A ball is launched from a slingshot. Its height, h(x), can be represented by a quadratic function in terms of time, x, in seconds.
After 1 second, the ball is 148 feet in the air, after 2 seconds the ball is 272 feet in the air.
Find the height in feet of the ball after 6 seconds in the air?
As we know,
h(x) = a+bx-16x^2
a+b-16 = 148
4a+2b-64 = 272
solve that and you end up with
h(x) = 4 + 160x - 16x^2
...
To find the height of the ball after 6 seconds, we need to determine the quadratic function that represents the height of the ball, given the information provided.
Let's assume the quadratic function is of the form: h(x) = ax^2 + bx + c, where h(x) represents the height of the ball in feet and x represents the time in seconds.
We can use the given information to form two equations:
1. When x = 1, h(x) = 148:
148 = a(1)^2 + b(1) + c => a + b + c = 148 (Equation 1)
2. When x = 2, h(x) = 272:
272 = a(2)^2 + b(2) + c => 4a + 2b + c = 272 (Equation 2)
Now, we have a system of two equations with three variables (a, b, and c). To solve this system, we need one more equation.
We can use the fact that the ball is launched from a slingshot, so at time x = 0, the height of the ball is 0. This gives us the third equation:
3. When x = 0, h(x) = 0:
0 = a(0)^2 + b(0) + c => c = 0 (Equation 3)
Now, we have a system of three equations with three variables:
a + b + c = 148 (Equation 1)
4a + 2b + c = 272 (Equation 2)
c = 0 (Equation 3)
Substituting Equation 3 into Equations 1 and 2, we get:
a + b + 0 = 148 => a + b = 148 (Equation 4)
4a + 2b + 0 = 272 => 4a + 2b = 272 (Equation 5)
Now, we can solve this system of equations by either substitution or elimination.
Let's use elimination method. Multiply Equation 4 by 2 to make the coefficients of b identical:
2a + 2b = 296 (Equation 6)
Now, subtract Equation 5 from Equation 6:
2a + 2b - (4a + 2b) = 296 - 272
-2a = 24
a = -12
Substitute the value of a into Equation 4 to find b:
-12 + b = 148
b = 148 + 12
b = 160
Now, we have the values of a and b. Substituting them into Equation 1, we can find c:
-12 + 160 + c = 148
c = 148 - 148
c = 0
Therefore, the quadratic function representing the height of the ball is: h(x) = -12x^2 + 160x.
To find the height of the ball after 6 seconds, substitute x = 6 into the equation:
h(6) = -12(6)^2 + 160(6)
h(6) = -12(36) + 960
h(6) = -432 + 960
h(6) = 528
Therefore, the height of the ball after 6 seconds in the air is 528 feet.