how do i solve: a manufacturer of calculators has determined that 10,000 calculators per week will be sold at price of $95 per calculator. At a price of $90, it is estimated that 12,000 calculators will be sold. Predict the number of calculators that will be sold at a price of $75.
It appears that the number of calculators sold, as a function of the amount of discount (x) is
p(x) = 10000+400x
so, just plug in x=20 for the demand at $75.
To predict the number of calculators that will be sold at a price of $75, we can use the concept of demand elasticity.
Demand elasticity measures the responsiveness of the quantity demanded to a change in price. It is calculated as the percentage change in quantity demanded divided by the percentage change in price.
In this case, we have two data points:
1. At a price of $95, 10,000 calculators are sold per week.
2. At a price of $90, 12,000 calculators are sold per week.
Let's first calculate the demand elasticity using these two data points:
Percentage change in quantity demanded = ((new quantity - old quantity) / old quantity) * 100
= ((12,000 - 10,000) / 10,000) * 100
= 20%
Percentage change in price = ((new price - old price) / old price) * 100
= ((90 - 95) / 95) * 100
= -5.3%
Demand elasticity = (percentage change in quantity demanded) / (percentage change in price)
= 20% / -5.3%
≈ -3.8
Now, let's use the concept of demand elasticity to predict the number of calculators sold at a price of $75.
Percentage change in price = ((new price - old price) / old price) * 100
= ((75 - 95) / 95) * 100
≈ -21.05%
Predicted percentage change in quantity demanded = Demand elasticity * Percentage change in price
≈ -3.8 * -21.05%
≈ 79.89%
Predicted quantity demanded = old quantity + (predicted percentage change in quantity demanded / 100) * old quantity
= 10,000 + (79.89 / 100) * 10,000
≈ 17,989
Therefore, it is predicted that approximately 17,989 calculators will be sold at a price of $75.
To solve this problem, we can use the concept of linear interpolation. Linear interpolation is a method of estimating an unknown value that lies between two known values based on a linear relationship.
Here's how you can use linear interpolation to find the number of calculators that will be sold at a price of $75:
Step 1: Determine the slope of the linear relationship
To find the slope of the linear relationship between price and the number of calculators sold, we can use the formula:
slope = (change in y) / (change in x)
In this case, the change in y is (12,000 - 10,000) and the change in x is ($90 - $95).
Plugging in the values: slope = (12,000 - 10,000) / ($90 - $95) = 2,000 / (-5) = -400
Step 2: Determine the equation of the line
We can use the point-slope form of a linear equation to find the equation of the line:
y - y1 = m(x - x1)
Using the known point (10,000, $95), we have:
y - 10,000 = -400(x - $95)
Simplifying the equation: y - 10,000 = -400x + $38,000
Step 3: Substitute the desired price and solve for the number of calculators sold
To find the number of calculators sold at a price of $75, substitute x = $75 into the equation and solve for y:
y - 10,000 = -400($75) + $38,000
Simplifying the equation: y - 10,000 = -30,000 + $38,000
Combining like terms: y - 10,000 = $8,000
Adding 10,000 to both sides: y = $18,000
Therefore, it is predicted that the number of calculators that will be sold at a price of $75 is 18,000.