Question

A market survey shows that for every $ 0.10 reduction in the​ price, 40 more sandwiches will be sold. How much should the deli charge in order to maximize​ revenue?

Answer 1

Answer:

$2.40

Explanation:

Here is the complete question: A deli sells 320 sandwiches per day at a price of $4 each.

A market survey shows that for every $ 0.10 reduction in the​ price, 40 more sandwiches will be sold. How much should the deli charge in order to maximize​ revenue?

Given: Number of sandwiches sold per day is 320.

           Price of each sandwich is $4.

           For every $0.10 reduction in the​ price, 40 more sandwiches will be sold.

Let´s assume price of each sandwich sold at new reduced price be "x" times.

∴ New price of sandwich is $(4-0.10 \times x)$

As given, for every $ 0.10 reduction in the​ price, 40 more sandwiches will be sold.

∴ Total number of sandwiches sold at new price= $(320+40x)$

We know, revenue= $price\ of\ sandwiches\times number\ of\ sandwiches$

⇒ revenue= $(4-0.10x)\times (320+40x)$

using distributive property of multiplication.

⇒ revenue= $-4x^{2} +128x+1280$

Considering the revenue to be R(x)

$R(x)= -4x^{2} +128x+1280$

Now taking derivatives of the revenue.

$R(x)$= $\int\limits {-4x^{2} +128x+1280} \, dx$

Solving it.

We get, $R(x)= -8x+128$

Finding critical points by setting derivative equal to 0

R´(x)⇒ $-8x+128= 0$

Subtracting both side by 128

⇒ $-8x= -128$

Dividing both side by -8

⇒ $x= \frac{128}{8}$

∴ x= 16

Hence, price of each sandwich is reduced by $0.10 is 16 times.

Next, finding cost of each sandwich to maximize revenue.

cost of each sandwich to maximize revenue= $(4-0.10 \times 16)$

∴ Cost of each sandwich to maximize revenue= $\ 2.40$

Hence, Deli should charges $2.40 for each sandwich to maximize revenue.