Chapter 4.7, Problem 31E

(a)

To determine

To find: the distance the hummingbird does feed.

Answer to Problem 31E

The hummingbird feeds at distance of o ft, 4ft, and 7 ft.

Explanation of Solution

Given information:

A hummingbird flies while feeding can be modeled by the function,

  $f\left(x\right)=-\frac{1}{5}x{\left(x-4\right)}^2\left(x-7\right),0\le x\le 7$

Where $x$ the horizontal is distance and $f\left(x\right)$ is the height. The hummingbird feeds each time it is at ground level.

Calculation:

To determine the distance at which the hummingbird feeds, substitute $f\left(x\right)=0$ and solve for $x$ .

  $\begin{array}{c}-\frac{1}{5}x{\left(x-4\right)}^2\left(x-7\right)=0\\ x=0,4,7\end{array}$

Therefore, the hummingbird feeds at distance of o ft, 4ft, and 7 ft.

(b)

To determine

To write: a function to model the path of the second hummingbird.

Answer to Problem 31E

Model for the path of second hummingbird is $g\left(x\right)=-\frac{2}{5}\left(x-1\right){\left(x-6\right)}^2\left(x-9\right)$ .

Explanation of Solution

Given information:

A hummingbird flies while feeding can be modeled by the function,

  $f\left(x\right)=-\frac{1}{5}x{\left(x-4\right)}^2\left(x-7\right),0\le x\le 7$

Where $x$ the horizontal is distance and $f\left(x\right)$ is the height. The hummingbird feeds each time it is at ground level.

A second hummingbird feeds 2 meters farther away than the first hummingbird and flies twice as high.

Calculation:

The second humming bird’s path can be modeled by translating the first humming bird’s model 2 unit’s right followed by a vertical stretch by a factor of 2.

  $\begin{array}{c}g\left(x\right)=2f\left(x-2\right)\\ =-\frac{2}{5}\left(x-1\right){\left(x-2-4\right)}^2\left(x-2-7\right)\\ =-\frac{2}{5}\left(x-1\right){\left(x-6\right)}^2\left(x-9\right)\end{array}$

Therefore, model for the path of second hummingbird is $g\left(x\right)=-\frac{2}{5}\left(x-1\right){\left(x-6\right)}^2\left(x-9\right)$ .