The hourly profit ($P) obtained from operating a fleet of n taxis is given by
P=-2n^2+84n-45
What is the profit if 20 taxis are on the road?
What is the maximum hourly profit?
What number of taxis gives the max hourly profit?
How much money is lost per hour if no taxis are on the road?
Hey, I told you use completing the square to find the vertex, boring.
2 n^2 - 84 n = - p - 45
n^2 - 42 n = -p/2 - 22.5
n^2 - 42 n + (21)^2 = -p/2 -22.5 + 441
(n-21)^2 = - (p/2 - 418.5)
(n-21)^2 = - (1/2) (p - 837)
so 21 taxis making 837
To find the profit when 20 taxis are on the road, substitute n=20 into the profit equation P=-2n^2+84n-45:
P = -2(20)^2 + 84(20) - 45
P = -2(400) + 1680 - 45
P = -800 + 1680 - 45
P = 835
Therefore, the profit when 20 taxis are on the road is $835.
To find the maximum hourly profit, we need to determine the vertex of the quadratic equation. The vertex can be found using the formula -b/2a, where the equation is in the form ax^2 + bx + c.
In this case, the quadratic equation is P = -2n^2 + 84n - 45, so a = -2, b = 84, and c = -45.
The x-coordinate of the vertex is given by -b/2a = -84 / (2 * -2) = -84 / -4 = 21.
To find the maximum hourly profit, substitute n=21 into the profit equation:
P = -2(21)^2 + 84(21) - 45
P = -2(441) + 1764 - 45
P = -882 + 1764 - 45
P = 837
Thus, the maximum hourly profit is $837.
The number of taxis that gives the maximum hourly profit is the x-coordinate of the vertex, which is 21 taxis.
To find the amount of money lost per hour if no taxis are on the road, substitute n=0 into the profit equation:
P = -2(0)^2 + 84(0) - 45
P = -2(0) + 0 - 45
P = -45
Therefore, if no taxis are on the road, the company loses $45 per hour.