The path of a football thrown by a quarterback can be modeled be the equation y=-0.04x^2+4x+5 where x is the horizontal distance in feet and y is the corresponding height in feet.find the interval on which the height is increasing.what is the average rate of change on this interval?What does the c value stand for in this equation?
y is increasing where y' > 0
y' = -0.08x+4
y' > 0 when x < 50
The average rate of change is
(y(50)-y(0))/(50-0) = 100/50 = 2
I don't know what c is supposed to be, but if you're angling toward the Mean Value Theorem, it is where y' = 2
To find the interval on which the height is increasing, we need to analyze the derivative of the equation. The derivative represents the rate of change or the slope of the function.
The equation given is y = -0.04x^2 + 4x + 5. To find the derivative, we differentiate with respect to x:
dy/dx = d/dx(-0.04x^2 + 4x + 5)
dy/dx = -0.08x + 4
To determine when the height is increasing, we set the derivative greater than zero and solve for x:
-0.08x + 4 > 0
-0.08x > -4
x < 50
Therefore, the height is increasing for x values less than 50. The interval on which the height is increasing is (-∞, 50).
Next, to find the average rate of change on this interval, we need to calculate the difference in y values divided by the difference in x values. In other words, we calculate the slope of the function over the given interval.
Average rate of change = Δy/Δx
Let's choose two arbitrary points within the interval (-∞, 50) to calculate the average rate of change. For example, let's choose x = 10 and x = 20.
y1 = -0.04(10)^2 + 4(10) + 5 = 45
y2 = -0.04(20)^2 + 4(20) + 5 = 65
Δy = y2 - y1 = 65 - 45 = 20
Δx = x2 - x1 = 20 - 10 = 10
Average rate of change = Δy/Δx = 20/10 = 2
The average rate of change on the interval (-∞, 50) is 2.
Finally, in the equation y = -0.04x^2 + 4x + 5, the "c" value represents the constant term, which is the y-intercept of the quadratic function. It determines the vertical shift of the parabolic curve. In this case, c is 5, meaning the graph of the function intersects the y-axis at the point (0, 5).