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?
(1 point)
Responses
2.7 cm high; 0.296 cm long
2.7 cm high; 0.296 cm long
6.2 cm high; 9.1 cm long
6.2 cm high; 9.1 cm long
4.6 cm high; 6.2 cm long
4.6 cm high; 6.2 cm long
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 parabola represented by the given equation. The vertex of a parabola in the form y = ax^2 + bx + c is given by the coordinates (-b/2a, f(-b/2a)).
In this case, a = -0.296 and b = 2.7.
Using the formula for the x-coordinate of the vertex, we get:
x = -b/2a = -2.7 / (2 * -0.296) ≈ 4.6
Plugging this value of x back into the equation, we can find the corresponding value of y (the height):
y = -0.296(4.6)^2 + 2.7(4.6) ≈ 6.2 cm
So, the maximum height that the rabbit can reach during its jump is 6.2 cm.
To find the total length of the jump, we need to find the x-intercepts of the parabola. These are the points where the equation y = 0.
Setting the equation -0.296x^2 + 2.7x = 0 and factoring out an x, we get:
x(-0.296x + 2.7) = 0
Solving for x, we get:
x = 0 or -0.296x + 2.7 = 0
x = 0 or -0.296x = -2.7
x = 0 or x ≈ 9.1
The total length of the jump is the distance between these two x-intercepts:
9.1 cm - 0 cm = 9.1 cm
Therefore, the correct answer is: 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 parabolic function.
The vertex of a parabola given by the equation y = ax^2 + bx + c is given by the coordinates (h, k), where h = -b/2a and k is the value of y at that point.
In this case, the equation is y = -0.296x^2 + 2.7x, so we have a = -0.296 and b = 2.7.
To find h, we use the formula h = -b/2a, so h = -2.7 / (2 * -0.296).
Calculating this, we find h = 9.121.
To find k, we substitute the value of h into the equation y = -0.296x^2 + 2.7x, so k = -0.296 * (9.121)^2 + 2.7 * 9.121.
Calculating this, we find k = 6.201.
Therefore, 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 graph, which represent when the function equals zero.
Setting y = 0 in the equation -0.296x^2 + 2.7x = 0, we can factor out x to get x(-0.296x + 2.7) = 0.
This gives us two solutions: x = 0 and x = 9.121 / -0.296.
Calculating this, we find x = -30.82.
Since we are dealing with positive values for length, we take the positive solution x = 9.121 / 0.296.
Calculating this, we find x = 30.82.
Therefore, the total length of the rabbit's jump is 30.82 cm.
So, the correct answer is:
Maximum height: 6.2 cm
Total length: 30.82 cm
To find the maximum height that the rabbit can reach, we need to find the vertex of the parabolic function. The vertex form of a quadratic function is given by y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
In this case, the given function is y = -0.296x^2 + 2.7x. To convert it into the vertex form, we need to complete the square.
1. Start by factoring out a common factor of -0.296 from both terms:
y = -0.296(x^2 - (2.7/-0.296)x)
2. Next, complete the square of the quadratic term inside the parentheses. Take half of the coefficient of x (-2.7/-0.296) and square it:
y = -0.296(x^2 - 9.1216x + (9.1216/2)^2)
3. Simplifying the expression inside the parentheses:
y = -0.296(x^2 - 9.1216x + 13.4049984)
4. Now, add and subtract the term (9.1216/2)^2 inside the parentheses, while also compensating outside the parentheses:
y = -0.296(x^2 - 9.1216x + 13.4049984 - 13.4049984) + 13.4049984
5. Simplify the expression inside the parentheses:
y = -0.296((x - 4.5608)^2 - 13.4049984) + 13.4049984
6. Distribute the -0.296 to both terms:
y = -0.296(x - 4.5608)^2 + 3.959998272 + 13.4049984
7. Combine the two constants:
y = -0.296(x - 4.5608)^2 + 17.364996672
From the vertex form, we can see that the vertex of the parabola is at (4.5608, 17.364996672). The x-coordinate represents the distance jumped by the rabbit, so the total length of its jump when it reaches the ground is approximately 4.5608 centimeters.
The maximum height reached by the rabbit is represented by the y-coordinate of the vertex, which is approximately 17.364996672 centimeters.
Thus, the correct answer is:
9.1 cm high; 6.2 cm long