The consumers will demand 32 units when the price of a product is $59, and 50 units when the price is $42. Find the demand function (express thr price p in terms of the quantity q), assuming it is linear.
To find the demand function, we can use the given price-quantity pairs to form two linear equations.
Let's denote the price as p and the quantity as q.
First, we have the point (p1, q1) = ($59, 32), which gives us the equation:
p1 = aq1 + b, where a and b are constants.
Substituting the values, we have:
$59 = 32a + b --(1)
Next, we have the point (p2, q2) = ($42, 50), yielding the equation:
p2 = aq2 + b.
Substituting the values, we have:
$42 = 50a + b --(2)
We now have a system of two linear equations (1) and (2). We can solve this system to find the values of a and b. Here's how you can do it:
Subtracting equation (2) from equation (1), we get:
$59 - $42 = 32a - 50a + b - b
Simplifying, we have:
$17 = -18a
Dividing both sides by -18, we find:
a = -$17 / 18
Substituting the value of a into equation (1), we can solve for b:
$59 = 32(-$17 / 18) + b
Simplifying, we have:
$59 = -$32/9 + b
Adding -$32/9 to both sides, we find:
$59 + $32/9 = b
Combining like terms, we have:
$59 + $32/9 = b
To simplify further, we can write $59 as $531/9:
$531/9 + $32/9 = b
Combining the fractions, we get:
$563/9 = b
Therefore, the demand function (expressing the price p in terms of the quantity q) is:
p = (-$17/18)q + $563/9.
In this linear demand function, the coefficient of q (-$17/18) represents the rate of change in price with respect to quantity, and the constant term ($563/9) represents the initial price intercept.
using the two-point form of a line, you have
(p-59) = (42-59)/(50-32) * (q-32)
Now just rearrange that into
p = 1/18 (1606-17q)