
x = - b / 2a 
where a = 1000 and b = 24000.

x = - 24000 / (2 x 1000) 
x = - 12

Since we cannot have a negative demand, we reject this answer. Therefore, the best choice is to produce 800 blenders per week at a price of $40 per blender, which will give a weekly revenue of: 
R = 1000 x 800 + 24000 = $32,000

A Company Is Planning to Manufacture a New Blender