Question

A hammock is suspended between two trees. The curve the hammock makes can bemodelled by the equation y = 0.2x² - 0.4x - 0.6, where x and y are measured inmetres.a) Find the x interceptsb) Find the vertex.c) What is the minimum height of the hammock?

Answer 1

We have the function that relates x and y expressed as:

$y=0.2x^2-0.4x-0.6$

a) We have to find the x-intercepts.

To do that we can use the quadratic equation:

$\begin{gathered} x=\frac{-(-0.4)\pm\sqrt{(-0.4)^2-4(0.2)(-0.6)}}{2(0.2)} \\ x=\frac{0.4\pm\sqrt{0.16+0.48}}{0.4} \\ x=\frac{0.4\pm\sqrt{0.64}}{0.4} \\ x=\frac{0.4\pm0.8}{0.4} \\ x=1\pm2 \\ x_1=1-2=-1 \\ x_2=1+2=3 \end{gathered}$

Then, we have x-intercepts at x = -1 and x = 3.

b) We have to find the vertex.

We can find the x-coordinate of the vertex using the linear coefficient b = -0.4 and the quadratic coefficient a = 0.2:

$x_v=\frac{-b}{2a}=\frac{-(-0.4)}{2(0.2)}=\frac{0.4}{0.4}=1$

It can also be calculated as the average of the x-intercepts.

Knowing the x-coordinate of the vertex, we can find the y-coordinate of teh vertex using the formula applied to x = 1:

$y=0.2(1)^2-0.4(1)-0.6=0.2-0.4-0.6=-0.8$

Then, the vertex is (1, -0.8).

c) The minimum height will be given by the y-coordinate of the vertex.

Relative to the horizontal axis (y = 0), the minimum height will be -0.8 meters below that level.

Answer:

a) The x-intercepts are x = -1 and x = 3.

b) The vertex is (1,-0.8)

c) The minimum height is 0.8 units below the horizontal axis.