Question

Your are driving away from New York. Your distance (in miles) away from New York x hours after 12:00 noon is given by f(t)=−4x3+27x2+80x+49. In how many hours are you the farthest north of home?

Answer 1

To find the time at which you are the farthest north of home, we need to find the maximum value of the function f(t) = -4x^3 + 27x^2 + 80x + 49. The highest point on a cubic function occurs at its vertex, which can be found using the formula -b/2a. In this case, the equation is in the form f(x) = ax^3 + bx^2 + cx + d, so a = -4, b = 27, c = 80. Plugging these values into the formula, we find the x-coordinate of the vertex to be -b/2a = -27/(2*(-4)) = 27/8. Therefore, you are the farthest north of home after approximately 27/8 hours.

Step-by-step explanation:

To find the time at which you are the farthest north of home, we need to find the maximum value of the function f(t) = -4x^3 + 27x^2 + 80x + 49. The highest point on a cubic function occurs at its vertex, which can be found using the formula -b/2a. In this case, the equation is in the form f(x) = ax^3 + bx^2 + cx + d, so a = -4, b = 27, c = 80. Plugging these values into the formula, we find the x-coordinate of the vertex to be -b/2a = -27/(2*(-4)) = 27/8. Therefore, you are the farthest north of home after approximately 27/8 hours.