3.) The demand equation for a certain product is
q=500-40p+p^2
here p is the price per unit (in dollars) and q is the quantity of units demanded (in thousands). Find the point elasticity of demand when p = 15. If this price of 15 is increased by 1/2%, what is the approximate change in demand?
So if I plug 15 into p. I get q(demand) = 125
Half of 15 is 7.5. Thus the price would be 22.5.. I think.
I replace p with 22.5 and I get q= 106.25
Q/P =8.33 for 15
Q/P = 4.722 for 22.5
Is the change in demand 3.608?
To find the point elasticity of demand when p = 15, you can use the formula for elasticity:
E = (dq/dp) * (p/q)
Where dq/dp is the derivative of the demand equation with respect to p, and (p/q) is the price-quantity ratio.
First, let's differentiate the demand equation.
dq/dp = -40 + 2p
Now, let's substitute p = 15 into dq/dp:
dq/dp = -40 + 2(15) = -40 + 30 = -10
Next, let's calculate the price-quantity ratio when p = 15:
(p/q) = 15/125 = 0.12
Finally, let's substitute the values into the elasticity formula:
E = (-10) * (0.12) = -1.2
The point elasticity of demand when p = 15 is -1.2.
Now, to find the approximate change in demand when the price of 15 is increased by 1/2%, we can use the point elasticity formula:
%∆Q ≈ E * %∆P
Where %∆Q is the percentage change in demand, E is the elasticity, and %∆P is the percentage change in price.
First, let's calculate the percentage change in price:
%∆P = (1/2)/100 = 0.005
Next, let's substitute the values into the formula:
%∆Q ≈ -1.2 * 0.005 = -0.006
The approximate change in demand is -0.006.
Therefore, the change in demand is approximately -0.006 (thousands of units), or -6 units.