Chapter 1.5, Problem 2MRPS

Solution

a.

To find: The domain and the range of the function $y=-0.0035x\left(x-143.9\right)$

Domain is $-\infty <x<\infty$ ad range is $y\le 18.1188$ .

Given:

Some harbor police departments have fire-fighting boats with water cannons. The boats are used to fight fires that occur within the harbor

  

Calculation:

The domain of a function is the set of $x$ -values that make that function true and the range of a function is the set of $y$ -values that make that function true.

Transform the function $y=-0.0035x\left(x-143.9\right)$ into vertex form as given below.

  $\begin{array}{c}y=-0.0035x\left(x-143.9\right)\\ y=-0.0035\left(x^2-143.9x\right)\\ y+\left(-0.0035\right)\left(5176.8025\right)=-0.0035\left(x^2-143.9x+5176.8025\right)\\ y=-0.0035{\left(x-71.95\right)}^2+18.1188\end{array}$

So, the vertex is $\left(71.95,18.1188\right)$

So, the graph of the function $y=-0.0035x\left(x-143.9\right)$ is shown below.

  

Since, the graph continues down town the left and down toward the right, the domain contains all real numbers from negative infinity to positive infinity.

Hence, domain of the function $y=-0.0035x\left(x-143.9\right)$ is $-\infty <x<\infty$ , or all real numbers

Since the maximum $y$ -value of the function is $18.1188$ and the graph continues to extend downward, the range contains all real numbers less than or equal to $18.1188$ .

Hence, range of the function $y=-0.0035x\left(x-143.9\right)$ is $y\le 18.1188$ .

b.

To find: How far does the water cannon shoot?

The water cannon can shoot up to a distance of $143.9$ feet.

Given:

Some harbor police departments have fire-fighting boats with water cannons. The boats are used to fight fires that occur within the harbor

  

Calculation:

First transform the function $y=-0.0035x\left(x-143.9\right)$ into intercept form as given below.

  $\begin{array}{c}y=-0.0035x\left(x-143.9\right)\\ =-0.0035(x-0)\left(x-143.9\right)\end{array}$

Now, compare the equation $y=-0.0035\left(x-0\right)\left(x-143.9\right)$ with the standard intercept form $y=a\left(x-p\right)\left(x-q\right)$ .

  $p=0,q=143.9$

Because $p=0$ and $q=143.9$ , the $x$ -intercepts are $0$ and $143.9$ .

So, the water cannon can shoot up to a distance of $143.9$ feet.

c.

To find: The maximum height of the water.

The maximum height of the water is $18.1188$ feet.

Given:

Some harbor police departments have fire-fighting boats with water cannons. The boats are used to fight fires that occur within the harbor

  

Calculation:

First transform the function $y=-0.0035x\left(x-143.9\right)$ into vertex form as given below.

  $\begin{array}{c}y=-0.0035\left(x^2-143.9x\right)\\ y+\left(-0.0035\right)\left(5176.8025\right)=-0.0035\left(x^2-143.9x+5176.8025\right)\\ y-18.1188=-0.0035{\left(x-71.95\right)}^2\\ y=-0.0035{\left(x-71.95\right)}^2+18.1188\end{array}$

So, the vertex is $\left(71.95,18.1188\right)$

The maximum height of the water is the $y$ -coordinate of the vertex $\left(71.95,18.1188\right)$ of the parabola with the given equation $y=-0.0035x\left(x-143.9\right)$ .

Hence, the maximum height of the water is $18.1188$ feet.