Final answer:
The value of x at the point where the price increases by $6 in the supply function p = 40 + 100√2x + 44 can be found by differentiating the supply function with respect to x and solving for x when the derivative equals 6. The value of x at that point is (25/2)√2.
Step-by-step explanation:
The supply function is given by the equation p = 40 + 100√2x + 44, where x is the number of units supplied and p is the price in dollars. We are told that the price is increasing at a rate of $6 per unit increase in x. To find the value of x at that point, we need to determine the rate at which x is increasing when the price increases by $6.
Since the derivative of the supply function with respect to x represents the rate at which x is changing, we can use the derivative to find the rate at which x is increasing when the price increases by $6. Differentiating the supply function with respect to x, we get:
p' = 100/(2√2) = 50/√2
Setting this equal to 6, we can solve for x:
50/√2 = 6
x = (50/√2) / 6 = 25/√2 = (25/√2) * (√2/√2) = (25√2/2) * (√2/√2) = (25√2/2) * 2/2 = 25√2/2^2 = 25√2/4 = 25√2/2 * 2/2 = (25/2)√2.