a projectile is laucnhed from the ground level and its height can be modeled as a function of time by the equation h=4-t-50t^2
What are the reasonable domain and range of this problem
Something not right here.
According to your equation, it is not launched from ground level, but from a height of 4 units.
Furthermore it is launched with a downwards velocity of 1 unit of distance/ 1 unit of time
domain: t ≥ 0
check for a typo
40t-5t^2
To determine the reasonable domain and range of the problem, we need to consider the context of the situation.
In this case, the equation h = 4-t-50t^2 represents the height of a projectile launched from the ground as a function of time.
Domain represents the possible values for the independent variable, which is time in this case. For a physical projectile launched from the ground, time cannot be negative because it must be measured from the moment of launch. Therefore, the reasonable domain of the problem is all non-negative real numbers.
Domain: t ≥ 0
Range represents the possible values for the dependent variable, which is height in this case. Since the projectile is launched from the ground, the height cannot be negative. Additionally, the maximum height reached by the projectile depends on factors such as initial velocity, angle of launch, and air resistance. However, without any additional information, we can assume that the height is limited by the maximum value of the function.
To find the maximum value of the quadratic function h = 4 - t - 50t^2, we can observe that it is a downward-opening parabola because the coefficient of the t^2 term is negative (-50). The vertex of the parabola gives us the highest point, which corresponds to the maximum height.
To find the vertex, we can use the formula x = -b/2a, where a and b are the coefficients of the quadratic equation. In this case, a = -50 and b = -1.
t = -(-1) / 2(-50)
t = 1 / 100
t = 0.01
By substituting t = 0.01 back into the equation, we can find the corresponding height:
h = 4 - (0.01) - 50(0.01)^2
h = 4 - 0.01 - 50(0.0001)
h ≈ 3.9899
Thus, the maximum height reached by the projectile is approximately 3.9899 units.
Therefore, the reasonable range of this problem is h ≥ 0 up to approximately h ≈ 3.9899.
Range: 0 ≤ h ≤ 3.9899 (approximately)