The demand equation for a microwave oven is given by:
p=500-0.5(e^0.004x)
Find the demand x for a price of (a) p=$600 and (b) p=$300
Any help would be greatly appreciated.
To find the demand x for a given price, we need to solve the demand equation for x. Let's start with part (a):
(a) To find the demand x for a price of p=$600, we need to solve the equation:
600 = 500 - 0.5(e^0.004x)
Step 1: Begin by isolating the exponential term on one side of the equation.
0.5(e^0.004x) = 500 - 600
Step 2: Simplify the right side of the equation.
0.5(e^0.004x) = -100
Step 3: Divide both sides of the equation by 0.5 to solve for the exponential term.
e^0.004x = -100 / 0.5
e^0.004x = -200
Step 4: Take the natural logarithm (ln) of both sides to eliminate the exponential term.
0.004x = ln(-200)
Step 5: Divide both sides of the equation by 0.004 to solve for x.
x = ln(-200) / 0.004
However, we encounter a problem here. The natural logarithm is undefined for negative values. Hence, we conclude that a price of $600 does not correspond to any real demand for the microwave oven.
Now let's move on to part (b):
(b) To find the demand x for a price of p=$300, we need to solve the equation:
300 = 500 - 0.5(e^0.004x)
Step 1: Isolate the exponential term on one side of the equation.
0.5(e^0.004x) = 500 - 300
Step 2: Simplify the right side of the equation.
0.5(e^0.004x) = 200
Step 3: Divide both sides of the equation by 0.5 to solve for the exponential term.
e^0.004x = 200 / 0.5
e^0.004x = 400
Step 4: Take the natural logarithm (ln) of both sides to eliminate the exponential term.
0.004x = ln(400)
Step 5: Divide both sides of the equation by 0.004 to solve for x.
x = ln(400) / 0.004
Using a calculator, evaluate ln(400) and divide by 0.004 to find the value of x that corresponds to a price of $300.