a) (3 points) Suppose a straight line passes through the points (6, 15) and (-6,9). Write downthe equation of this line.b) (3 points) The demand function is given by P = a − 8Q. We know that the demand in unit-elastic at Q = 70. What is the value of a?2
Question
a) (3 points) Suppose a straight line passes through the points (6, 15) and (-6,9). Write downthe equation of this line.b) (3 points) The demand function is given by P = a − 8Q. We know that the demand in unit-elastic at Q = 70. What is the value of a?2
Solution
a) To find the equation of a line passing through two points, we first need to find the slope (m) of the line. The formula for the slope is (y2 - y1) / (x2 - x1).
Here, (x1, y1) = (6, 15) and (x2, y2) = (-6, 9).
So, m = (9 - 15) / (-6 - 6) = -6 / -12 = 0.5
The equation of a line is y = mx + c, where c is the y-intercept. We can find c by substituting one of the points and the slope into this equation.
Let's use the point (6, 15). So, 15 = 0.5 * 6 + c. Solving for c gives c = 15 - 3 = 12.
Therefore, the equation of the line is y = 0.5x + 12.
b) The demand function is given by P = a - 8Q. We know that the demand is unit-elastic at Q = 70.
Unit elasticity means that the percentage change in quantity demanded is equal to the percentage change in price. Mathematically, this is represented as (dQ/Q) / (dP/P) = 1.
In this case, we can write the elasticity as (dQ/70) / (dP/(a - 8*70)) = 1.
Solving this equation for a gives a = 8*70 + dP * 70 / dQ.
Without additional information about the changes in price (dP) and quantity (dQ), we cannot solve for a.
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