Question

The percentage of hearing loss experienced by people throughout their adult years can bemodeled by the equationy=0.0086x2 – 0.4267x + 6.6434, with 18 5x 584,where y represents the percent of hearing loss and x is the age of a person.1. By the age of 18, what percent of hearing loss can a person experience?2. Find the percent range of hearing loss for the 30–60 year age group.3. Find the percent range of hearing loss for the 60–84 year age group.4. Find the age group where the percent of hearing loss is less than 5%.5. Find the age group where the percent of hearing loss is greater than 25%.6. Find the age group where the percent of hearing loss is greater than 15%, but lessthan 30%.7. Sketch a graph of the equation. Use your graph to check your answers for Exercises1-6.

Answer 1

We have and equation for hearing loss:

$\begin{gathered} y=0.0086x^2-0.4267x+6.6434 \\ \text{Where x is the age and y the percent of hearing loss} \end{gathered}$

1. We want to know the percent of hearing loss at 18 years old, so we have to evaluate the equation at x=18:

$y(18)=0.0086\cdot18^2-0.4267\cdot18+6.6434=1.7492$

2. We want to find the percent range of hearing loss for the 30 - 60 years all. To do this we need to find the minimum and maximun of the equation at x between [30,60].

First we know the equation is a parabola, so we can find the minimun of the parabola, this is the vertex:

$\begin{gathered} \text{General equation of a parabola is:} \\ y=ax^2+bx+c \\ x_{\text{vertex}}=-\frac{b}{2a},y_{\text{vertex}}=y(x_{\text{vertex}}) \end{gathered}$

In this case, a=0.0086, b=-0.4267 and c=6.6434:

$\begin{gathered} x_{\text{vertex}}=-\frac{(-0.4267)}{2\cdot0.0086}=\frac{0.4267}{2\cdot0.0086}=24.8 \\ y_{\text{vertex}}=0.0086\cdot24.8^2-0.4267\cdot24.8+6.6434=\text{1}.35 \end{gathered}$

So, the minimun hearing loss is 1.35% at 24.8 years old, because 24.8 is not between 30 -60 years and it's less than 30, the minumun ot the range is in 30 years and the maximun at 60 years:

$\begin{gathered} y(30)=0.0086\cdot30^2-0.4267\cdot30+6.6434=1.58 \\ y(60)=0.0086\cdot60^2-0.4267\cdot60+6.6434=12.0 \end{gathered}$

So, the range of hearing loss is [1.58, 12.0] at ages 30-60 years.

3. This is the same as point 2, we already has the value for 60 years, so we need to evaluate the equation at 84 years:

$y(84)=0.0086\cdot84^2-0.4267\cdot84+6.6434=31.48$

The range of hearing loss is [12.0, 31.48] ar ages 60-84 years.

4. Now we want to find the values of x where the y < 5, so:

$\begin{gathered} y=5=0.0086x^2-0.4267x+6.6434 \\ 0=0.0086x^2-0.4267x+6.6434-5 \\ 0=0.0086x^2-0.4267x+1.6434 \end{gathered}$

Now we need to find the roots with the formula:

$\begin{gathered} \text{General parabola form is:} \\ ax^2+bx+c \\ \text{and the roots are:} \\ x_{1,2}=\frac{-b\pm\sqrt[]{b^2-4\cdot a\cdot c}}{2a} \end{gathered}$

In this case, a=0.0086, b=-0.4267 and c=1.6434

$\begin{gathered} x_{1,2}=\frac{0.4267\pm\sqrt[]{(-0.4267)^2-4\cdot0.0086\cdot1.6434}}{2\cdot0.0086} \\ x_{1,2}=\frac{0.4267\pm0.3543}{0.0172} \\ x_1=\frac{0.4267+0.3543}{0.0172}=45.4 \\ x_2=\frac{0.4267-0.3543}{0.0172}=4.2 \end{gathered}$

So, the people between 4.2 and 45.4 years has a hearing loss less than 5%.