The demand function for a product is given by
p = 10,000 [1 − (5/5 + e^−0.001x)]
where p is the price per unit (in dollars) and x is the number of units sold. Find the numbers of units sold for prices of
p = $1000
and
p = $1500.
(Round your answers to the nearest integer.)
(a) p = $1000 _____units
(b) p = $1500 _____units
To find the number of units sold for a given price, we need to solve the demand function equation for x. Let's start with the first price:
(a) p = $1000:
Substitute p = $1000 into the demand function equation:
1000 = 10,000 [1 − (5/5 + e^−0.001x)]
Simplify the equation by dividing everything by 10,000:
0.1 = 1 − (5/5 + e^−0.001x)
Let's focus on the exponential term e^−0.001x first. To isolate it, subtract 1 from both sides:
-0.9 = -5/5 + e^−0.001x
Now, add 5/5 to both sides:
0.1 = e^−0.001x
To solve for x, take the natural logarithm of both sides:
ln(0.1) = ln(e^−0.001x)
Simplify the right side using logarithm properties:
ln(0.1) = -0.001x
Now, divide both sides by -0.001:
x = ln(0.1) / -0.001
Using a calculator, calculate ln(0.1) / -0.001 to find the approximate value of x.
(b) p = $1500:
Following the same steps as above, substitute p = $1500 into the demand function equation, simplify, and solve for x.
1500 = 10,000 [1 − (5/5 + e^−0.001x)]
Simplify the equation:
0.5 = e^−0.001x
Take the natural logarithm of both sides:
ln(0.5) = ln(e^−0.001x)
Simplify the right side:
ln(0.5) = -0.001x
Divide both sides by -0.001:
x = ln(0.5) / -0.001
Again, use a calculator to find the approximate value of x.