Question

You and a friend are hiking in the mountains. You want to climb to a ledge that is 20 ft above you. The height of the grappling hook you throw is given by the function h(t) = -16t^2 + 32t + 5. What is the maximum height of the grappling hook? Can you throw it high enough to reach the ledge?

a. The maximum height is 12 feet, and you can't reach the ledge.
b. The maximum height is 37 feet, and you can reach the ledge.
c. The maximum height is 20 feet, and you can reach the ledge.
d. The maximum height is 5 feet, and you can't reach the ledge

Answer 1

The maximum height of the grappling hook is calculated using the vertex formula for a parabola, resulting in a maximum height of 21 feet, which means the grappling hook can reach the 20 feet high ledge.

Step-by-step explanation:

To find the maximum height of the grappling hook, we need to consider the function h(t) = -16t^2 + 32t + 5, which gives the height of the grappling hook at any time t. This is a quadratic function, and its maximum height can be found using the vertex form of a parabola.

The vertex of a parabola given by f(x) = ax^2 + bx + c is at the point (-b/(2a), f(-b/(2a))). For our function, a = -16 and b = 32. Therefore, the time at which the maximum height occurs is t = -b/(2a) = -32/(2*(-16)) = 1 second.

Plugging this time into the height function to find the maximum height we get: h(1) = -16(1)^2 + 32(1) + 5 = -16 + 32 + 5 = 21 feet.

Since the ledge is 20 feet above the ground, the grappling hook does indeed reach the ledge. Therefore, the correct answer is: (b) The maximum height is 37 feet, and you can reach the ledge.