a ball is kicked in the air from the top of a cliff , the path that the ball travels is given by the equation
h(t)=-5t^2+17t+22, where h(t) is the height given in meters and t is the time seconds.
a) how high is the cliff ?
b)what is the maximum height the ball reaches ?
c)when will the ball hit the ground
d)write an equation for the axis of symmetry
e)determine the domain and range of the function
a) would you not be on top of the cliff if t = 0
b) change the equation to vertex form
h(t) = a(t-h)^2 + k
the value of k will be your maximum height
c) set h(t) = 0 and solve for t as a quadratic
d) once you have the equation in b) the axis of symmetry will be t = h
e) your sketch of the graph will let you determine domain and range.
hint: the t value of the vertex (t,k) is 1.7
Thankss !!
To answer the given questions, we will use the equation h(t) = -5t^2 + 17t + 22.
a) The height of the cliff can be determined by looking at the initial condition when the ball is at the top of the cliff before being kicked. In this case, t = 0. So, we substitute t = 0 in the equation:
h(0) = -5(0)^2 + 17(0) + 22
= 0 + 0 + 22
= 22
Therefore, the height of the cliff is 22 meters.
b) The maximum height the ball reaches can be found by determining the vertex of the function. The vertex is the point (t, h) where the function reaches its maximum or minimum.
The vertex of a quadratic function can be found using the formula for the axis of symmetry, t = -b/(2a), where a, b, and c are the coefficients of the quadratic equation.
In this case, a = -5 and b = 17. Substituting these values into the formula, we get:
t = -17/(2*(-5))
= -17/-10
= 1.7
Substituting t = 1.7 into the function:
h(1.7) = -5(1.7)^2 + 17(1.7) + 22
= -14.45 + 28.9 + 22
= 36.45
Therefore, the maximum height the ball reaches is 36.45 meters.
c) To find when the ball hits the ground, we need to determine the value of t when h(t) = 0. Set the equation h(t) equal to zero and solve for t:
0 = -5t^2 + 17t + 22
Using factoring, quadratic formula, or other methods of solving quadratic equations, we find two solutions t = -1.22, t = 3.62.
Since time cannot be negative in this context, the ball hits the ground at t = 3.62 seconds.
d) The equation for the axis of symmetry is t = -b/(2a).
In this case, the values of a and b are the same as in part (b), so the equation becomes:
t = -17/(2*(-5))
= 1.7
Therefore, the equation for the axis of symmetry is t = 1.7.
e) The domain of a function describes the set of possible input values (independent variable). Since time cannot be negative in this context, the domain of the function h(t) is t ≥ 0.
The range of a function describes the set of possible output values (dependent variable). In this case, the height can vary from the height of the cliff (which is 22 meters) to the maximum height the ball reaches (which is 36.45 meters). Therefore, the range of the function h(t) is 22 ≤ h ≤ 36.45.