Chapter 1.5, Problem 1MRPS

Solution

a.

To write: A function that models the height of the pinecone as it falls.

A function that models the height of the pinecone as it falls is $h=-16t^2+20$

Given:

A pinecone falls from a tree branch that is $20$ feet above the ground.

Calculation:

For a falling object, the height $h$ above the ground after $t$ seconds can be modeled by the function $h=-16t^2+h_0$ where $h_0$ is the object's initial height.

In this case, the initial height is $20$ feet. Substitute $20$ for $h_0$ to write a function that models the height of the pinecone as it falls.

  $h=-16t^2+20$

b.

To graph: The function.

Domain is $\left(-\infty ,\infty \right)$ ad the range is $\left(-\infty ,16\right)$

Given:

A pinecone falls from a tree branch that is $20$ feet above the ground.

Graph:

First, find some points to graph the function. For this, make table of values for the given function.

Substitute any value for $x$ , say, $-2$ and evaluate $y$ .

  $\begin{array}{c}y=-16{\left(-2\right)}^2+20\\ =-16\left(4\right)+20\\ =44\end{array}$

Choose some more $x$ -values and find the corresponding $y$ -values. Organize the results in a table.

  $\begin{array}{cccccc}x& -2& -1& 0& 1& 2\\ y& 44& 4& 20& 4& 44\end{array}$

Plot the points as follows.

  

Join the points as follows.

  

c.

To find: Time taken by the pinecone to hit the ground

The pinecone takes about $1.1$ seconds to hit the ground.

Given:

A pinecone falls from a tree branch that is $20$ feet above the ground.

Calculation:

Height when the falling object hits the surface is $0$ .

Substitute $0$ for $h$ in the function that models the height of the pinecone as it falls.

  $0=-16t^2+20$

Subtract $20$ from each side.

  $\begin{array}{c}0-20=-16t^2+20-20\\ -20=-16t^2\end{array}$

Now, divide each side by $-16$ .

  $\begin{array}{c}\frac{-20}{-16}=\frac{-16t^2}{-16}\\ \frac{20}{16}=t^2\end{array}$

Take the square root on each side.

  $\begin{array}{c}\sqrt{\frac{20}{16}}=\sqrt{t^2}\\ \pm \frac{\sqrt{20}}{4}=t\end{array}$

Use a calculator to evaluate.

  $t\approx \pm 1.1$

Since time cannot be negative, reject the negative value. Thus, the pinecone takes about $1.1$ seconds to hit the ground.

d.

To find: Domain and range of the function.

Domain is $\left(-\infty ,\infty \right)$ ad the range is $\left(-\infty ,16\right)$

Given:

A pinecone falls from a tree branch that is $20$ feet above the ground.

Calculation:

Consider the following graph of the function.

  

So,

Domain is $\left(-\infty ,\infty \right)$ ad the range is $\left(-\infty ,16\right)$