algebra 2 trig: A food vendor at a stadium sells hot dogs for $2. At this price, she can sell about 500 hot dogs per day. For every $.25 increase in price, she will sell 25 less hot dogs. The vendor's revenue can be modeled by R=(500-25x)(2+0.25x) Use vertex form to find how the vendor can maximize her daily revenue.
Don't see much trig here.
R = (500-25x)(2+.25x)
= 1000 + 125x - 50x - 6.25x^2
= 1000 + 75x - 6.25x^2
R = -6.25(x^2 - 12x - 160)
= -6.25(x^2 - 12x + 36 - 160 - 36)
= -6.25(x-6)^2 - 600
So, the max occurs where x = 6
R(6) = 1225 when dogs cost $2 + 6*.25 = $3.50
R=(500-25x)(2+0.25x)
To find the maximum daily revenue, we need to express the quadratic equation in vertex form. The vertex form of a quadratic equation is given by:
y = a(x-h)^2 + k
where (h, k) represents the vertex of the parabola.
In this case, we have the quadratic equation:
R = (500 - 25x)(2 + 0.25x)
Expanding the equation gives:
R = (1000 + 12.5x - 50x - 6.25x^2)
Rearranging the equation and combining like terms:
R = -6.25x^2 - 37.5x + 1000
To express the equation in vertex form, we can complete the square.
First, divide the equation by -6.25 to make the coefficient of x^2 equal to 1:
R = -6.25(x^2 + 6x - 160)
Next, we need to focus on the expression inside the parentheses (x^2 + 6x) and complete the square. To complete the square, we take half of the coefficient of x (6), square it (36), and add it to the expression. To balance the equation, we also need to subtract 160 (which we add) multiplied by -6.25.
R = -6.25(x^2 + 6x + 36 - 36 - 160)
R = -6.25(x^2 + 6x + 36 - 196)
R = -6.25(x^2 + 6x + 36) + 1225
Now we can rewrite the equation in vertex form:
R = -6.25(x + 3)^2 + 1225
Comparing this with the vertex form equation, y = a(x-h)^2 + k, we can see that the vertex is at the point (-3, 1225).
Therefore, the food vendor can maximize her daily revenue by selling the hot dogs for $2.75, where the maximum daily revenue is $1225.
To find the vendor's maximum daily revenue, we need to convert the given revenue function to vertex form. The vertex form of a quadratic function is in the form: y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
The revenue function given is R = (500 - 25x)(2 + 0.25x). First, let's expand this equation:
R = 1000 + 12.5x - 50x - 6.25x^2
Rearranging the terms, we get:
R = -6.25x^2 - 37.5x + 1000
Now, we can rewrite the function in vertex form by completing the square. To complete the square, we need to factor out the coefficient of x^2, which is -6.25. This gives us:
R = -6.25(x^2 + 6x) + 1000
Next, we add and subtract the square of half the coefficient of x (6/2)^2 = 9 to the equation:
R = -6.25(x^2 + 6x + 9 - 9) + 1000
This allows us to rewrite the equation as:
R = -6.25(x^2 + 6x + 9) + 56.25 + 1000
Simplifying this equation further:
R = -6.25(x + 3)^2 + 1056.25
The equation is now in vertex form, y = a(x - h)^2 + k, where the vertex is given by (h, k). Comparing this with our equation, we can see that the vertex is (-3, 1056.25).
Therefore, to maximize her daily revenue, the vendor should sell hot dogs at a price of $2.75 per hot dog, resulting in a maximum revenue of $1056.25.