Chapter 9.3, Problem 30PPS

(a)

To determine

Find which sprinkler will send water the farthest.

Answer to Problem 30PPS

The parabola for Sprinkler C is the most compressed vertically, so it will send water the farthest.

Explanation of Solution

Given:

The path of water from sprinkler can be modeled by quadratic functions. The following functions model paths for 3 different sprinklers .

  $\begin{array}{l}\text{Sprinkler}\text{A:}y=-0.35x^2+3.5\\ \text{Sprinkler}\text{B:}y=-0.21x^2+1.7\\ \text{Sprinkler}\text{C:}y=-0.08x^2+2.4\end{array}$

Concept Used:

Al three of the quadratic functions are of the form $f(x)=a{x}^2+c$

  

The value of a in the function for sprinkler C is a smallest of the three.

Therefore, the parabola for Sprinkler C is the most compressed vertically, so it will send water the farthest.

Thus, the parabola for Sprinkler C is the most compressed vertically, so it will send water the farthest.

(b)

To determine

Find which sprinkler will send water the highest.

Answer to Problem 30PPS

The parabola for sprinkler A is translated up the most, so it will send water the highest.

Explanation of Solution

Given:

The path of water from sprinkler can be modeled by quadratic functions. The following functions model paths for 3 different sprinklers.

  $\begin{array}{l}\text{Sprinkler}\text{A:}y=-0.35x^2+3.5\\ \text{Sprinkler}\text{B:}y=-0.21x^2+1.7\\ \text{Sprinkler}\text{C:}y=-0.08x^2+2.4\end{array}$

Concept Used:

Al three of the quadratic functions are of the form $f(x)=a{x}^2+c$

  

The value of c in the function for Sprinkler A is the largest of the three.

Therefore, the parabola for sprinkler A is translated up the most, so it will send water the highest.

Thus, the parabola for sprinkler A is translated up the most, so it will send water the highest.

(c)

To determine

Find which sprinkler will produce the narrowest path.

Answer to Problem 30PPS

The parabola for Sprinkler A is the most stretched vertically, so it will expand the least, and therefore have the narrowest path.

Explanation of Solution

Given:

The path of water from sprinkler can be modeled by quadratic functions. The following functions model paths for 3 different sprinklers.

  $\begin{array}{l}\text{Sprinkler}\text{A:}y=-0.35x^2+3.5\\ \text{Sprinkler}\text{B:}y=-0.21x^2+1.7\\ \text{Sprinkler}\text{C:}y=-0.08x^2+2.4\end{array}$

Concept Used:

Al three of the quadratic functions are of the form $f(x)=a{x}^2+c$

When comparing with $f(x)=a{x}^2+c$ , the equation with highest value of $\left|a\right|$ is the most vertically compressed.

  

The value of a in the function of sprinkler A is the largest of the three.

Therefore, the parabola for Sprinkler A is the most stretched vertically, so it will expand the least, and therefore have the narrowest path.

Thus, the parabola for Sprinkler A is the most stretched vertically, so it will expand the least, and therefore have the narrowest path.