Question

technology required. a small marshmallow is launched straight up in the air with a slingshot. the function h, given by the equation h(t) = 5 + 20t - 5t², describes the height of the marshmallow in meters as a function of time, t, in seconds since it was launched. 1. use graphing technology to graph the function h. 2. about when does the marshmallow reach its maximum height? 3. about how long does it take before the marshmallow hits the ground? 4. what domain makes sense for the function h in this situation? (from unit 6, lesson 5.)

Explanation:

Sub - question 2

Step1: Recall the formula for vertex of a quadratic function

For a quadratic function in the form \(h(t)=at^{2}+bt + c\) (here \(a=- 5\), \(b = 20\), \(c = 5\)), the time \(t\) at which the vertex (maximum or minimum) occurs is given by \(t=-\frac{b}{2a}\).

Step2: Substitute the values of \(a\) and \(b\)

We have \(a=-5\) and \(b = 20\). Substituting into the formula \(t =-\frac{b}{2a}\), we get \(t=-\frac{20}{2\times(-5)}\).
First, calculate the denominator: \(2\times(-5)=- 10\). Then, \(t =-\frac{20}{-10}=2\).

Step1: Set \(h(t)=0\)

We need to solve the equation \(0=5 + 20t-5t^{2}\). First, we can rewrite the equation as \(5t^{2}-20t - 5=0\). Divide the entire equation by 5 to simplify: \(t^{2}-4t - 1=0\).

Step2: Use the quadratic formula

For a quadratic equation \(ax^{2}+bx + c = 0\) (here \(a = 1\), \(b=-4\), \(c=-1\)), the quadratic formula is \(t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
Substitute \(a = 1\), \(b=-4\), \(c=-1\) into the formula: \(t=\frac{4\pm\sqrt{(-4)^{2}-4\times1\times(-1)}}{2\times1}=\frac{4\pm\sqrt{16 + 4}}{2}=\frac{4\pm\sqrt{20}}{2}=\frac{4\pm2\sqrt{5}}{2}=2\pm\sqrt{5}\).
Since time cannot be negative, we take the positive root. \(\sqrt{5}\approx2.24\), so \(t=2+\sqrt{5}\approx2 + 2.24 = 4.24\) seconds.

Step1: Consider the context

The time \(t\) starts at \(t = 0\) (when the marshmallow is launched) and ends when the marshmallow hits the ground (at \(t=2+\sqrt{5}\approx4.24\) seconds). So the domain of the function \(h(t)\) in this context is the set of all real numbers \(t\) such that \(0\leq t\leq2+\sqrt{5}\) (or approximately \(0\leq t\leq4.24\)).

Answer:

The marshmallow reaches its maximum height at \(t = 2\) seconds.

Sub - question 3