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?
To find the maximum height that the rabbit can reach during its jump, we need to determine the vertex of the parabolic function. The vertex is given by the formula x = -b/2a, where a and b are coefficients of x in the equation y = ax^2 + bx + c.
In the given equation y = -0.296x^2 + 2.7x, a = -0.296 and b = 2.7. Plugging these values into the formula, we get:
x = -(2.7) / (2*(-0.296))
x = -2.7 / -0.592
x = 4.563
Now that we have the x-coordinate of the vertex, we can substitute it back into the equation to find the maximum height:
y = -0.296(4.563)^2 + 2.7(4.563)
y = -0.296(20.830) + 12.337
y = -6.157 + 12.337
y = 6.18
Therefore, the maximum height that the rabbit can reach during its jump is 6.18 centimeters.
To find the total length of the jump, we need to find the x-coordinate where y is equal to zero (since that indicates when the rabbit reaches the ground). We can solve the equation -0.296x^2 + 2.7x = 0 by factoring out x:
x(-0.296x + 2.7) = 0
This equation is satisfied when either x = 0 or -0.296x + 2.7 = 0. Solving the second equation, we get:
-0.296x = -2.7
x = -2.7 / -0.296
x = 9.12
Therefore, the rabbit's jump covers a total length of 9.12 centimeters.
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 and b are the coefficients of the quadratic equation. Comparing the equation y = -0.296x^2 + 2.7x to the standard quadratic equation y = ax^2 + bx + c, we can see that a = -0.296 and b = 2.7.
x = -(2.7) / (2 * -0.296)
x = -2.7 / -0.592
x ≈ 4.563
To find the y-coordinate of the vertex, substitute the value of x into the original equation:
y = -0.296(4.563)^2 + 2.7(4.563)
y ≈ -0.296 * 20.825 + 12.328
y ≈ -6.148 + 12.328
y ≈ 6.18
Therefore, the maximum height that the rabbit can reach during its jump is approximately 6.18 centimeters.
To find the total length of its jump, we need to find the two points where the height of the rabbit is zero (y = 0). We can solve the quadratic equation for x to find these points.
-0.296x^2 + 2.7x = 0
Factoring out x, we get:
x(-0.296x + 2.7) = 0
Setting each factor equal to zero:
x = 0 and -0.296x + 2.7 = 0
For x = 0, it represents the starting point of the jump.
Solving -0.296x + 2.7 = 0 for x:
-0.296x = -2.7
x ≈ -2.7 / -0.296
x ≈ 9.122
Therefore, the rabbit jumps a total length of approximately 9.122 centimeters.
To find the maximum height that the rabbit can reach during its jump, we can use the equation of the function provided:
y = -0.296x^2 + 2.7x
The maximum height occurs at the vertex of the quadratic function. The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)
In this case, a = -0.296 and b = 2.7. Plugging these values into the formula, we have:
x = -2.7 / (2 * (-0.296))
Calculating this, x ≈ 4.56
Now, to find the maximum height (y-coordinate) reached by the rabbit, substitute this x-value back into the equation:
y = -0.296 * (4.56)^2 + 2.7 * 4.56
Evaluating this expression, we find that the maximum height reached by the rabbit is approximately 4.92 centimeters.
To find the total length of the rabbit's jump, we need to find the x-coordinates where the rabbit reaches the ground. The rabbit reaches the ground when y = 0. Therefore, we can solve the equation:
0 = -0.296x^2 + 2.7x
This equation is quadratic, so we can solve it by factoring or using the quadratic formula. However, it's worth noting that this equation does not have real solutions, meaning the rabbit doesn't reach the ground during its jump according to this model.
So, according to the given function, the rabbit maximally reaches a height of approximately 4.92 centimeters during its jump, but it doesn't land on the ground, so there is no total length of the jump in this scenario.