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?
The given function is a quadratic function in the form y = ax^2 + bx.
From the given function y = -0.296x^2 + 2.7x, we can see that the coefficient of the x^2 term is negative (-0.296).
For a quadratic function in the form y = ax^2 + bx, the graph of the function is a parabola that opens downwards if a < 0. In this case, a = -0.296 < 0, so the parabola opens downwards.
Since the parabola opens downwards, the highest point on the parabola represents the maximum height of the jump.
To find the maximum height, we can find the vertex of the parabola.
The x-coordinate of the vertex, denoted as x1, is given by the formula x1 = -b / (2a).
For this function, a = -0.296 and b = 2.7, so x1 = -2.7 / (2 * -0.296).
Calculating x1, we get x1 ≈ 4.572 cm.
To find the y-coordinate of the vertex, denoted as y1, we substitute the x1 value into the given function:
y1 = -0.296(4.572)^2 + 2.7(4.572).
Calculating y1, we get y1 ≈ 6.697 cm.
Therefore, the maximum height that the rabbit can reach during its jump is approximately 6.697 cm.
To find the total length of the rabbit's jump, we need to find the values of x where the function y = -0.296x^2 + 2.7x crosses the x-axis (the ground).
When the rabbit reaches the ground, the height y is equal to zero. So, we solve the quadratic equation -0.296x^2 + 2.7x = 0.
Factorizing the equation, we get x(-0.296x + 2.7) = 0, which gives two solutions: x = 0 and x ≈ 9.121 cm.
Therefore, the total length of the rabbit's jump is approximately 9.121 cm.