Select the correct answer.

The height of a pelican as it swoops down to catch a fish in the lake is modeled by this equation, where [tex]$x$[/tex] represents the time, in seconds, and [tex]$y$[/tex] represents the height of the pelican, in feet.

[tex] y = x^2 - 6x + 8.75 [/tex]

At the same time, a second pelican flies in a straight line, modeled by this equation.

[tex] y = \frac{1}{4}x + 5 [/tex]

Which statement describes the number and viability of the system's solutions?

A. The system has only one solution, but it is not viable because it results in a negative value of time.
B. The system has only one solution, and it is viable because it results in positive values of time.
C. The system has two solutions, but only one is viable because the other results in a negative value of time.
D. The system has two solutions, and both are viable because they result in positive values of time.



Answer :

To determine the viability and number of solutions for the given system of equations:

[tex]\[ y = x^2 - 6x + 8.75 \][/tex]
[tex]\[ y = \frac{1}{4}x + 5 \][/tex]

we need to solve this system of equations.

1. Set the equations equal to each other to find the points where they intersect:

[tex]\[ x^2 - 6x + 8.75 = \frac{1}{4}x + 5 \][/tex]

2. Move all terms to one side of the equation to set it to zero:

[tex]\[ x^2 - 6x + 8.75 - \frac{1}{4}x - 5 = 0 \][/tex]

Combine like terms:

[tex]\[ x^2 - 6.25x + 3.75 = 0 \][/tex]

3. Solve the quadratic equation [tex]\( x^2 - 6.25x + 3.75 = 0 \)[/tex] for [tex]\( x \)[/tex]:

This is a standard quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex].

[tex]\[ a = 1, \, b = -6.25, \, c = 3.75 \][/tex]

Using the quadratic formula:

[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:

[tex]\[ x = \frac{-(-6.25) \pm \sqrt{(-6.25)^2 - 4(1)(3.75)}}{2(1)} \][/tex]
[tex]\[ x = \frac{6.25 \pm \sqrt{39.0625 - 15}}{2} \][/tex]
[tex]\[ x = \frac{6.25 \pm \sqrt{24.0625}}{2} \][/tex]
[tex]\[ x = \frac{6.25 \pm 4.9051}{2} \][/tex]

Therefore, the two solutions for [tex]\( x \)[/tex] are:

[tex]\[ x = \frac{6.25 + 4.9051}{2} \approx 5.58 \][/tex]
[tex]\[ x = \frac{6.25 - 4.9051}{2} \approx 0.672 \][/tex]

4. Check the viability of both solutions (i.e., whether they are positive):

Both [tex]\( x = 5.58 \)[/tex] and [tex]\( x = 0.672 \)[/tex] are positive values of time.

Thus, the system has two solutions, and both are viable because they result in positive values of time.

Therefore, the correct statement that describes the number and viability of the system's solutions is:

D. The system has two solutions, and both are viable because they result in positive values of time.