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.