Question

A dome tent’s arch is modeled by y= -0.18(x-6)(x+6)where x and y are measured in feet.

Answer 1

The student's question pertains to solving a quadratic function that models a dome tent's arch, given by the equation y = -0.18(x-6)(x+6) in Mathematics.

Step-by-step explanation:

The student's question involves a quadratic function in the form of y = -0.18(x-6)(x+6), which models the arch of a dome tent. This equation represents a parabola that opens downwards, as indicated by the negative coefficient of the quadratic term. In this context, the variables x and y represent the dimensions of the arch in feet. The roots of the quadratic equation are at x = 6 and x = -6, corresponding to the points where the arch meets the ground (the 'x-axis'), and the vertex of the parabola is at the midpoint of the two roots, which is at x = 0. The vertex represents the highest point of the dome's arch. To find the maximum height y of the dome, one would evaluate the quadratic function at the vertex.

Answer 2

The height of the highest point of the arch is 3 feet.

Step-by-step explanation:

The complete question is:

A dome tent’s arch is modeled by y= -0.18(x-6)(x+6) where x and y are measured in feet. To the nearest foot, what is the height of the highest point of the arch.

Solution:

The expression provided is:

`y= -0.18(x-6)(x+6)\\y=-0.18(x^(2)-36)\\y=-0.18x^(2)+6.48x`

The equation is of a parabolic arch.

The general equation of a parabolic arch is:

`y=ax^(2)+bx+c`

So,

a = -0.18

b = 6.48

c = 0

Highest point of the parabolic arch is the vertex of the parabolic equation if a < 0 .

As a = -0.18 < 0, the ordinate of vertex of equation will give the height of highest point of arch.

For a parabola the abscissa of vertex is given as follows:

`x=-(b)/(2a)`

⇒

`x=-(6.48)/(2* (-0.18))\\\\x=18`

Compute the value of y as follows:

`y=-0.18x+6.48`

`=(-0.18* 18)+6.48\\=3.24\\\approx 3`

Thus, the height of the highest point of the arch is 3 feet.