at a price of x dollars, the supply function for a music player is q = 80e^{0.003x}, where q is in thousands…
at a price of x dollars, the supply function for a music player is q = 80e^{0.003x}, where q is in thousands of units.\nhow many music players will be supplied at a price of 150? (round to the nearest thousand.)\nthousand units\nfind the marginal supply\nmarginal supply(x) =\nwhich is the best interpretation of the derivative?\nthe rate of change of the quantity supplied as the price increases\nthe rate of change of the price as the quantity supplying increases\nthe price at a given supply of units\nthe quantity supplied if the price increases\nthe number of units that will be demanded at a given price\nadd work
Answer
Explanation:
Step1: Find quantity supplied at $x = 150$
Substitute $x = 150$ into $q=80e^{0.003x}$. So $q = 80e^{0.003\times150}=80e^{0.45}$. Using a calculator, $e^{0.45}\approx1.56831$, then $q = 80\times1.56831 = 125.4648\approx125$ (rounded to nearest thousand).
Step2: Find marginal supply
Differentiate $q = 80e^{0.003x}$ with respect to $x$. Using the rule $\frac{d}{dx}(e^{ax})=ae^{ax}$, we have $\frac{dq}{dx}=80\times0.003e^{0.003x}=0.24e^{0.003x}$. So Marginal supply$(x)=0.24e^{0.003x}$.
Step3: Interpret derivative
The derivative of the supply - function with respect to price $x$ represents the rate of change of the quantity supplied as the price increases.
Answer:
125 $0.24e^{0.003x}$ The rate of change of the quantity supplied as the price increases