<span>x^2/8 - x/4 + 1/8 = 0
A parabola is defined as the set of all points such that each point has the same distance from the focus and the directrix. Also the parabola's equation will be a quadratic equation of the form ax^2 + bx + c. So if we can determine 3 points on the parabola, we can use those points to calculate the desired equation.
First, let's draw the shortest possible line from the focus to the directrix. The midpoint of that line will be a point on the desired parabola. Since the slope of the directrix is 0, the line will have the equation of x=1. This line segment will be from (1,2) to (1,-2) and the midpoint will be ((1+1)/2, (2 + -2)/2) = (2/2, 0/2) = (1,0).
Now for the 2nd point, let's draw a line that's parallel to the directrix and passing through the focus. The equation of that line will be y=2. Any point on that line will have a distance of 4 from the directrix. So let's give it an x-coordinate value of (1+4) = 5. So another point for the parabola is (5,2). And finally, if we subtract 4 instead of adding 4 to the x coordinate, we can get a third point of 1-4 = -3. So that 3rd point is (-3,2).
So we now have 3 points on the parabola. They are (1,0), (5,2), and (-3,2). Let's create some equations of the form ax^2 + bx + c = y and then substitute the known values into those equations. SO
ax^2 + bx + c = y
(1) a*1^2 + b*1 + c = 0
(2) a*5^2 + b*5 + c = 2
(3) a*(-3)^2 + b*(-3) + c = 2
Let's do the multiplication for those expressions. So
(4) a + b + c = 0
(5) 25a + 5b + c = 2
(6) 9a - 3b + c = 2
Equations (5) and (6) above look interesting. Let's subtract (6) from (5). So
25a + 5b + c = 2
- 9a - 3b + c = 2
= 16a + 8b = 0
Now let's express a in terms of b.
16a + 8b = 0
16a = -8b
a = -8b/16
(7) a = -b/2
Now let's substitute the value (-b/2) for a in expression (4) above. So
a + b + c = 0
-b/2 + b + c = 0
And solve for c
-b/2 + b + c = 0
b/2 + c = 0
(8) c = -b/2
So we know that a = -b/2 and c = -b/2. Let's substitute those values for a and c in equation (5) above and solve for b.
25a + 5b + c = 2
25(-b/2) + 5b - b/2 = 2
-25b/2 + 5b - b/2 = 2
2(-25b/2 + 5b - b/2) = 2*2
-25b + 10b - b = 4
-16b = 4
b = -4/16
b = -1/4
So we now know that b = -1/4. Using equations (7) and (8) above, let's calculate a and c.
a = -b/2 = -(-1/4)/2 = 1/4 * 1/2 = 1/8
c = -b/2 = -(-1/4)/2 = 1/4 * 1/2 = 1/8
So both a and c are 1/8. So the equation for the parabola is
x^2/8 - x/4 + 1/8 = 0
Let's test to make sure it works. First, let's use an x of 1.
x^2/8 - x/4 + 1/8 = y
1^2/8 - 1/4 + 1/8 = y
1/8 - 1/4 + 1/8 = y
1/8 - 2/8 + 1/8 = y
0 = y
And we get 0 as expected. Let's try x = 2
x^2/8 - x/4 + 1/8 = y
2^2/8 - 2/4 + 1/8 = y
4/8 - 1/2 + 1/8 = y
4/8 - 1/2 + 1/8 = y
1/2 - 1/2 + 1/8 = y
1/8 = y.
Let's test if (2,1/8) is the same distance from both the focus and the directrix. The distance from the directrix is 1/8 - (-2) = 1/8 + 2 = 1/8 + 16/8 = 17/8
The distance from the focus is
d = sqrt((2-1)^2 + (1/8-2)^2)
d = sqrt(1^2 + -15/8^2)
d = sqrt(1 + 225/64)
d = sqrt(289/64)
d = 17/8
And the distances match again. So we do have the correct equation of:
x^2/8 - x/4 + 1/8 = 0</span>
Answer:
The ball will be 84 feet above the ground 1.125 seconds and 4.5 seconds after launch.
Step-by-step explanation:
Statement is incorrect. Correct form is presented below:
<em>The height </em>
<em> of an ball that is thrown straight upward from an initial position 3 feet off the ground with initial velocity of 90 feet per second is given by equation </em>
<em>, where </em>
<em> is time in seconds. After how many seconds will the ball be 84 feet above the ground. </em>
We equalize the kinematic formula to 84 feet and solve the resulting second-order polynomial by Quadratic Formula to determine the instants associated with such height:
(1)
By Quadratic Formula:
, 
The ball will be 84 feet above the ground 1.125 seconds and 4.5 seconds after launch.
Answer:
y= -5/3x-2
Step-by-step explanation:
Bobby should grow an amount greater than 70-61 = 9 inches to become taller than 70 inches. Therefore, the correct answer is that Bobby must grow an amount of x> 9 inches to be taller than 70 inches.
