Answer:
a. $25
b. $2025
Step-by-step explanation:
The profit the club makes is
profit = number of customers x amount charged per customer
Now, we are told that if we increase the price by $1, we will lose one customer.
Therefore, if we increase the price by $p then the number of customers we will have is
remaining customers = 70 customers - p = 70 - p
and the amount charged will become
amount charged per customer = 20 + p
Therefore, the profit will be
profit = ( 70 - p) ( 20 + p)
Now, the question is, for what value of p is the profit maximum?
Note that the equation for profit is a parabola (more precisely, a concave down parabola). Therefore, the vertex of the parabola gives the value of the maximum profit.
The parabola (70 - p)(20 + p) is graphed below.
As can be seen, the vertex of the parabola is at (25, 2025). This means that for p = 25, the profit is maximum.
In other words, if the club increases its price by $25, it will earn the maximum profit.
This maximum profit will be $2025.
Hence, to summarise.
a. $25
b. $2025