Question

The demand function for a certain commodity is given by p = 100e⁻ᵠ/², where p is the price per unit and q is the number of units. At what price per unit will the quantity demanded be half of the original demand? In other words, find the price p when q = 1/2ᵩₒ, where qo is the original quantity demanded.

Answer 1

The price per unit when the quantity demanded is half of the original demand
`(\(q_0\))` is given by
`\(p = 50\).`

Step-by-step explanation:

The given demand function is
`\(p = 100e^{-(q)/(2)}\)`, where
`\(p\)`is the price per unit and
`\(q\)` is the number of units.

To find the price per unit when the quantity demanded is half of the original demand
`(\(q_0\))`, set
`\(q = (1)/(2)q_0\)` in the demand function and solve for
`\(p\):`

`\[p = 100e^{-(1)/(2)\cdot(q_0)/(2)}\]`

Simplifying this expression gives:

`\[p = 100e^{-(q_0)/(4)}\]`

Now, since
`\(e^{-(q_0)/(4)}\)` is the same as
`\(\frac{1}{e^{(q_0)/(4)}}\)`, we can rewrite the expression as:

`\[p = \frac{100}{e^{(q_0)/(4)}}\]`

Now, when
`\(q_0\)` is substituted by
`\(\ln(2)\)` (since half of the original demand corresponds to
`\(q = (1)/(2)\)),` we get:

`\[p = (100)/(e^(\ln(2)/4))\]`

Simplifying further gives:

`\[p = \frac{100}{\sqrt[4]{2}}\]`

`\[p = 50\]\alpha`

Therefore, the price per unit when the quantity demanded is half of the original demand is
`\(p = 50\).`