Maximum/X intercept of a parabola?

Question

Math question on the study guide I can't get for the life of me:

9] The height of an incomplete pass thrown by W. Paddock can be modeled by (3 pts each)
y(x) = -0.024x2 + 1.39x + 6
where y(x) is the height of the ball x feet from the W. Paddock.

(a) At how many feet from W. Paddock does the ball hit the maximum height?
(Round to the nearest foot)

(b) What is the maximum height that the ball reaches?
(Round to the nearest foot)

(c) How far is the point at which the ball hits the ground from the W. Paddock?
(Round to the nearest foot)

Thanks for any help.

Answer 1

It’s useful to know if you know calculus when you post these problems because they are much easier done with calculus than with algebra. I’ll do it with calculus first; then with algebra.
(a) At how many feet from W. Paddock does the ball hit the maximum height?
(Round to the nearest foot)
1) Take the first derivative and set it = 0 and solve for x.
y’= 2(-0.024) x+ 1.39
0 = -0.048x +1.39
0.048x = 1.39
x = 1.39/0.048
= 28.95
29 feet rounded.
In case you don’t know calculus, use the formula:
X = -b/(2a)
=-1.39(-0.048)
= 28.95

(b) What is the maximum height that the ball reaches?
(Round to the nearest foot)
1) Substitute the value found above into the equation for y and solve for y.
Y = -0.024x² +1.39x +6
= -0.024(28.95)² + 1.39(28.95) +6
= 26.12
= 26 feet rounded.

(c) How far is the point at which the ball hits the ground from the W. Paddock?
(Round to the nearest foot)
1) Set y = 0 and solve for x.
Y = -0.024x² +1.39x +6
0 = -0.024x² +1.39x +6
Solve this using the quadratic formula.
X = [-b+-√(b²-4ac)]/2a
= [-1.39+-√(1.39²-4(-.024*6)]/2(-0.024)
= (-1.39 -1.58)/(-0.048)
Plug that into your calculator and you’ll get:
61.875
=62 feet rounded.

FE

Answer 2

a) The distance from W. Paddock is the x coordinate of the vertex (maximum) of the parabola. The distance is 29 feet.

b) The height is the y coordinate of the vertex (maximum) of the parabola. The height is 26 feet

c) The distance away from W. Paddock is the "zero" or root of the function (the height is zero when the ball is on the ground). This is 62 feet.

Answer 3

You just have to find the vertex and the rightmost x-intercept.

The X coordinate of the vertex would be -1.39 / 2(-0.024) This gives you 28.9583333 Which, rounded to the nearest foot, is 29 ft. which is the answer to letter (a).

Now you plug this back into the original equation to get 45.61400 which is the Y coordinate of the vertex. Rounded up the answer to letter (b) is 46 ft. high.

Now to answer (c) you must either convert the equation to intercept form or guess and check by imputing different x values into your equation until one of them comes out to 0.

P.S. You can also graph it to find the X-Inercepts.

Answer 4

Set first derivative of Y to zero to find the top of the parabola: .048x = 1.39 x=29ft a)
ymax=-.024[29*29] +1.39[29]+6=26' b)
c)xmax= twice the distance to the highest point = 58'