Suppose that the selling price P of an item for the quantity X sold is given by the function P=-1/3x+44
Express the revenue R as a function of x (R=x*p ) how many items must be sold to Mackenzie the revenue and what is the maximum revenue that can be obtained from this model
Every business should try very hard to "Mackenzie the revenue" LOL
R = x(-1/3x + 44) = (-1/3)x^2 + 44x
this is a parabola, its max is the R value of its vertex
for the x of the vertex, -44/(-2/3) = 66
so 66 items must be sold to obtain the "MACKENZIE" and that would be
66( (-1/3)(66) + 44) = 1452
To express the revenue R as a function of x, we need to use the given selling price equation P = -1/3x + 44.
First, let's recall that the revenue R is the product of the quantity sold (x) and the selling price (P). Therefore, we have:
R = x * P
Substituting the given selling price equation P = -1/3x + 44 into the revenue equation, we get:
R = x * (-1/3x + 44)
Simplifying, we have:
R = (-1/3)x^2 + 44x
To find the number of items that must be sold to maximize the revenue, we need to determine the value of x that corresponds to the maximum point of the parabola given by the revenue function.
The revenue function R(x) = (-1/3)x^2 + 44x represents a downward opening parabola. The x-value of the maximum point can be found using the formula x = -b/2a, where a = -1/3 and b = 44.
Substituting the values into the formula, we have:
x = -(44)/(2*(-1/3))
x = -132/-2
x = 66
So, 66 items must be sold to maximize the revenue.
To find the maximum revenue, we substitute x = 66 into the revenue function:
R = (-1/3)(66)^2 + 44(66)
R = (-1/3)(4356) + 2904
R = -1452 + 2904
R = 1452
Therefore, the maximum revenue that can be obtained from this model is $1452.
To express the revenue R as a function of x, we need to multiply the quantity sold, x, by the selling price P. From the given equation, P = -1/3x + 44, we can substitute this value into the revenue function:
R = x * P
R = x * (-1/3x + 44)
Simplifying this equation, we get:
R = -1/3x^2 + 44x
To find the number of items that must be sold to maximize the revenue, we need to find the x-value that corresponds to the maximum value of the revenue function. In this case, we have a quadratic function R = -1/3x^2 + 44x, which has a downward opening parabola (since the coefficient of x^2 is negative).
The maximum value of the quadratic function can be found at the vertex, which occurs at x = -b/2a, where a and b are coefficients in the general quadratic equation (ax^2 + bx + c). In our case, a = -1/3 and b = 44.
x = -(44) / (2 * (-1/3))
x = -(44) / (-2/3)
To simplify this expression, we can multiply by the reciprocal of the fraction:
x = -(44) * (3 / -2)
x = -66 / -2
x = 33
Therefore, 33 items must be sold to maximize the revenue.
To determine the maximum revenue that can be obtained, we substitute the value of x into the revenue function:
R = -1/3(33)^2 + 44(33)
R = -1/3(1089) + 1452
R = -363 + 1452
R = 1089
The maximum revenue that can be obtained from this model is 1089 units.