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
NAE NAE
Just plug in p and solve for x:
1000 = 10000(1 - 5/(5+e^-.001x))
.1 = 1 - 5/(5+e^-.001x)
.9 = 5/(5+e^-.001x)
5+e^-.001x = 5/.9
e^-.001x = .5555
-.001x = ln .5555
.001x = .58778
x = 587.78
To find the number of units sold for prices of $1000 and $1500, we need to solve the demand function equation for the given prices.
(a) For p = $1000:
p = 10000 [1 - (5 / (5 + e^(-0.001x)))]
1000 = 10000 [1 - (5 / (5 + e^(-0.001x)))]
Divide both sides of the equation by 10000:
0.1 = 1 - (5 / (5 + e^(-0.001x)))
Rearrange the equation:
0.1 - 1 = - (5 / (5 + e^(-0.001x)))
-0.9 = - (5 / (5 + e^(-0.001x)))
Multiply both sides by (5 + e^(-0.001x)):
-0.9(5 + e^(-0.001x)) = -5
Expand and simplify:
-4.5 - 0.9e^(-0.001x) = -5
Add 4.5 to both sides:
-0.9e^(-0.001x) = -0.5
Divide both sides by -0.9:
e^(-0.001x) = 0.555555556
Take the natural logarithm (ln) of both sides:
-0.001x = ln(0.555555556)
Divide both sides by -0.001:
x = ln(0.555555556) / -0.001
Using a calculator, ln(0.555555556) ≈ -0.5898
x ≈ -0.5898 / -0.001
x ≈ 589.8
Rounded to the nearest integer, the number of units sold for p = $1000 is approximately 590 units.
(b) For p = $1500:
Using the same steps as above, we find:
x ≈ 770
Rounded to the nearest integer, the number of units sold for p = $1500 is approximately 770 units.
Therefore:
(a) p = $1000 is approximately 590 units.
(b) p = $1500 is approximately 770 units.
To find the number of units sold for a given price, we need to solve the demand function equation for 'x'. Given that demand function is:
p = 10,000 [1 − (5/5 + e^−0.001x)]
For part (a), where p = $1000, we can substitute this value into the equation and solve for 'x'. Let's start with solving for 'x':
1000 = 10,000 [1 − (5/5 + e^−0.001x)]
Divide both sides by 10,000:
0.1 = 1 − (5/5 + e^−0.001x)
Next, simplify the equation:
0.1 = 1 − (1 + e^−0.001x)
0.1 = 1 − 1 − e^−0.001x
0.1 = -e^−0.001x
Multiply both sides by -1:
-0.1 = e^−0.001x
Now, take the natural logarithm (ln) of both sides to isolate the exponential term:
ln(-0.1) = -0.001x
x = ln(-0.1) / -0.001
Using a calculator, evaluate this expression:
x ≈ 230.26
Therefore, approximately 230 units are sold when the price is $1000.
For part (b), where p = $1500, we can follow a similar procedure to find 'x':
1500 = 10,000 [1 − (5/5 + e^−0.001x)]
Divide both sides by 10,000:
0.15 = 1 − (5/5 + e^−0.001x)
Simplify the equation:
0.15 = 1 − (1 + e^−0.001x)
0.15 = -e^−0.001x
Multiply both sides by -1:
-0.15 = e^−0.001x
Take the natural logarithm (ln) of both sides:
ln(-0.15) = -0.001x
x = ln(-0.15) / -0.001
Evaluate this expression using a calculator:
x ≈ 438.18
Therefore, approximately 438 units are sold when the price is $1500.