Question

Kristin owns a bakery called Kristin's Cakes n' Such and is considering lowering the price of her cakes. Kristin polls her customers and determines that she can sell 100 cakes each week when she chargers $25 each. She also discovers that for every $1 decrease in the price of the cake, she will sell 5 more cakes. At what price should Kristin sell her cakes in order to reach he maximum potential revenue?

Answer 1

The total revenue that is gained from the sales of the cakes is determined by multiplying the number of cakes by the price. If we let x be the number of $1 that should be deducted from the price and y be the total revenue,
                               y = (25 - x)(100 + 5x)
Simplifying, 
                              y = 2500 + 25x - 5x²
We get the value of x that will give us the maximum revenue by differentiating the equation and equating the differential to zero.
                             dy/dx = 0 = 25 - 10x
The value of x is 2.5. 
 The price of the cake should be 25 - 2.5 = 22.5. 
Thus, the price of the cake that will give the maximum potential revenue is $22.5. 

Answer 2

The maximum revenue is the highest point of the revenue function. It is

the revenue when the price is $22.5.

Response:

• The price Kristin should sell her cakes in order to reach maximum potential revenue is $22.5.

How is a quadratic function used in the current context?

A quadratic equation is one which is formed by product two linear

functions in the input variable, such as the function for the price and the

function for the quantity sold.

Given:

The number of cakes Kristin can sell each week at $25 = 100

Increase in the number sold for each $1 decrease in price = 5 cakes

Let x represent the number of extra cakes sold, we have;

Price at which 5·x + initial amount of cake are sold = Initial price - x

Initial amount = 100 cakes

Initial price = $25

Quantity sold following a reduction of price to (25 - x) = (100 + 5·x)

• Revenue, R = Quantity × Price

Therefore;

R = (100 + 5·x) × (25 - x) = 2,500 - 100·x + 125 - 5·x² = 2,500 + 25 - 5·x²

R = 2,500 - 25 - 5·x²

The revenue is a quadratic function of the form, y = a·x² + b·x + c

Given that the leading coefficient of the revenue function, R, is negative,

the function has a maximum value.

At the maximum value of a quadratic function, we have;

$x = \mathbf{\dfrac{-b}{2 \cdot a}}$

From the function for R, we have;

$x = \dfrac{-(25)}{2 \times (-5)} = \mathbf{2.5}$

The price that gives the maximum revenue, is therefore;

Price = 25 - x

At maximum revenue, x = 2.5

Price at maximum revenue =25 - 2.5 = 22.5

• The price that Kristin should sell her cakes to reach maximum potential revenue is $22.5

Learn more about quadratic functions here:

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