confusing need fast help don't understand

Question

LawnSprinkler The arc of the water sprayed from a lawn sprinkler can be modeled by

the function y = —0.04x(x —24) wherex is the distance (in feet) across the lawn and y is

the height (in feet) of the arc.

a.) How far does the arc travel?

b.) What is the maximum height of the arc?

Answer 1

Step 1: To find how far the arc travels, we need to find the places in which y = 0, or where the arc of the sprinkler hits the ground. Set y = 0 and solve for x.

0 = -0.04x (x - 24)

0 = x - 24

x = 24 feet

Step 2: To find the maximum height of the arc, first take the derivative of the function. Then, set this derivative equal to zero to find the maximum point of the arc.

y = -0.04x² + 0.96x

y' = -0.08x + 0.96

0 = -0.08x + 0.96

0.08x = 0.96

x = 12 feet

So at position 12 feet, the arc is at its highest. This makes sense because the arc is at y = 0 at x = 0 (beginning of arc) and x = 24 (end of arc). It would make sense that the arc's peak is in the middle of these two point. Now, let's find what y equals at x = 12 feet

y = -0.04(12)(12-24)

y = 5.76 feet

Answer 2

Setting y=o when the water hits the ground,

-.04x(x-24)=0

x=24, distance travelled

To solve for height, find vertex which is given by

-b/2a=x, y=f(-b/2a)

from the given equation,

-.04x^2-.96x,

-.96/-.04(2)

x=12

-.04(12)(12-24)

=5.76 ft.