Chapter 9.5, Problem 48E

a.

To determine

To determine the number of days having 17 hours of day light.

Answer to Problem 48E

There will be no days having 17 hours of daylight.

Explanation of Solution

Given:

Equation:

  $y=-0.00046x^2+0.076x+13$

Formula used:

Quadratic formula:

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

According to question

  $y=17$

Therefore,

  $17=-0.00046x^2+0.076x+13$

Converting the given equation in standard form i.e. $a{x}^2+bx+c=0$

  $0.00046x^2-0.076x-13+17=0$

So,

  $0.00046x^2-0.076x+4=0$

On comparing the equation with $a{x}^2+bx+c$

  $a=0.00046,b=-0.076,c=4$

Finding the value of $x$

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Substituting the values,

  $x=\frac{-(-0.076)\pm \sqrt{{(-0.076)}^2-4(0.00046)(4)}}{2(0.00046)}$

Opening brackets,

  $x=\frac{0.076\pm \sqrt{0.0057-0.00736}}{0.00092}$

So,

  $x=\frac{0.076\pm \sqrt{-0.00166}}{0.00092}$

Since the discriminant came out to be negative so there will be no days having 17 hours of daylight.

b.

To determine

To determine the number of days having 17 hours of day light.

Answer to Problem 48E

There will 15 or 150 days having 14 hours of daylight.

Explanation of Solution

Given:

Equation:

  $y=-0.00046x^2+0.076x+13$

Formula used:

Quadratic formula:

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

According to question

  $y=14$

Therefore,

  $14=-0.00046x^2+0.076x+13$

Converting the given equation in standard form i.e. $a{x}^2+bx+c=0$

  $0.00046x^2-0.076x-13+14=0$

So,

  $0.00046x^2-0.076x+1=0$

On comparing the equation with $a{x}^2+bx+c$

  $a=0.00046,b=-0.076,c=4$

Finding the value of $x$

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Substituting the values,

  $x=\frac{-(-0.076)\pm \sqrt{{(-0.076)}^2-4(0.00046)(1)}}{2(0.00046)}$

Opening brackets,

  $x=\frac{0.076\pm \sqrt{0.0057-0.00184}}{0.00092}$

So,

  $x=\frac{0.076\pm \sqrt{0.00386}}{0.00092}$

Solving root,

  $x=\frac{0.076\pm 0.0621}{0.00092}$

So,

  $x=\frac{0.076+0.0621}{0.00092}$ or $x=\frac{0.076-0.0621}{0.00092}$

So,

  $x=\frac{0.1381}{0.00092}$ or $x=\frac{0.0139}{0.00092}$

So,

  $x=150.10$ or $x=15.10$

Since the discriminant came out to be positive so there will 15 or 150 days having 14 hours of daylight.