"A woodland jumping mouse hops along a parabolic path given by y=-0.2x(squared)+1.3x where x is the mouse's horizontal position(in feet) and y is the corresponding height(in feet). Can the mouse jump over a fence that is 3 feet high? Explain in quadratic function."
so you would want
3 = -.2x^2 + 1.3x
.2x^2 - 1.3x + 3 = 0
times 10
2x^2 - 13x + 30 = 0
x = (13 ±√-71)/4
which is not a real number,
so NO, the mouse cannot jump over that fence
or
the x of the vertex is 1.3/(2(.2)) = 1.3/.4 = 13/4
when x = 13/4
y = -.2(13/4) + 1.3(13/4) = appr 2.1
since the parabola opens downwards the max value of y is 2.1
but we needed y = 3
NO WAY!
To determine if the woodland jumping mouse can jump over a fence that is 3 feet high, we need to examine the height of the mouse's path at its maximum point.
First, we need to find the x-coordinate of the maximum point of the parabolic path. The x-coordinate of the maximum or minimum point of a quadratic function can be calculated using the formula x = -b / (2a), where a and b are the coefficients of x^2 and x, respectively, in the equation of the quadratic function.
In this case, the equation of the parabolic path is y = -0.2x^2 + 1.3x, so the coefficient of x^2 is -0.2 and the coefficient of x is 1.3.
Using the formula, we can find the x-coordinate of the maximum point: x = -1.3 / (2 * -0.2) = 3.25.
Now that we know the x-coordinate of the maximum point, we can substitute it into the quadratic function to find the corresponding height. Plugging x = 3.25 into the equation y = -0.2x^2 + 1.3x:
y = -0.2 * (3.25)^2 + 1.3 * 3.25
y = -0.2 * 10.5625 + 4.225
y = -2.1125 + 4.225
y = 2.1125
Therefore, the height of the mouse at the maximum point is approximately 2.1125 feet.
Since the height of the fence is 3 feet and the maximum height of the mouse's path is less than 3 feet, the mouse cannot jump over the 3-foot fence.
To determine whether the mouse can jump over a 3-foot high fence, we need to find the height of the mouse's jump at its maximum horizontal distance.
First, we should note that the given equation y = -0.2x² + 1.3x is a quadratic function in the form of y = ax² + bx + c, where "a" is the coefficient of the quadratic term (x²), "b" is the coefficient of the linear term (x), and "c" is the constant term.
In this case, a = -0.2, b = 1.3, and c = 0.
To find the maximum height of the jump, we can use the vertex formula for quadratic functions, given by:
x = -b / (2a)
Once we find the value of x using the above formula, we can substitute it back into the original equation to find the corresponding height y.
Let's calculate:
x = -b / (2a)
x = -(1.3) / (2(-0.2))
x = -(1.3) / (-0.4)
x = 3.25
The value of x at the maximum height is 3.25 feet.
Now, substitute this value of x back into the equation y = -0.2x² + 1.3x to find the corresponding height:
y = -0.2(3.25)² + 1.3(3.25)
y = -0.2(10.5625) + 4.225
y = -2.1125 + 4.225
y = 2.1125
The maximum height of the mouse's jump is 2.1125 feet.
Since the maximum height is less than the height of the fence (which is 3 feet), the mouse cannot jump over the fence.
In conclusion, using the quadratic function y = -0.2x² + 1.3x, we determined that the maximum height of the mouse's jump is 2.1125 feet. Since this height is less than the height of the fence (which is 3 feet), the mouse cannot jump over the fence.