A trader realizes 25x − x
2 naira from the
sale of x bags of rice. Find the sum of bags
that will give him the maximum profit.
A. 14
B. 11
C. 12½
D. 22
To find the sum of bags that will give him the maximum profit, we need to maximize the profit function.
The profit function is given by P(x) = 25x - x².
To find the maximum of this quadratic function, we can take the derivative and set it equal to zero.
P'(x) = 25 - 2x = 0
Solving for x, we get:
25 - 2x = 0
2x = 25
x = 25/2
x = 12.5
The trader will have the maximum profit when he sells 12.5 bags of rice.
However, since the question asks for the sum of bags, we need to round this value to the nearest whole number. The sum of bags that will give him the maximum profit is 13.
Therefore, the answer is not among the given options.
To find the sum of bags that will give the trader the maximum profit, we need to determine the value of x that maximizes the expression 25x - x^2.
To find the maximum value, we can use the concept of the vertex of a parabola. The vertex represents the point where the function reaches its maximum (or minimum) value.
The expression 25x - x^2 represents a quadratic equation, so we know its graph will be a downward-opening parabola.
The formula for the x-coordinate of the vertex of a quadratic equation in the form ax^2 + bx + c is given by:
x = -b / (2a)
In this case, a = -1, b = 25, and c = 0. Plugging those values into the formula, we get:
x = -25 / (2*(-1))
x = -25 / (-2)
x = 12.5
So, the sum of bags that will give the trader the maximum profit is 12.5 bags.
However, since the number of bags must be a whole number, we have to round the answer. The closest whole numbers to 12.5 are 12 and 13. To decide which one gives the maximum profit, we can evaluate the expression for both values.
For x = 12, the profit is:
25(12) - 12^2 = 300 - 144 = 156
For x = 13, the profit is:
25(13) - 13^2 = 325 - 169 = 156
The profits for both values are the same, so either 12 or 13 bags will give the maximum profit.
Therefore, the correct answer is C. 12½