Question

When a gray kangaroo jumps, its path through the air can be modeled by y= -0.0267x2 + 0.8x where x is the kangaroo's horizontal distance traveled (in feet) wand y is the corresponding height (in feet). HOw high can a gray kangaroo jump? How far can it jump?

Answer 1

A gray kangaroo jumped and it's path through the air is given by

  y =$-0.0267 x^2 +0.8 x$

      where x is the kangaroo's horizontal distance traveled (in feet) wand y is the corresponding height (in feet).

For maximum distance we have to differentiate this expression.

$\frac{\mathrm{d}y }{\mathrm{d} x}=-0.0267\frac{\mathrm{d} x^2}{\mathrm{d} x}+0.8\frac{\mathrm{d} x}{\mathrm{d} x}\\ \frac{\mathrm{d} y}{\mathrm{d} x}=-0.0267\times2x + 0.8$

For maxima or minima

$\frac{\mathrm{d} y}{\mathrm{d} x}$=0

-0.0267×2 x + 0.8=0

⇒-0.0534 x + 0.8=0

⇒0.8=0.0534 x

⇒x=0.8/0.0534

x =14.98 (approx)

Now differentiating the expression again

$\frac{\mathrm{d^{2}{y}} }{\mathrm{d} x^{2}}$=-0.0534

Since double derivative is negative , so

x=14.98 will be the point of Maxima.

The  Kangaroo can go the maximum distance of 14.98 feet (approx)

and the height that can kangaroo go through=-0.0267×14.98×14.98+0.8×14.98

                                                                          = -5.9914 +11.984

                                                                          =   5.9925 feet

                                                                            = 6 feet (approx)

So, X=Horizontal distance covered=14.98×2=29.96 feet,[∵ at x=14.98 it attains maximum height, So total distance Travelled by kangaroo will be two times of the distance covered that it has gone through to achieve maximum height.]⇒ The path is parabolic. = 30 feet (approx)

Y= Vertical distance covered= 5.9925=6 feet(approx)