The vertical distance of a given point on Mount HImalayas is $(-x^2+4x+6)$ km. It has a horizontal distance of $x$km from a reference point O. A plane flew above Mount Himalayas and its average vertical height was given by $(\frac{1}{4}x^2+kx+\frac{5}{4}k)$ km, where k is a real number. The horizontal distance of the plane was also x km from O.

Given that the vertical safety distance between any plane and Mount Himalayas should be at least 1 km at any point in time, can the plane meet the safety requirements at all times? Explain your answer with the help of mathematical concepts learnt in the topic of quadratic functions.

2. Hello acc100jt
Originally Posted by acc100jt
The vertical distance of a given point on Mount HImalayas is $(-x^2+4x+6)$ km. It has a horizontal distance of $x$km from a reference point O. A plane flew above Mount Himalayas and its average vertical height was given by $(\frac{1}{4}x^2+kx+\frac{5}{4}k)$ km, where k is a real number. The horizontal distance of the plane was also x km from O.

Given that the vertical safety distance between any plane and Mount Himalayas should be at least 1 km at any point in time, can the plane meet the safety requirements at all times? Explain your answer with the help of mathematical concepts learnt in the topic of quadratic functions.
The vertical distance between the plane and the ground is the difference between the two functions:

$d=(\tfrac{1}{4}x^2+kx+\tfrac{5}{4}k) - (-x^2+4x+6)$

$= \tfrac54x^2 +(k-4)x + (\tfrac54k-6)$

Now find the minimum value of this function (either by completing the square or differentiating), and interpret your answer in the context of the question.

Can you do this?