Clovis is standing at the edge of a cliff, which slopes 4 feet downward from him for every 1 horizontal foot. He launches a small model rocket from where he is standing. With the origin of the coordinate system located where he is standing, and the x-axis extending horizontally, the path of the rocket is described by the formula y = −2x² + 160x.
(a) Give a function h = f(x) relating the height h of the rocket above the sloping ground to its x-coordinate.
(b) Find the maximum height of the rocket above the sloping ground. What is its x-coordinate when it is at its maximum height?
(c) Clovis measures its height h of the rocket above the sloping ground while it is going up. Give a function x = g(h) relating the x-coordinate of the rocket to h.
(d) Does this function still work when the rocket is going down? Explain.

a) -2x^2 + 164x

b) 3362 feet

c) (82 , -328)

d) yes

Step-by-step explanation:

y = -2x^2 + 160x

Slope = 4 feet downward for every 1 horizontal foot.

a) h(x) = -2x^2 + 160x - (-4x)

= -2x^2 + 160x + 4x

= -2x^2 + 164x

b) The highest point occurs at the vertex of the parabolic equation. x is the same as the number of the axis of symmetry.

x = -b/2a

From the equation, a = -2 , b= 164

x = -164/ 2(-2)

x = -164/-4

x = 41

Put x = 41 into the value of h(x)

h(x) = -2x^2 + 164x

= -2(41^2) + 164(41)

= -2(1681) + 6724

= -3362 + 6724

= 3362 feet.

The maximum height occurs at 41 feet out from the top of the sloping ground at a height of 3362ft about the top edge of the cliff.

c) h(x) = -2x^2 + 164x

2x^2 - 164x + h = 0 when 0 ≤ x ≤ 41

Solve the equation using the formula (-b+/-√b^2 - 4ac) / 2a

a = 2, b= -164 , c = h

= [-(-164) +/- √(-164)^2 - 4(2)(h) ] / 2(2)

= (164 +/- √26896 - 8h)/ 4

This gives the value of -328 ≤ h ≤ 3362 is used because the rocket hits the sloping ground of (82 , -328)

d) the function still works when it is going down