I need help to start this problem please,
A company can sell 4500 pairs of sunglases monthly when the price is $5. When the price of a pair of sunglasses is increased by 10%, the demand drops to 4250 pairs a month. Assume that the demand equation is linear.
Find the demand equation
Assume the demand equation is linear.
We have two known points:
P1(5,4500), and P2(5.50,4250)
To fit a straight line between two known points, we can apply the two-point form of line, namely,
(x-x1)/(x2-x1)=(y-y1)/(y2-y1)
The equation seems long, but its symmetry makes it very easy to memorize.
Substitute x1=5,y1=4500,x2=5.5,y2=4250, we get
(x-5)/(5.50-5)=(y-4500)/(4250-4500)
-250(x-5)=0.5(y-4500)
y=-500x+7000
To find the demand equation, we need to first identify two points on the linear demand curve: (quantity, price) pairs. We are given two data points in the problem:
Point 1: (quantity = 4500 pairs, price = $5)
Point 2: (quantity = 4250 pairs, price increased by 10%)
To find the price after the 10% increase, we can calculate the new price as follows:
Price increased by 10% = $5 + ($5 * 10%) = $5 + ($5 * 0.1) = $5 + $0.50 = $5.50
So, Point 2 becomes: (quantity = 4250 pairs, price = $5.50).
Now, we have two points, and we can use the point-slope form of the equation of a line to find the demand equation. The point-slope form is given by:
(y - y1) = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope of the line.
Let's use Point 1 as (x1, y1) = (4500, $5) and Point 2 as (x2, y2) = (4250, $5.50).
Using the point-slope form, we have:
(quantity - 4500) = m(price - $5)
Substituting the values of Point 1, we get:
(quantity - 4500) = m($5 - $5)
Which simplifies to:
(quantity - 4500) = 0
Thus, the slope (m) of the demand equation is 0.
Now we can write the equation of the demand curve as:
(quantity - 4500) = 0(price - $5)
Simplifying further, we have:
quantity - 4500 = 0
And finally:
quantity = 4500
So, the demand equation for this scenario is: quantity = 4500.