The height of a pelican as it swoops down to catch a fish in the lake is modeled by this equation, where x represents the time, in seconds, and y represents the height of the pelican, in feet.
y=x^(2)-6x+8.75
At the same time, a second pelican flies in a straight line, modeled by this equation.
y=(1)/(4)x+5
Which statement describes the number and viability of the system’s solutions?
A.
The system has two solutions, but only one is viable because the other results in a negative value of time.
B.
The system has two solutions, and both are viable because they result in positive values of time.
C.
The system has only one solution, but it is not viable because it results in a negative value of time.
D.
The system has only one solution, and it is viable because it results in positive values of time.
To find the number and viability of the system's solutions, we need to find the points where the two equations intersect.
Setting the two equations equal to each other:
x^2 - 6x + 8.75 = (1/4)x + 5
Combining like terms:
x^2 - 6x - (1/4)x + 8.75 - 5 = 0
x^2 - (25/4)x + 3.75 = 0
This equation is a quadratic equation. We can determine the number of solutions by considering the discriminant.
The discriminant, b^2 - 4ac, is given by:
(-25/4)^2 - 4(1)(3.75)
625/16 - 60/16
565/16
Since the discriminant is positive, the quadratic equation has two real solutions.
Therefore, the statement that describes the number and viability of the system's solutions is:
A. The system has two solutions, but only one is viable because the other results in a negative value of time.