Question

An arrow is shot vertically upward from a platform 45 feet high at a rate of 168 ft/sec. Use the quadratic function

h(t) = -16t2 + 168t + 45 to find how long it will take for the arrow to reach its maximum height, and then find the

maximum height. Round your answers to the nearest tenth.

Answer 1

To determine the maximum height and the time it takes to reach it, we use the vertex of the parabola represented by the quadratic function. The vertex formula gives us 5.25 seconds, and plugging this value back into the function yields a maximum height of approximately 513.25 feet. After rounding, it's 5.3 seconds to reach a maximum height of 513.3 feet.

Step-by-step explanation:

To find how long it takes for the arrow to reach its maximum height using the quadratic function h(t) = -16t2 + 168t + 45, we need to find the vertex of the parabola, because the vertex will give us both the time at which the arrow reaches its maximum height and the maximum height itself. The vertex can be found at time t = -b/(2a), where a and b are coefficients from the quadratic equation h(t) = at2 + bt + c. Plugging in the values from the given function, we have t = -168/(2*(-16)) = 5.25 seconds. This is the time at which the arrow will reach its maximum height.

Next, we plug t = 5.25 seconds back into the original equation to find the maximum height: h(5.25) = -16(5.25)2 + 168(5.25) + 45. Calculating this gives us the maximum height the arrow reaches, which is approximately 513.25 feet.

So, the arrow will reach its maximum height in 5.3 seconds (rounded to the nearest tenth), and the maximum height will be approximately 513.3 feet (also rounded to the nearest tenth).

Answer 2

486 feet is reached after 5.25 seconds, which would round to 5.3

Step-by-step explanation:

Since a is negative, the parabola opens downward. The quadratic has a maximum. The equation is in general form, y=a
`x^(2)`+bx+c, so use the formula −
`(b)/(2a)` to find the axis of symmetry.

The vertex is on the line t=5.25. The maximum height is reached after 5.25 seconds.

To find the maximum height, evaluate h(5.25).

h(5.25)=−16
`t^(2)`
`(5.25)^(2)`)+168(5.25)+45=486

The maximum height of 486 feet is reached after 5.25 seconds.