SOLUTION: The demand equation for a certain product is given by p=5000(1-(4/(4+e^-0.002x))) Find the demands x for prices of a) p=$600 b) p=$400

Question 373981: The demand equation for a certain product is given by
p=5000(1-(4/(4+e^-0.002x)))
Find the demands x for prices of
a) p=$600
b) p=$400
Answer by jsmallt9(3759) (Show Source): You can put this solution on YOUR website!

To find x for p = 600:

We need "peel away" from the outside on the right side. First we get rid of the 5000 by dividing both sides be 5000:

Next the 1 must go. Subtract 1 from each side:

Multiply (or divide) both sides by -1 to eliminate the minus sign:

Next we'll eliminate the fraction by multiplying both sides by :

Using the Distributive Property on the left side:

Subtracting 3.52 from each side:

Divide by 0.88:

Now that the exponential part of the equation is isolated, we use logarithms to proceed. Since the base of the exponent is e, it is best to use base e logarithms (aka lm):

On the left side we can use a property of logarithms, , to "move" the exponent out in front of the logarithm. (It is this property which is the very reason we use logarithms. It gives us a way to get the variable out of the exponent.) Using the property on the left side we get:

By definition, so this becomes:

Last of all we divide by -0.002:

This is an exact expression for your answer. Use your calculator if you want a decimal approximation:

x = 430.9012732950426674
So for a price of $699 the demand will be approximately 431.

For a price of $400, replace the p with 400 and repeat these steps to find the demand at that price.
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