Dalto Pizza currently sells 1000 pizzas per week at $18 per pizza. It is planning to reduce the unit price of each pizza. It estimates that for every $1 discount in price, it can sell 100 more pizzas each week.
(a) Form the weekly revenue function of Dalto Pizza in terms of p, the new unit price of the pizza.
(b) What should the new unit price be in order to maximize weekly revenue? What is the maximum weekly revenue?
To answer these questions, we need to understand the relationship between the unit price of the pizza and the number of pizzas sold each week. We can then use this information to calculate the revenue and find the optimal price for maximizing revenue.
Let's start by defining some variables:
- Let p be the new unit price of the pizza.
- Let q be the number of pizzas sold each week.
Based on the given information, we know that for every $1 discount in price, 100 more pizzas are sold each week. Therefore, the equation relating the price discount and the increase in quantity sold is given by:
q = 100(p - 18)
(a) Now, let's form the weekly revenue function in terms of the new unit price p. The revenue function is calculated by multiplying the unit price by the number of pizzas sold:
R(p) = p * q
Substituting the equation for q, we get:
R(p) = p * 100(p - 18)
Simplifying further:
R(p) = 100p^2 - 1800p
(b) To find the price that maximizes the weekly revenue, we need to find the maximum point of the revenue function R(p). Since R(p) is a quadratic function (100p^2 - 1800p), the maximum occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula -b/2a, where a = 100 and b = -1800.
Using this formula:
p = -(-1800) / (2 * 100)
p = 18
Therefore, the new unit price should be $18 in order to maximize the weekly revenue.
To find the maximum weekly revenue, substitute the value of p back into the revenue function R(p):
R(p) = 100(18)^2 - 1800(18)
R(p) = 32,400 - 32,400
R(p) = $0
Hence, the maximum weekly revenue is $0, which means that reducing the price to $18 would result in zero revenue.