The function y equals negative 0.296 x squared plus 2.7 x models the length x and height y that your sister's pet rabbit can jump, in centimeters. What is the maximum height that the rabbit can reach during its jump? Once the rabbit reaches the ground, what is the total length of its jump?
To find the maximum height, we need to find the vertex of the parabola. The vertex of a parabola of the form y = ax^2 + bx + c is given by the formula x = -b/(2a).
In this case, a = -0.296, b = 2.7, and c = 0.
x = -2.7/(2*(-0.296)) = -2.7/(-0.592) = 4.56
To find the maximum height, we substitute this value of x into the equation:
y = -0.296 * 4.56^2 + 2.7 * 4.56 = -6.67 + 12.31 = 5.64
Thus, the maximum height that the rabbit can reach during its jump is 5.64 centimeters.
To find the total length of the jump, we need to find the x-intercepts of the parabola, which occur when y = 0.
0 = -0.296 x^2 + 2.7 x
We can solve this equation by factoring:
0 = x (-0.296 x + 2.7)
Setting each factor equal to zero, we find:
x = 0 or -0.296 x + 2.7 = 0
The first solution, x = 0, corresponds to the starting point of the jump.
To find the second solution, we solve the equation:
-0.296 x + 2.7 = 0
-0.296 x = -2.7
x = -2.7 / -0.296 = 9.12
Thus, once the rabbit reaches the ground, the total length of its jump is 9.12 centimeters.