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
The maximum height that the rabbit can reach during its jump is found at the vertex of the quadratic function. We can find the x-coordinate of the vertex by using the formula x = -b / (2a), where a and b are the coefficients of the quadratic function in the form ax^2 + bx + c.
In this case, a = -0.296 and b = 2.7. Plugging these values into the formula, we get x = -2.7 / (2(-0.296)) = -2.7 / (-0.592) = 4.569
To find the y-coordinate of the vertex (or the maximum height), we can substitute this value of x back into the function:
y = -0.296(4.569)^2 + 2.7(4.569) ≈ 6.2
So, the maximum height that the rabbit can reach during its jump is approximately 6.2 cm.
Once the rabbit reaches the ground, its height y is 0. So, we set y = 0 and solve for x to find the x-coordinate of the other point where the function crosses the x-axis (the ground):
0 = -0.296x^2 + 2.7x
We can factor out an x from both terms:
x(-0.296x + 2.7) = 0
Either x = 0 or -0.296x + 2.7 = 0
Solving -0.296x + 2.7 = 0, we get x = 2.7 / 0.296 ≈ 9.1
So, the rabbit jumps approximately 9.1 cm.
Therefore, the answer is B. 6.2 cm high; 9.1 cm long.
To find the maximum height that the rabbit can reach during its jump, we need to determine the vertex of the parabola represented by the given function. The vertex of a parabola in the form of y = ax^2 + bx + c is given by (-b/2a, f(-b/2a)).
In this case, the equation y = -0.296x^2 + 2.7x represents the height of the rabbit's jump. Comparing this to the general form of a quadratic equation, we have a = -0.296, b = 2.7, and c = 0.
The x-coordinate of the vertex is -b/2a, which is calculated as follows:
x = -2.7 / (2 * -0.296) = -2.7 / (-0.592) ≈ 4.57
Substituting this value of x into the equation, we can find the y-coordinate (the maximum height):
y = -0.296(4.57)^2 + 2.7(4.57) ≈ 6.179
So the maximum height that the rabbit can reach during its jump is approximately 6.179 cm.
To find the total length of the rabbit's jump, we need to determine the x-intercepts of the parabola. This can be done by setting y = 0 and solving for x.
0 = -0.296x^2 + 2.7x
Using the quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a), we can find the values of x:
x = (-2.7 ± √(2.7^2 - 4*(-0.296)*0)) / (2*(-0.296))
x = (-2.7 ± √(7.29)) / (-0.592)
x ≈ 9.11 or x ≈ 0.26
The rabbit starts at x = 0 and lands at either x = 0.26 or x = 9.11, so the total length of its jump can be calculated as:
length = |9.11 - 0| = 9.11 or length = |0.26 - 0| = 0.26
Therefore, the correct answer is:
D. 9.1 cm high; 6.2 cm long
To find the maximum height that the rabbit can reach during its jump, we need to determine the vertex of the parabolic function y = -0.296x^2 + 2.7x.
The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a = -0.296 and b = 2.7.
Substituting the values, we get x = -2.7 / (2 * (-0.296)) = 9.121
Next, we substitute this value of x back into the original equation to find the corresponding y-coordinate of the vertex:
y = -0.296 * 9.121^2 + 2.7 * 9.121 = 6.154
Therefore, the maximum height that the rabbit can reach during its jump is 6.154 cm.
To find the total length of the jump, we need to find the x-intercepts of the equation, which occur when y = 0.
Setting y = 0 in the original equation, we get:
0 = -0.296x^2 + 2.7x
To solve this quadratic equation, we can factor it as follows:
0.296x^2 - 2.7x = 0
x(0.296x - 2.7) = 0
Thus, x = 0 or 0.296x - 2.7 = 0.
Solving the second equation, we get:
0.296x = 2.7
x = 2.7 / 0.296 = 9.12
We discard the x = 0 solution since it doesn't make sense in the context of the problem.
Therefore, the total length of the rabbit's jump is 9.12 cm.
The correct answer is option B. The rabbit can reach a maximum height of 6.2 cm during its jump, and the total length of the jump is 9.1 cm.