The function y=-0.296x^2+2.7x 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?
A. 2.7 cm high; 0.296 cm long
B. 6.2 cm high; 9.1 cm long
C. 4.6 cm high; 6.2 cm long
D. 9.1 cm high; 6.2 cm long
To find the maximum height that the rabbit can reach, we need to find the vertex of the quadratic function. The x-coordinate of the vertex is given by -b/2a, where a and b are the coefficients of the quadratic function. In this case, a = -0.296 and b = 2.7.
x = - (2.7) / (2 * -0.296)
x = 4.574324324324324
Substituting this back into the equation, we can find the maximum height:
y = -0.296(4.574324324324324)^2 + 2.7(4.574324324324324)
y = -0.296(20.912046393355624) + 12.355405405405405
y = -6.17536549197995 + 12.355405405405405
y = 6.180039913425455
So, the maximum height that the rabbit can reach is 6.2 cm.
To find the total length of the jump, we need to find the x-intercepts of the quadratic function. Set y = 0:
0 = -0.296x^2 + 2.7x
0 = x(-0.296x + 2.7)
x = 0 or -2.7/ -0.296
x = 0 or 9.121621621621622
The rabbit jumps from x = 0 to x = 9.121621621621622. The total length of its jump is the difference between these two values:
9.121621621621622 - 0 = 9.121621621621622 cm
Therefore, the correct answer is option B. 6.2 cm high; 9.1 cm long.