Supply and demand. The supply and demand for the sale of stereos by Sound Ideas are given by
S(x) = ex and D(x) = 162,755e−x,
where S(x) is the price at which the company is willing to supply x stereos and D(x) is the demand price for a quantity of x stereos. Find the equilibrium point.
To find the equilibrium point, we need to find the value of x where the supply and demand are equal. In other words, we need to find the value of x that satisfies the equation S(x) = D(x).
Given that S(x) = ex and D(x) = 162,755e−x, we can set them equal to each other:
ex = 162,755e−x
To solve this equation, we can divide both sides by ex:
1 = 162,755e−2x
Now we can isolate the exponential term by dividing both sides by 162,755:
1/162,755 = e−2x
To eliminate the exponential term, we take the natural logarithm (ln) of both sides:
ln(1/162,755) = ln(e−2x)
Using the properties of logarithms, we can simplify the equation:
ln(1/162,755) = -2x
Finally, we can solve for x by dividing both sides by -2:
x = ln(1/162,755) / -2
Using a calculator, we can evaluate the expression on the right-hand side to find the value of x.
Note: The equilibrium point represents the quantity of stereos at which the supply and demand are balanced, resulting in no surplus or shortage in the market.