the dart from a high-powered dart gun follows a path given by h(t)=-4t^2+62t+10 where h is the height in inches t seconds after it is fired. What is the maximum height reached? how many seconds did it take to reach this height?
max height reached when t = -62/-8
To find the maximum height reached by the dart and the time it takes to reach this height, you need to determine the vertex of the parabolic function h(t) = -4t^2 + 62t + 10.
The vertex of a parabola in the form y = ax^2 + bx + c is given by the formula (-b/2a, f(-b/2a)), where f(x) represents the value of y at the given x-coordinate.
In this case, the function is h(t) = -4t^2 + 62t + 10, where a = -4, b = 62, and c = 10.
To find the x-coordinate of the vertex, use the formula: t = -b/2a.
t = -62 / (2 * -4) = 62 / 8 = 15/2 = 7.5 seconds.
Now, substitute this value back into the original equation to find the maximum height:
h(7.5) = -4(7.5)^2 + 62(7.5) + 10
= -180 + 465 + 10
= 295 inches.
Therefore, the maximum height reached by the dart is 295 inches, and it takes 7.5 seconds to reach this height.