The demand function for a product is modeled by :
p=10,000(1-(3/3+e^-0.001x)
Find the price of the product if the quantity demanded is (a) x=1000 units and (b) x=1500 units. What is the limit of the price as x increases without bound?
To find the price of the product given a certain quantity demanded, we can substitute the value of x into the demand function and evaluate it. Let's solve for each case:
(a) x = 1000 units:
p = 10,000(1 - (3/3 + e^(-0.001x)))
p = 10,000(1 - (3/3 + e^(-0.001 * 1000)))
p = 10,000(1 - (3/3 + e^(-1)))
p = 10,000(1 - (3/3 + 0.3679))
p = 10,000(1 - (1 + 0.3679))
p = 10,000(1 - 1.3679)
p = 10,000 * (-0.3679)
p = -3,679
Therefore, if the quantity demanded is 1000 units, the price of the product would be -3,679 units. However, negative prices are not practical, so it is best to assume that the price cannot be negative and consider the absolute value. Hence, the price would be 3,679 units.
(b) x = 1500 units:
p = 10,000(1 - (3/3 + e^(-0.001x)))
p = 10,000(1 - (3/3 + e^(-0.001 * 1500)))
p = 10,000(1 - (3/3 + e^(-1.5)))
p = 10,000(1 - (1 + 0.2231))
p = 10,000(1 - 1.2231)
p = 10,000 * (-0.2231)
p = -2,231
Similar to the previous case, we should use the absolute value to obtain a positive price. Therefore, if the quantity demanded is 1500 units, the price of the product would be 2,231 units.
Now, to find the limit of the price as x increases without bound, we need to evaluate the limit of the demand function as x approaches infinity (x -> ∞).
lim(x -> ∞) [p] = lim(x -> ∞) [10,000(1 - (3/3 + e^(-0.001x)))]
To evaluate this limit, we need to analyze the behavior of the exponential term e^(-0.001x) as x approaches infinity. Since the base of the exponential function is e (Euler's number, approximately 2.718), and the exponent approaches negative infinity, the exponential term diminishes to zero.
Therefore, the limit of the price as x increases without bound is:
lim(x -> ∞) [p] = 10,000(1 - (3/3 + 0))
lim(x -> ∞) [p] = 10,000(1 - 1)
lim(x -> ∞) [p] = 10,000 * 0
lim(x -> ∞) [p] = 0
Hence, the limit of the price as x increases without bound is zero.